Functions of Matrices and Nearest Correlation Matrices - NAG
54
Functions of Matrices and Nearest Correlation Matrices Nick Higham School of Mathematics The University of Manchester [email protected] http://www.ma.man.ac.uk/~higham/ NAG Quant Event, New York, Dec 7, 2011
Transcript of Functions of Matrices and Nearest Correlation Matrices - NAG
Research Matters
February 25 2009
Nick HighamDirector of Research
School of Mathematics
1 6
Functions of Matrices andNearest Correlation Matrices
Nick HighamSchool of Mathematics
The University of Manchester
highammamanacukhttpwwwmamanacuk~higham
NAG Quant Event New York Dec 7 2011
What is a Matrix Function
Itrsquos not
det(A) or trace(A)elementwise evaluation f (aij)AT matrix factor (eg A = LU)
It is
Aminus1radicA
eA
University of Manchester Nick Higham Matrix functions amp correlation matrices 2 35
What is a Matrix Function
Itrsquos not
det(A) or trace(A)elementwise evaluation f (aij)AT matrix factor (eg A = LU)
It is
Aminus1radicA
eA
University of Manchester Nick Higham Matrix functions amp correlation matrices 2 35
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Matrix functions amp correlation matrices 3 35
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Matrix functions amp correlation matrices 4 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Matrices in Applied Mathematics
Frazer Duncan amp Collar Aerodynamics Division ofNPL aircraft flutter matrix structural analysis
Elementary Matrices amp Some Applications toDynamics and Differential Equations 1938Emphasizes importance of eA
Arthur Roderick Collar FRS(1908ndash1986) ldquoFirst book to treatmatrices as a branch of appliedmathematicsrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 6 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
What is a Matrix Function
Itrsquos not
det(A) or trace(A)elementwise evaluation f (aij)AT matrix factor (eg A = LU)
It is
Aminus1radicA
eA
University of Manchester Nick Higham Matrix functions amp correlation matrices 2 35
What is a Matrix Function
Itrsquos not
det(A) or trace(A)elementwise evaluation f (aij)AT matrix factor (eg A = LU)
It is
Aminus1radicA
eA
University of Manchester Nick Higham Matrix functions amp correlation matrices 2 35
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Matrix functions amp correlation matrices 3 35
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Matrix functions amp correlation matrices 4 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Matrices in Applied Mathematics
Frazer Duncan amp Collar Aerodynamics Division ofNPL aircraft flutter matrix structural analysis
Elementary Matrices amp Some Applications toDynamics and Differential Equations 1938Emphasizes importance of eA
Arthur Roderick Collar FRS(1908ndash1986) ldquoFirst book to treatmatrices as a branch of appliedmathematicsrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 6 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
What is a Matrix Function
Itrsquos not
det(A) or trace(A)elementwise evaluation f (aij)AT matrix factor (eg A = LU)
It is
Aminus1radicA
eA
University of Manchester Nick Higham Matrix functions amp correlation matrices 2 35
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Matrix functions amp correlation matrices 3 35
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Matrix functions amp correlation matrices 4 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Matrices in Applied Mathematics
Frazer Duncan amp Collar Aerodynamics Division ofNPL aircraft flutter matrix structural analysis
Elementary Matrices amp Some Applications toDynamics and Differential Equations 1938Emphasizes importance of eA
Arthur Roderick Collar FRS(1908ndash1986) ldquoFirst book to treatmatrices as a branch of appliedmathematicsrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 6 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Matrix functions amp correlation matrices 3 35
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Matrix functions amp correlation matrices 4 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Matrices in Applied Mathematics
Frazer Duncan amp Collar Aerodynamics Division ofNPL aircraft flutter matrix structural analysis
Elementary Matrices amp Some Applications toDynamics and Differential Equations 1938Emphasizes importance of eA
Arthur Roderick Collar FRS(1908ndash1986) ldquoFirst book to treatmatrices as a branch of appliedmathematicsrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 6 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Matrix functions amp correlation matrices 4 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Matrices in Applied Mathematics
Frazer Duncan amp Collar Aerodynamics Division ofNPL aircraft flutter matrix structural analysis
Elementary Matrices amp Some Applications toDynamics and Differential Equations 1938Emphasizes importance of eA
Arthur Roderick Collar FRS(1908ndash1986) ldquoFirst book to treatmatrices as a branch of appliedmathematicsrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 6 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Matrices in Applied Mathematics
Frazer Duncan amp Collar Aerodynamics Division ofNPL aircraft flutter matrix structural analysis
Elementary Matrices amp Some Applications toDynamics and Differential Equations 1938Emphasizes importance of eA
Arthur Roderick Collar FRS(1908ndash1986) ldquoFirst book to treatmatrices as a branch of appliedmathematicsrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 6 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Two Definitions
Definition (Taylor series)
If f has a Taylor series expansion f (z) =suminfin
k=0 akzk withradius of convergence r and ρ(A) lt r then
f (A) =infinsum
k=0
akAk
Definition (Cauchy integral formula)
f (A) =1
2πi
intΓ
f (z)(zI minus A)minus1 dz
where f analytic on and inside closed contour Γ enclosingλ(A)
University of Manchester Nick Higham Matrix functions amp correlation matrices 5 35
Matrices in Applied Mathematics
Frazer Duncan amp Collar Aerodynamics Division ofNPL aircraft flutter matrix structural analysis
Elementary Matrices amp Some Applications toDynamics and Differential Equations 1938Emphasizes importance of eA
Arthur Roderick Collar FRS(1908ndash1986) ldquoFirst book to treatmatrices as a branch of appliedmathematicsrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 6 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Matrices in Applied Mathematics
Frazer Duncan amp Collar Aerodynamics Division ofNPL aircraft flutter matrix structural analysis
Elementary Matrices amp Some Applications toDynamics and Differential Equations 1938Emphasizes importance of eA
Arthur Roderick Collar FRS(1908ndash1986) ldquoFirst book to treatmatrices as a branch of appliedmathematicsrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 6 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent credit ratings or stockprices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = PWhat is P23 What is P09
Ps = exp(s log(P))
University of Manchester Nick Higham Matrix functions amp correlation matrices 8 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Solving Ordinary Differential Equations
A isin Cntimesn dydt
= Ay y(0) = y0 rArr y(t) = eAty0
d2ydt2 + Ay = 0 y(0) = y0 y prime(0) = y prime0
has solution
y(t) = cos(radic
At)y0 +(radic
A)minus1 sin(
radicAt)y prime0
But also [y prime
y
]= exp
([0 minustA
t In 0
])[y prime0y0
]
University of Manchester Nick Higham Matrix functions amp correlation matrices 9 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Phi Functions Definition
ϕ0(z) = ez ϕ1(z) =ez minus 1
z ϕ2(z) =
ez minus 1minus zz2
ϕk+1(z) =ϕk(z)minus 1k
z
ϕk(z) =infinsum
j=0
z j
(j + k)
University of Manchester Nick Higham Matrix functions amp correlation matrices 10 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Phi Functions Solving ODEs
y isin Cn A isin Cntimesn
dydt
= Ay y(0) = y0 rArr y(t) = eAty0
dydt
= Ay + b y(0) = 0 rArr y(t) = t ϕ1(tA)b
dydt
= Ay + ct y(0) = 0 rArr y(t) = t2ϕ2(tA)c
University of Manchester Nick Higham Matrix functions amp correlation matrices 11 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Exponential Integrators
Considery prime = Ly + N(y)
N(y(t)) asymp N(y(0)) implies
y(t) asymp etLy0 + tϕ1(tL)N(y(0))
Exponential Euler method
yn+1 = ehLyn + hϕ1(hL)N(yn)
Lawson (1967) recent resurgence
University of Manchester Nick Higham Matrix functions amp correlation matrices 12 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Toolbox of Matrix Functions
Want software for evaluating interesting f at matrix argsas well as scalar argsMATLAB has expm logm sqrtm funmThe Matrix Function Toolbox (H 2008)NAG Library
f01ecf (f01ecc) for matrix exponentialf01efff01fff for function ofsymmetricHermitian matrixMore on the way
University of Manchester Nick Higham Matrix functions amp correlation matrices 13 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Scaling and Squaring Method
Scale B larr A2s so Binfin asymp 1Approximate rm(B) = [mm] Padeacute approximant to eB
Square X = rm(B)2s asymp eA
Moler amp Van Loan (1978) ldquoNineteen dubious ways tocompute the exponential of a matrixrdquomdashmethodology forchoosing s and mH (2005) sharper analysis giving optimal s and mAl-Mohy amp H (2009) further improvements
Newer problem action of matrix exponential on a vector
University of Manchester Nick Higham Matrix functions amp correlation matrices 14 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Compute eAb
Exploit for integer s
eAb = (esminus1A)sb = esminus1Aesminus1A middot middot middot esminus1A︸ ︷︷ ︸s times
b
Choose s so Tm(sminus1A) =summ
j=0(sminus1A)j
jasymp esminus1A Then
bi+1 = Tm(sminus1A)bi i = 0 s minus 1 b0 = b
yields bs asymp eAb
Al-Mohy amp H (2011) SIAM J Sci Comp
University of Manchester Nick Higham Matrix functions amp correlation matrices 15 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
ExperimentCompute etAb for HarwellndashBoeing matrices
orani678 n = 2529 t = 100 b = [11 1]T bcspwr10 n = 5300 t = 10 b = [10 01]T
2D Laplacian matrix poisson tol = 6times 10minus8
Alg AH ode15stime cost error time cost error
orani678 013 878 4e-8 136 7780+ middot middot middot 2e-6bcspwr10 0021 215 7e-7 292 1890+ middot middot middot 5e-5poisson 376 29255 2e-6 248 402+ middot middot middot 8e-6
4poisson 15 116849 9e-6 324 49+ middot middot middot 1e-1
University of Manchester Nick Higham Matrix functions amp correlation matrices 16 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
General Functions
SchurndashParlett algorithm (Davies amp H 2003)computes f (A) given the ability to evaluate f (k)(x) forany k and x Implemented in MATLABrsquos funmBeware unstable diagonalization algorithmfunction F = funm_ev(Afun)FUNM_EV Evaluate general matrix function via eigensystem[VD] = eig(A)F = V diag(feval(fundiag(D))) V
University of Manchester Nick Higham Matrix functions amp correlation matrices 17 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 Chap 3 for survey of regularizationmethods
University of Manchester Nick Higham Matrix functions amp correlation matrices 18 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Matrix functions amp correlation matrices 19 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy and H2011) mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Matrix functions amp correlation matrices 20 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Knowledge Transfer Partnership 1
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSBKTP Associate Edvin DeadmanDeveloping suite of NAG Library codes for matrixfunctions (NAG Library already has several eg for eA)At least six new codes to appear in next release ofNAG LibraryImprovements to existing state of the art faster andmore accurateSuggestions for prioritizing code developmentwelcome
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Matrix functions amp correlation matrices 21 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Some NAG Toolbox Timings
All eirsquovals amp eirsquovectors of real symmetric matrixf08fc divide and conquer eig QR
n f08fc eig500 0062 0072
1000 0294 05092000 1907 3915
Matrix logarithm using the SchurndashParlett alg
n f01ej logm10 34e-4 10e-2
100 025 256500 291 9011000 212 501
University of Manchester Nick Higham Matrix functions amp correlation matrices 22 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Knowledge Transfer Partnership 2
University of Manchester and NAG (2012ndash2013)funded by NAG and TSBLead academic Jack Dongarra (UT Knoxville OakRidge National Laboratory amp U Manchester)Developing tuning and integrating key components ofthe Parallel Linear Algebra for Scalable MulticoreArchitectures (PLASMA) library to support NAGproducts
University of Manchester Nick Higham Matrix functions amp correlation matrices 23 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Questions From Finance Practitioners
ldquoGiven a real symmetric matrix A which is almost acorrelation matrix what is the best approximating(in Frobenius norm) correlation matrixrdquo
ldquoI am researching ways to make our companyrsquoscorrelation matrix positive semi-definiterdquo
ldquoCurrently I am trying to implement some realoptions multivariate models in a simulationframework Therefore I estimate correlationmatrices from inconsistent data set whicheventually are non psdrdquo
University of Manchester Nick Higham Matrix functions amp correlation matrices 24 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Correlation Matrix
An n times n symmetric positive semidefinite matrix A withaii equiv 1
Properties
symmetric1s on the diagonaloff-diagonal elements between minus1 and 1eigenvalues nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Matrix functions amp correlation matrices 26 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Alternating Projections
von Neumann (1933) for subspaces
S1
S2
Dykstra (1983) incorporated corrections for closed convexsets
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Newton Method
Qi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Newton MethodQi amp Sun (2006) convergent Newton method basedon theory of strongly semismooth matrix functionsGlobally and quadratically convergentAlgorithmic improvements by Borsdorf amp H (2010)Implemented in NAG codes g02aaf (g02aac) andg02abf (weights lower bound on eirsquovalsmdashMark 23)
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
k Factor Problem
minXisinRntimesk
f (X ) = Aminus C(X )2F subject to
ksumj=1
x2ij le 1
Nonlinear objective function with convex quadraticconstraintsSome existing algs ignore the constraints
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Algorithms
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2011)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23)
Principal factors method (Andersen et al 2003) hasno convergence theory and can converge to anincorrect answer
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
Conclusions
Matrix functions a powerful and versatile tool withexcellent algs availableBeware unstableimpractical algs in literatureWorking with NAG to implement state of the art f (A)algs for NAG Library
Excellent algs available in NAG Library for nearestcorrelation matrix problems Further improvementscomingBeware algs in literature that may not converge orconverge to wrong solution
Keen to hear about your matrix problems
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References I
A H Al-Mohy and N J HighamA new scaling and squaring algorithm for the matrixexponentialSIAM J Matrix Anal Appl 31(3)970ndash989 2009
A H Al-Mohy and N J HighamComputing the action of the matrix exponential with anapplication to exponential integratorsSIAM J Sci Comput 33(2)488ndash511 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 27 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References II
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmMIMS EPrint 201183 Manchester Institute forMathematical Sciences The University of ManchesterUK Oct 201118 pp
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
University of Manchester Nick Higham Matrix functions amp correlation matrices 28 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References III
R Borsdorf and N J HighamA preconditioned Newton algorithm for the nearestcorrelation matrixIMA J Numer Anal 30(1)94ndash107 2010
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
University of Manchester Nick Higham Matrix functions amp correlation matrices 29 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References IV
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
P I Davies and N J HighamA SchurndashParlett algorithm for computing matrixfunctionsSIAM J Matrix Anal Appl 25(2)464ndash485 2003
University of Manchester Nick Higham Matrix functions amp correlation matrices 30 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References V
R A Frazer W J Duncan and A R CollarElementary Matrices and Some Applications toDynamics and Differential EquationsCambridge University Press Cambridge UK 1938xviii+416 pp1963 printing
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
N J HighamThe Matrix Function Toolboxhttpwwwmamanacuk~highammftoolbox
University of Manchester Nick Higham Matrix functions amp correlation matrices 31 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References VI
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM J Matrix Anal Appl 26(4)1179ndash1193 2005
N J HighamFunctions of Matrices Theory and ComputationSociety for Industrial and Applied MathematicsPhiladelphia PA USA 2008ISBN 978-0-898716-46-7xx+425 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 32 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References VII
N J HighamThe scaling and squaring method for the matrixexponential revisitedSIAM Rev 51(4)747ndash764 2009
N J Higham and A H Al-MohyComputing matrix functionsActa Numerica 19159ndash208 2010
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
University of Manchester Nick Higham Matrix functions amp correlation matrices 33 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References VIII
J D LawsonGeneralized Runge-Kutta processes for stable systemswith large Lipschitz constantsSIAM J Numer Anal 4(3)372ndash380 Sept 1967
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
University of Manchester Nick Higham Matrix functions amp correlation matrices 34 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
References IX
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
University of Manchester Nick Higham Matrix functions amp correlation matrices 35 35
