Post on 27-Dec-2015
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Recent advances in Recent advances in Global Sensitivity Analysis Global Sensitivity Analysis
techniquestechniquesS. Kucherenko
Imperial College London, UKs.kucherenko@imperial.ac.uk
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Introduction of Global Sensitivity Analysis and Introduction of Global Sensitivity Analysis and Sobol’ Sobol’ Sensitivity IndicesSensitivity Indices Why Quasi Monte Carlo methods (Why Quasi Monte Carlo methods (Sobol’ sequence sampling) Sobol’ sequence sampling) are much more efficient than are much more efficient than MonteMonte Carlo (random sampling) ? Carlo (random sampling) ?
Effective dimensions and their link with Sobol’ Effective dimensions and their link with Sobol’ Sensitivity Sensitivity IndicesIndices
Classification of functions based on global sensitivity indicesClassification of functions based on global sensitivity indices
Link between Link between Sobol’ Sobol’ Sensitivity IndicesSensitivity Indices and Derivative based and Derivative based Global Sensitivity MeasuresGlobal Sensitivity Measures
Quasi Randon Sampling - Quasi Randon Sampling - High Dimensional Model High Dimensional Model Representation with polynomial approximationRepresentation with polynomial approximation
Application of parametric GSA for optimal experimental design Application of parametric GSA for optimal experimental design
Outline
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Model Output
......
xi : input factors
Propagation of uncertainty
Input
x1
x2
…
x3
…1 2 n
…
x4
xk
y
)(xp
)(xfY
)(Yp
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Consider a model
x is a vector of input variables
Y is the model output.
1 1
0 1,2,..., 1 21
1
...
0
( ) , ... , ,..., ,
( , ,..., ) 0, , 1i s is k
k
i i ij i j k ki i j i
i i i
Y f x f f x f x x f x x x
f x x dx k k s
klji
ijlji
ij
k
ii SSSS ,...,2,1
1
...1
Variance decomposition:
Sobol’ SI:
Sensitivity Indices (SI)Sensitivity Indices (SI)
2 2 2 2, 1,2,...,i iji i j n
ANOVA decomposition (HDMR):
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Sobol’ Sensitivity Indices (SI)Sobol’ Sensitivity Indices (SI)
Definition:
- partial
variances
- variance
Requires 2n integral evaluations for calculations
Sensitivity indices for subsets of variables:
Introduction of the total variance:
Corresponding global sensitivity indices:
1 1
2 2... ... /
s si i i iS
1 1 1 1
12 2... ...
0
,..., ,...,s si i i i i is i isf x x dx x
1
220
0
f x f dx 1
2... si i
,x y z
1
1
2 2, ,
1 ...s
s
m
y i is i i
222
ztoty
,/ 22 yyS ./ 22 toty
totyS
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0 1toty yS S
How to use Sobol’ Sensitivity Indices?How to use Sobol’ Sensitivity Indices?
ytoty SS
iS totiS
0totiS f x
accounts for all interactions between y and z, x=(y,z).
The important indices in practice are and
does not depend on ;
does only depend on ;
corresponds to the absence of interactions between
and other variables
If then function has additive structure:
Fixing unessential variables
If does not depend on so it can be fixed
complexity reduction, from to variables
ix
1iS f x ixtotii SS ix
n
siS
1
,1 0 i ii
f x f f x
1totzS f x z
0,f x f y z znn n
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Evaluation of Sobol’ Sensitivity Evaluation of Sobol’ Sensitivity IndicesIndices
1
0
1 2
0
1 2 200
1( , ) ( , ') ',
1[ ( , ) ( ', )] ',
2
( , )
y
toty
S f y z f y z dydzdzD
S f y z f y z dydzdzD
D f y z dydz f
Straightforward use of Anova decomposition requires Straightforward use of Anova decomposition requires
22nn integral evaluations – not practical ! integral evaluations – not practical !
There are efficient formulas for evaluationThere are efficient formulas for evaluation
of Sobol’ Sensitivity Indices ( Sobol’ 1990):of Sobol’ Sensitivity Indices ( Sobol’ 1990):
Evaluation is reduced to high-dimensional Evaluation is reduced to high-dimensional
integration. Monte Carlo method is the only integration. Monte Carlo method is the only
way to deal with such problemsway to deal with such problems
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Original vrs Improved formulae for Original vrs Improved formulae for evaluation ofevaluation of
Sobol’ Sensitivity IndicesSobol’ Sensitivity Indices
1 200
1 2 200
1 200
112
0 00
1
0
( , ) ( , ') '
( , )
for small indices 1
( , ) ( , ') '
loss of accuracy
Notice that ( , ) ( ', ')
( , ) ( , ') ( ', ') '
( ,
' '
y
y
y
f y z f y z dydzdz fS
f y z dydz f
S
f y z f y z dydzdz f
f f y z dyd
f y z f y z f y z dydzdz
z f y z d dz
f
y
Sy
1
0 00
much more accurate ( Kucherenko, Mauntz, 2002)
Requires ( +2) model evalution original Sobol' formulas (2 +1)
The same model evaluations ca
) ( , )
n be used for computing second
z f f y z f dydz
N n N n
order indices
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Improved formula for Sobol’ Sensitivity Improved formula for Sobol’ Sensitivity IndicesIndices
Comparison between improved and original formulas for Si
-25
-20
-15
-10
-5
0
11 13 15 17 19 21
log 2
(Err
or)
improved Sobol original Sobol
log2(N)
1
71
6( ) ,
( 1)(2 1)
180, 5.110
nT
i ii
f x ix S Sn n n
n S
Comparison between improved and original formulas for Si
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
16 18 20 22 24
log
2(S
1)
improved Sobol original Sobol analytical value
log2(N)
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Comparison deterministic and Monte Carlo Comparison deterministic and Monte Carlo integration methodsintegration methods
n
n/p
2
50d
[ ] ( )
Deterministic integration method of p-order,
k points in each direction: N = k
Error: ( ), N =O(1/ ) .
Estimate: 10 , 2, 50
N =10 the total number of particles in the unive
nH
p
I f f x dx
O k
p n
rse
[ ] is impossible to evalua
"Curse of Dimensiona
te
li "
!
ty
I f
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Monte Carlo integration methods Monte Carlo integration methods
1
22
2 1/ 21/ 2
[ ] [ ( )]
1Monte Carlo : [ ] ( )
{ } is a sequence of random points in
Error: [ ] [ ]
( )Expectation:
Convergence does not d
( )
( )= ( ( ))
epent on dimensiona
N
N ii
ni
N
N
I f E f x
I f f zN
z H
I f I f
fE
N
fE
N
lity
but it is slow
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How to improve MC ?How to improve MC ?
1/ 2
II. Use better ( more uniformly di
( )Slow convergence: =
How to
stributed )
sequences.
improve MC ?
I. Decrease ( ) variance reduction.
Discrepancy is a measure of deviation from uniformity:
( )
N
f
N
f
Q y
n1 2
( )
( )
1/ 2 1/ 2
, ( ) [0, ) [0, ) ... [0, ),
( ) volume of
sup ( ) ,
random sequences: (ln ln ) / ~ 1/
n
n
Q yN
Q y H
N
H Q y y y y
m Q Q
ND m Q
N
D N N N
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Sobol’ Sequences vrs Random numbersSobol’ Sequences vrs Random numbersand regular gridand regular grid
Unlike random numbers, successive Sobol’ points “know" about the position of previously sampled points and fill the gaps between them
Regular Grid/ 64 Points Random Numbers/ 64 Points Sobol Numbers/ 64 Points
Sobol Numbers/ 256 PointsRandom Numbers/ 256 PointsRegular Grid/ 256 Points
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Quasi random sequencesQuasi random sequences
(ln )( ) Low discrepancy sequences (LDS)
Convergence: [ ] [ ] ( )
( )
Assymptotically ~ (1/ ) much higher than
Variation of
(ln
~
)
(1/ )
n
N
QMC N
Q
N
M
n
QM
C
MC
C
ND c d
NI f I f V f D
V f f
O N
NO N
O N
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What is the optimal way to arrange N What is the optimal way to arrange N points in two dimensions?points in two dimensions?
Regular Grid Sobol’ Sequence
Low dimensional projections of low discrepancy sequences are better distributed than higher dimensional projections
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Comparison between Sobol sequencesComparison between Sobol sequences
and random numbersand random numbers
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Normally distributed Sobol’ Normally distributed Sobol’ SequencesSequences
Normal probability plots Histograms
Uniformly distributed Sobol’ sequences can be Uniformly distributed Sobol’ sequences can be
transformed to any other distribution with a known transformed to any other distribution with a known
distribution functiondistribution function
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Are QMC efficient for high dimensional Are QMC efficient for high dimensional problems ?problems ?
21
(ln )
Assymptotically ~ (1/ )
but increseas with u
not achievable for prac
ntil exp( )
50, 5 10 tical applications
n
QMC
QMC
QMC
O N
NO N
N N n
n N
“For high-dimensional problems (n > 12),
QMC offers no practical advantage over Monte Carlo”
( Bratley, Fox, and Niederreiter (1992)) ?!
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DiscrepancyDiscrepancyI. Low DimensionsI. Low Dimensions
0.0001
0.001
1024 2048 4096 8192
T
N
Discrepancy, n=5
RandomHaltonSobol
1e-06
1e-05
0.0001
0.001
0.01
128 256 512 1024 2048 4096 8192 16384 32768 65536
T
N
Discrepancy, n=20
RandomHaltonSobol
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DiscrepancyDiscrepancyII. High DimensionsII. High Dimensions
1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
128 256 512 1024 2048 4096 8192 16384 32768
T
N
Discrepancy, n=50
RandomHaltonSobol
1e-20
1e-19
1e-18
1e-17
1e-16
128 256 512 1024 2048 4096 8192 16384
T
N
Discrepancy, n=100
RandomSobol
MC in high-dimensions has smaller discrepancy
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Is MC more efficient for high-dimensional Is MC more efficient for high-dimensional problems than QMC ?problems than QMC ?
Pros: MC in high-dimensions has smaller discrepancy Some studies show degradation of the convergence
rate of QMC methods in high-dimensions to O(1/√N)
Cons:
Huge success of QMC methods in finance: QMC methods were proven to be much more efficient than MC even for problems with thousands of variables
Many tests showed superior performance of QMC methods for high-dimensional integration
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Effective dimensionEffective dimension
0
The effective dimension of ( ) in the superposition sense
is the smallest integer such that
Let u be a car
(1 ),
dinality of a set of variables .
It means that ( ) is almost a sum
1S
S
uu d
f x
d
S
u
f x
1
{1,2,..., }
1,
The function ( ) has effective dimension in the truncation sense if
E
of -dimensional functions.
does not depend on the
xa
order in which
mple:
1
(1 ),T
n
i
S
i TS
u
S
T
ud
n
d
f x
f x
x d d
d
d
S
can be reduced
the input variables are sampled,
- depends on the order by reodering variables T Td d
___________________________________________________________
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For many problems only low order terms in the ANOVA decomposition are important
Consider an approximation error
Theorem 1:
Link between an approximation error and
effective dimension in superposition sense
Approximation errorsApproximation errors
Set of variables can be regarded as not important if If and
Consider an approximation error
Theorem 2:
Link between an approximation error and effective dimension in truncation sense
1 1
1
1 1
1
0 ...1 ...
2
...1 ...
( ) ( ,..., )
1( , ) ( ) ( )
( , ) 1 ( ,..., )
( , )
s s
s
s s
s
d s
i i i is i i
d s
i i i is i i
h x f f x x
f h f x h x dx
f h S x x
f h
20 0
0
0
1( ) ( ) ( , )
( ) 2
( ) 2
totz
z f x f y z dxD
z S
z
E
E
:S dd
nd
0( ) ,f x f y z1totzS
z0z z
( , )f h
:T dd
0( )z
1( ,..., )nx x x
1 1( , ) : ( ,..., ), ( ,..., )d d nx y z y x x z z x ___________________________________________________________________
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Type B. Dominant low order indices
1
1n
ii
SS nd
Classification of functionsClassification of functions
Type B,C. Variables are equally important
T Ti j TS S nd
Type A. Variables are not equally important
T Ty z
y zT
S Sn
n nd
Type C. Dominanthigher order indices
1
1n
ii
SS nd
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Sensitivity indices for type A functionsSensitivity indices for type A functions
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Integration error vs. N. Type AIntegration error vs. N. Type A(a) f(x) = ∑n
j=1(-1)i ij=1 xj, n = 360, (b) f(x) = s
i=1 │4xi-2│/(1+a
i), n = 100
{1,2} {3,4,...360}
{1,2} {3,4,...100}
0.94, 0.1
0.64
T T
T T
S S
S S
1/ 2
2
1
1( )
KkN
k
I IK
(a)
(b)
-20
-15
-10
-5
0
8 11 14 17 20
log2(N)
log
2( ) QMC (-0.94)
MC (-0.52)
-14
-10
-6
-2
8 11 14 17 20
log2(N)
log
2(
)
QMC (-0.69)
MC (-0.49)
~ , 0 1N
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Sensitivity indices for type B functionsSensitivity indices for type B functions Dominant low order indices
1
1n
ii
SS nd
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Integration error vs. N. Type BIntegration error vs. N. Type B Dominant low order indices
360
)11()(1
/1
n
xnxfn
i
ni
360
5.0)(
1
n
n
xnxf
n
i
i
-22
-18
-14
-10
8 11 14 17 20
log2(N)
log
2( ) QMC (-0.66)
MC (-0.50)
-19
-15
-11
-7
8 11 14 17 20
log2(N)
log
2(
)
QMC (-0.66)
MC (-0.53)
(a)
(b)
1
1n
ii
SS nd
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Sensitivity indices for type C functionsSensitivity indices for type C functions Dominant higher order indices
1
1n
ii
SS nd
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The integration error vs. N. Type CThe integration error vs. N. Type C Dominant higher order indices:
1/
1
( ) (1/ 2)
20
nn
ii
f x x
n
-8
-6
-4
-2
0
8 11 14 17 20
log2(N)
log
2(
)
QMC (-0.46)
MC (-0.45)
-7
-5
-3
-1
8 11 14 17 20
log2(N)
log
2)
QMC (-0.44)
MC (-0.44)
(a)
(b)
1
1
4 2( ) , 0
1
4 2
20
ni i
ii i
n
ii
x af x a
a
x
n
1
1n
ii
SS nd
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The Morris method
Model ),..,( 1 kxxyy
Elementary Effect for the ith input factor in a point Xo
),...,(),..,,,..,,(
),...,(00000
,000
00 111211
kkiiik
xxyxxxxxxyxxEEi
x1
x2
(x01, x0
2) (x01+, x0
2)
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x1
x2
x1
x2
x..
xr
r elem. effects EE1i EE2
i … EEri are
computed at X1 , … , Xr and then averaged.
Average of EEi’s (xi)
Standard deviation of the EEi’s σ (xi)
The EEi is still a local measure Solution: take the average of several EE
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A graphical representation of results
DK5 ZJ3
DK3
DJ3
ZK5
ZK4
DJ4
DK2
ZJ6
ZK1
0,00E+00
1,00E-01
2,00E-01
3,00E-01
4,00E-01
5,00E-01
6,00E-01
7,00E-01
8,00E-01
9,00E-01
0,00E+00 5,00E-02 1,00E-01 1,50E-01 2,00E-01 2,50E-01 3,00E-01 3,50E-01 4,00E-01 4,50E-01
mu
sigm
a
Factors can be screened on the (xi), σ (xi) plane
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Implemention of the Morris method
r trajectories of (k+1) sample points are generated, each providing one EE per input
x1
x2
x3
Y1 Y2
Y3
Y4
A trajectory of the EE design
Total cost = r (k + 1)r is in the range 4 -10
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.]1,0[~
0
1
24)(
UX
a
a
aXXg
i
i
i
iiii
)(1
i
k
ii Xgy
*(xi) and STi give similar ranking
Problems: large Δ -> incorrect *(xi)
a=99a=9a=0.9
A comparison with variance-based methods:*(xi) is related to STi
Test: the g-function of Sobol’
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Derivative based Global Sensitivity Measures
Morris measure in the limit Δ → 0
1 1
1
0 0 0 0
0
0 0
*
2
( ,..., ) lim ( ,..., ).
( ,..., ) .
.
k k
k
n
n
i i
ii
i iH
iiH
E x x EE x x
fE x x
x
M E dx
fdx
x
x1
x2
x1
x2
x..
xr
Sample X1 , … , Xr Sobol points, estimate finite differences E1
i ,E2i
… Eri and then averaged.
Average of Ei’s M*(xi)
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The integration error vs. N. Type AThe integration error vs. N. Type A g-function of Sobol’ .
(a)
(b)
1
1 2 3
4 2( ) ,
1
0, 6.52
ni i
i i
n
x af x
a
a a a a
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Comparison of Sobol’ SI and Derivative Comparison of Sobol’ SI and Derivative based Global Sensitivity Measuresbased Global Sensitivity Measures
(a)
(b)
1
1 2 3
4 2( ) ,
1
0, 6.52
ni i
i i
n
x af x
a
a a a a
(c) There is a link between and i tot
iS
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Comparison of Sobol’ SI and Comparison of Sobol’ SI and Derivative based Global Sensitivity Derivative based Global Sensitivity
MeasuresMeasures
iv1. Small values of imply small values of . 2. For highly nonlinear functions ranking based on global SI can be very different from that based on derivative based sensitivity measures
2
2
Assume that
then
i
tot ii
f x L
vS
D
Theorem
totiS
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For many problems only low order terms in the ANOVA decomposition are important.
01
( ) ,n
i i ij i ji i j i
h x f f x f x x
1 1
1,
1 ,n n
i iji i i j
nTi i ij
j j i
S S
S S S
Sobol’ SI:
Quasi Randon Sampling Quasi Randon Sampling HDMRHDMR
is a metamodel (HDMR), Rabitz et al:
1 1
1
1 1
1
0 ...1 ...
2
...1 ...
( ) ( ,..., )
1( , ) ( ) ( )
( , ) 1 ( ,..., )
s s
s
s s
s
d s
i i i is i i
d s
i i i is i i
h x f f x x
f h f x h x dx
f h S x x
( )h x
It is assumed that effective dimension in superposition sense ds=2.
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Polynomial Approximation
Properties:
Orthonormal polynomial
base
First few Legendre polynomials:
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Global Sensitivity Analysis Global Sensitivity Analysis (HDMR)(HDMR)
The number of function evaluations is N(n+2) for original Sobol’ method N for sensitivity indices based on RS-HDMR
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How to define maximum polynomial How to define maximum polynomial order ?order ?
Homma-Saltelli function
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RMSE for Homma-Saltelli functionRMSE for Homma-Saltelli function
Root mean square error:
QMC outperforms MC
RS-HDMR hashigher convergencethan Sobol SI method
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g-function: with 2 important and 8 unimportant variables
Sobol g-functionSobol g-function
QRS-HDMRconverges
faster
Values of Sitot
can be inaccurate.
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Sobol g-functionSobol g-function
Error measure:Error measure:
Function ApproximationFunction Approximation
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QRS-HDMR method requires 10 to 103 times less model evaluations than Sobol SI method !
Computational costsComputational costs
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Optimal experimental design (OED) for Optimal experimental design (OED) for parameter estimationparameter estimation
Find values of experimentally manipulable variables (controls) and the
time sampling strategy for a set of Nexp experiments which provides
maximum information for the subsequent parameter estimation problem
UL uuu
Non-linear programming
problem (NLP) with partial
differential-algebraic
(PDAEs) constraints
subject to:
System dynamics (ODEs, DAEs)
Other algebraic constraints
Upper and lower bounds:
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Case study: fed-batch reactorCase study: fed-batch reactor
11111 ypyur
dt
dym
2212
12 yuup
yr
dt
dy m
2
2
5.0
5.0
y
yrm
Biomass:
Substrate:
Reaction rate:
Parameters to be estimated: p1, p2
0.05 < p1 < 0.98, 0.05 < p2 < 0.98
Control variables: u1, u2
Dilution factor: 0.05 < u1 < 0.5
Feed substrate concentration:
5 < u2 < 50
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OED traditional approachOED traditional approach
2
1
A-optimality
E-optimality
D-optimality
Fisher Information Matrix ( FIM ) based criteria:
A criterion =
D criterion =
E criterion =
Modified-E criterion =
FIMFIM
min
maxmin
FIMminmax
FIMdetmax
1min FIMtrace
N
iii
T
i tp
yWt
p
yFIM
1
Main drawback: based on local SI non-realistic linear and local assumptions
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Parametric GSAParametric GSA
Optimal experimental design: identification of a set of experiments with conditions that
deliver measurement data that are the most sensitive to the unknown parameters
Nonlinear dynamic model: Y ( , , )
- uncertain parameters,
- control variables,
- time
Fi depend on parameters !nd ( , ), ( , )
Solve: max ( ( , ))
OED *for parameter estimation
Ti i
iu
f p u t
p
u
t
S u t S u t
F S u t
u
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Application of ParametricApplication of Parametric GSA for GSA for parameter optimizationparameter optimization
Main advantage: based on global SI allows to consider a range of
values for the parameters to be estimated
objective function:
Application of Global Optimization method
1
, ,N T
i i i ii
GSIM Q u t W Q u t
1,1 1,2 1,
2,1 2,2 2,
,1 ,2 ,
, , ,
, , ,
, , ,
i i p i
i i p ii
s i s i s p i
S u t S u t S u t
S u t S u t S u tQ t
S u t S u t S u t
1 1 1
1 2
1 2
, , ,
, , ,
i i ip
i
s s si i i
p
y y yu t u t u t
p p p
Q t
y y yu t u t u t
p p p
GSIMu
detmax
1
, ,N T
i i i ii
FIM Q u t W Q u t
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.
Case study: fed-batch reactorCase study: fed-batch reactor
11111 ypyur
dt
dym
2212
12 yuup
yr
dt
dy m
2
2
5.0
5.0
y
yrm
Biomass:
Substrate:
Reaction rate:
Parameters to be estimated: p1, p2
0.05 < p1 < 0.98, 0.05 < p2 < 0.98
Control variables: u1, u2
Dilution factor: 0.05 < u1 < 0.5
Feed substrate concentration:
5 < u2 < 50
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Optimal Experimental DesignOptimal Experimental Design
Problem constraints: Experiment duration: 10 h Number of measurement times: 10 Controls varied every 2 hours
Results:
0 2 4 6 8 100
10
20
30
40
Time (h)
Co
nce
ntr
atio
n (
g/l)
y1y2
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Time (h)
u1
(h
-1)
0 2 4 6 8 100
10
20
30
40
50
u2
(g
/l)
Optimal input profile for u1 and u2 :
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Setting of the Parameter Estimation Setting of the Parameter Estimation ProblemProblem
Steps to find p:
Take experimental or generated pseudo-experimental points
Maximum likelihood optimization
subject to:
System dynamics (ODEs, DAEs)
Other algebraic constraints
Upper and lower bounds:
UL ppp
Non-linear programming
problem (NLP) with partial
differential-algebraic
(PDAEs) constraints
p: set of parameters to be estimated : model prediction
: measurements variance : experimental measures
pyki
kiy~2ki
NE
i
NV
j
NM
k ijk
ijkijkijkml
i ypypJ
1 1 1
2
212~
2
1exp2
y
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Results of Results of parameter parameter estimationestimation
p1 = 0.37 ± 0.02, p2 = 0.72 ± 0.12
p1 = 0.5 ± 0.05 , p2 = 0.5 ± 0.11
0 0.2 0.4 0.6 0.8 10
5
10
15
20
25
p1
Den
sity
0 0.2 0.4 0.6 0.8 10
5
10
15
20
p2
Den
sity
0 0.2 0.4 0.6 0.8 10
5
10
15
p1
Den
sity
0 0.2 0.4 0.6 0.8 10
5
10
15
20
p2
Den
sity
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PublicationsPublications
Hung WY, Kucherenko S., Samsatli N.J. and Shah N., The Proceedings of the 2003 Summer Computer Simulation Conference, Canada. Simulation Series, V 35, N3, pp. 101-106 (2003)Hung W.Y., Kucherenko S., Samsatli N.J. and Shah N (2004). Journal of the Operational Research Society 55, 801-813.Sobol’ I., Kucherenko S. Monte Carlo Methods and Simulation, 11, 1, 1-9 (2005).Sobol’ I., Kucherenko S. Wilmott, 56-61, 1 (2005).Kucherenko S., Shah N. Wilmott, 82-91, 4 (2007). Sobol, I.M., S. Tarantola, D. Gatelli, S.S. Kucherenko, W. Mauntz Reliability Engineering & System Safety, 957-960, 92 (2007 ). Rodriguez-Fernandez M., Kucherenko S., Pantelides C., Shah N. Proc. ESCAPE17, V. Plesu and P.S. Agachi (Editors), p66-71, (2007)Kucherenko S., Mauntz W. Submitted to Journal of Comp. Physics (2007). S. Kucherenko. Fifth International Conference on Sensitivity Analysis of Model Output, Budapest, (2007)S. Kucherenko, M. Rodriguez-Fernandez, C. Pantelides, N. Shah. Submitted to Reliability Engineering Systems Safety (2007)D. Gatelli, S. Kucherenko, M. Ratto, S. Tarantola, Submitted to Reliability Engineering Systems Safety (2007)I.M. Sobol’, S. Kucherenko. Submitted to Journal of Comp. Physics (2008).
Application of Global Sensitivity Analysis to Biological ModelsA.Kiparissides, M.Rodriguez-Fernandez, S. Kucherenko, A. Mantalaris, E.Pistikopoulos Application of Global Sensitivity Analysis to Biological Models, Submitted to ESCAPE18 (2008).
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SummarySummary
Quasi MC methods based on Sobol’ sequences outperform MCuasi MC methods based on Sobol’ sequences outperform MC
The error generated by the factors fixing is bounded by the total The error generated by the factors fixing is bounded by the total sensitivitysensitivity index of the fixed factors index of the fixed factors Functions can be classified according to their effective dimensionFunctions can be classified according to their effective dimension
The method of derivative based global sensitivity measures The method of derivative based global sensitivity measures (DGSM) is more efficient than the Morris and the Sobol’ SI (DGSM) is more efficient than the Morris and the Sobol’ SI methods. There is a link between DGSM and Sobol’ SImethods. There is a link between DGSM and Sobol’ SI
Quasi Randon Sampling - Quasi Randon Sampling - High Dimensional Model Representation High Dimensional Model Representation with polynomial approximation can be orders of magnitude more with polynomial approximation can be orders of magnitude more efficient than Sobol’ SI for evaluation of main effectsefficient than Sobol’ SI for evaluation of main effects
Application of global SI to OED results in the reduction of the Application of global SI to OED results in the reduction of the required experimental work and the increased accuracy of required experimental work and the increased accuracy of parameter estimationparameter estimation
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Thank you for inviting me !
AcknowledgmentsAcknowledgments
Prof. Sobol’
Imperial College London, UK:
N. Shah, M. Rodríguez Fernández, B. Feil, W. Mauntz,
C. Pantelides
Joint Research Centre, ISPRA, Italy:
S. Tarantola, D. Gatelli, M. Ratto
Financial support:
EPSRC Grant EP/D506743/1