METHODS OF TRANSFORMING NON-POSITIVE DEFINITE

CORRELATION MATRICES

Katarzyna Wojtaszek

student number 1118676

CROSS

I will try to answer questions:  How can I estimate correlation matrix when I have

data?

  What can I do if matrices are non-PD?

Shrinking method

Eigenvalues method

Vines method

 How can we calculate distances between original and transformed matrices?

Which method is the best?

comparing

conclusions

  How can I estimate correlation matrix if I have data?

I can estimate the correlation matrices from data as follows:

1. I can estimate each off-diagonal element separately

n

ii

n

ii

n

iii

yyxx

yyxx

1

2_

1

2_

1

__

)()(

))((

2. I can also estimate whole data together:

with

i=1,…,s ; j=1,…,n

__

1_

_

1

......)( n

n

T

xx

x

x

s

XXxC

s

xx

s

kik

i

1

_

][ ijxX

What can I do when matrices are non- PD?

 We can use some methods for transforming these matrices to PD correlation matrices using:

Shrinking method

Eigenvalues method

Vines method

How can we calculate distances between original and transformed matrices?  

There are many methods which we can use to calculate the distance between matrices .

In my project I used formula:

n

i

n

iijij rrRRd

1 1

2~~

2 )() ,(

1. SHRINKING METHOD 

linear shrinking

Assumptions:

• Rnxn is given non-PD pseudo correlation matrix

• is arbitrary correlation matrix

Define: ( [0,1]) =R+ (R* - R) is a pseudo correlation matrix.

*nxnR

~

R

Idea:

find the smallest such that matrix will be PD. Since R is non-PD then the smallest eigenvalue of R is negative , so we have to choose such that will be positive. Hence:

~

~

*)1()*)1((minmin1

~

1

xRxRxxxRx TT

xx

T

xx tt

And 0 if - / (*-).

So we find matrix which is PD matrix given

non-PD matrix R.

~

~

R

~

R

non-linear shrinking

Assumption:

Rnxn is given non-PD pseudo correlation matrix

Procedure:

)())((

)(0

)())((

)(

11

1

11

frifrff

frif

frifrff

rg

ijij

ij

ijij

ij

where f is strictly increasing odd function with f(0)=0 and >0.

I considered the following four functions:

)(tanh)(1 xxf

)(tanh)( 12 xxf

)arctan(2

)(3 xxf

2tan)(4

xxf

In

Rnxn

nxnR~

nxnR~

SET OF PD-MATRICES

Linear shrinking

Non-linear shrinking

Comparison of the linear and non-linear shrinking methods

 2.THE EIGENVALUE METHOD.

 Assumptions:

• Rnxn non-PD pseudo correlation matrix

•P -orthogonal matrix such that R=PDPT

•D matrix which the eigenvalues of R on the diagonal

is some constant 0

Idea:

Replaced negative values in matrix D by .

We obtain:

R*=PD*PT

= where is a diagonal matrix

with diagonal elements equal

for i=1,2,…,n.

~

RT

DRD~

*~ ~

D

*

1

ijr

3.VINES METHOD. Assumptions:

•Rnxn pseudo correlation matrix

Idea:

First we have to check if our matrix is PD

21,...,3;2

21,...,3;1

1,...,3;21,..,3;11,...,3;12,...,4,3;12

11

nnnn

nnnnnn

If some (-1,1) we change the value

V( ) (-1,1)) and recalculate partial correlation using:

V( ) =V( )

+

We obtain new matrix , witch we have check again.

n,...,3;12

n,...,3;12

1,...,3;12 n n,...,4,3;12 21,...,3;2

21,...,3;1 11 nnnn

1,...,3;21,..,3;1 nnnn

Example

Let say that we have matrix R4x4

44434241

34333231

24232221

14131211

Very useful is making graphical model

1

2

3

4

1;23

1;24

1413

12;34

12

Which method is the best?Comparing. Using Matlab I chose randomly 500 non-PD matrices, transformed them and calculated the average distances between non-PD and PD matrices. This table shows us my results.

n 3 4 5 6 7 8 9 10

Lin. shrinking 2.7868 4.371 6.7233 9.8977 14.0027 18.4047 23.7102 29.6013

Shrinking f1 0.1388 0.4028 1.1251 2.5161 4.3623 6.76 9.8484 13.8416

Shrinking f2 0.2756 0.9696 2.382 4.6464 8.1327 11.4816 16.3835 20.5501

Shrinking f3 0.1441 0.4589 1.1432 2.5153 4.4483 6.9127 10.176 13.7543

Shrinking f4 0.4091 1.4379 3.3365 5.7357 8.6839 11.7034 15.686 18.9959

Eigenvalues 0.0861 0.2039 0.451 0.913 1.5799 2.3263 3.3845 4.7033

Vines 0.2285 1.2999 3.3251 6.6395 11.3295 17.813 24.7021 34.4963

ILUSTATION: average distance

2 3 4 5 6 7 8 9 10 11

0

5

10

15

20

25

30

35

SIZE OF MATRICES

ME

AN

DIS

TA

NC

E

MEAN DISTANCE BETWEEN ORIGINAL AND TRANSFORMED MATRICES

linear shrinkink shrinking f1 shrinking f2 shrinking f3 shrinking f4 eigenvalue vines

Conclusions:

1. The reason that the linear shrinking is very bad method is that we shrink all elements by the same relative amount

2. The eigenvalues method performes fast and gives very good results regardless matrices dimensions

3. For the non-linear shrinking method the best choice of the projection function are and )(tanh)(1 xxf )arctan(

2)(3 xxf