METHODS OF TRANSFORMING NON-POSITIVE DEFINITE CORRELATION MATRICES Katarzyna Wojtaszek student number 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

2. I can also estimate whole data together: with i=1,…,s ; j=1,…,n

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:

1. SHRINKING METHOD  linear shrinking Assumptions: R nxn is given non-PD pseudo correlation matrix is arbitrary correlation matrix Define:  (  [0,1]) =R+ (R* - R) is a pseudo correlation matrix.

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: And  0 if  - / ( *- ). So we find matrix which is PD matrix given non-PD matrix R.

 non-linear shrinking Assumption: R nxn is given non-PD pseudo correlation matrix Procedure: where f is strictly increasing odd function with f(0)=0 and  >0.

I considered the following four functions:    

InIn R nxn SET OF PD- MATRICES Linear shrinking Non-linear shrinking Comparison of the linear and non-linear shrinking methods

2.THE EIGENVALUE METHOD. Assumptions: R nxn non-PD pseudo correlation matrix P -orthogonal matrix such that R=PDP T 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*P T = where is a diagonal matrix with diagonal elements equal for i=1,2,…,n.

3.VINES METHOD. Assumptions: R nxn pseudo correlation matrix Idea:  First we have to check if our matrix is PD

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.

 Example  Let say that we have matrix R 4x4 Very useful is making graphical model

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 Lin. shrinking Shrinking f Shrinking f Shrinking f Shrinking f Eigenvalues Vines

 ILUSTATION: average distance

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