Methods of Transforming Non-Positive Definite Correlation Matrices

1. METHODS OF TRANSFORMING NON-POSITIVE DEFINITE CORRELATION MATRICES Katarzyna Wojtaszek student number 1118676 CROSS

2. 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

3. 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

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

5. 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

6. 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:

7. 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.

8. 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.

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

10. I considered the following four functions:    

11. Comparison of the linear and non-linear shrinking methods Non-linear shrinking Rnxn SET OF PD-MATRICES Linear shrinking In

12. 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

13. 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.

14. 3.VINES METHOD. • Assumptions: • Rnxn pseudo correlation matrix • Idea: • First we have to check if our matrix is PD

15. 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.

16. Example • Let say that we have matrix R4x4 Very useful is making graphical model 1 2 4 3

17. 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.

18. ILUSTATION: average distance

19. Conclusions: • The reason that the linear shrinking is very bad method is that we shrink all elements by the same relative amount • The eigenvalues method performes fast and gives very good results regardless matrices dimensions • For the non-linear shrinking method the best choice of the projection function are and