1. METHODS OF TRANSFORMING NON-POSITIVE DEFINITE CORRELATION MATRICES
Katarzyna Wojtaszek
student number
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 Assumptions:
 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
Rnxn
SET OF PD-MATRICES
Linear shrinking In

12. P -orthogonal matrix such that R=PDPT
 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. = where is a diagonal matrix
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. Rnxn pseudo correlation matrix Idea:
 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. Let say that we have matrix R4x4
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. n 3 4 5 6 7 8 9 10
Lin. shrinking
2.7868
4.371
6.7233
9.8977
Shrinking f1
0.1388
0.4028
1.1251
2.5161
4.3623
6.76
9.8484
Shrinking f2
0.2756
0.9696
2.382
4.6464
8.1327
Shrinking f3
0.1441
0.4589
1.1432
2.5153
4.4483
6.9127
10.176
Shrinking f4
0.4091
1.4379
3.3365
5.7357
8.6839
15.686
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
17.813

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