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