Re: How to parameterize the positive semi-definite correlation matrix?

From: Herman Rubin (hrubin_at_odds.stat.purdue.edu)
Date: 12/13/04 

Date: 13 Dec 2004 09:27:01 -0500

In article <s7o4jelq1uoq@legacy>,
T. Wakamatsu <WakamatsuT@pac.dfo-mpo.gc.ca> wrote:
>Hi there

>I am trying to compute random vectors obeying a certain correlation
>matrix parameterized by the following function.

>C(x_i,y_i;x_j,y_j)=exp(-(x_i-x_j)**2/L-(y_i-y_j)**2/L)

>Where, L is the decorrelation length scale and (x_i,y_i) is
>a point on a 2D plane. So, I am trying to construct 2D random field.
>The eigenvalue-eigenvector decomposition of the correlation matrix
>works in perfect to construct square root of the matrix in a sample
>problem with smaller size, say 10*10 matrix. But when I apply
>the same procedure for my real problem of 10000*10000 matrix,
>it starts to reply negative eigen values.
>Questions are:
>1. Does the above function not guarantee the positive
>semi-difinitiness of the resultant correlation matrix?
>2. If not, is there better simple function to parameterize
>a correlation matrix?

It is a correlation matrix for any size. Further, if you
take a rectangular grid, it is a Kronecker product, as the
correlations are the product of a function of x_i-x_j and
a function of y_i-y_j. This makes the characteristic values
and vectors easy to compute.

So if x_i, i=1, ..., I, and y_j, j=1, ..., J, are the points
of interest, compute the characteristic roots r_1, ..., r_I
for the first index and the corresponding vectors v_i, and
similarly s_j and w_j for the second. Then for the 2D array,
the roots are the products r_i*s_j and the vectors v_i*w_j. 

-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558