topic Re: Kronecker Product in Statistical Procedures

Kronecker Product

JOADUTWUM — Tue, 03 May 2022 23:54:53 GMT

I used UN@AR(1) correlation structure for a spatiotemporal dataset in Proc Mixed. How can I get the spatiotemporal correlation (the Kronecker product of the two correlations)? Thank you.

Re: Kronecker Product

Ksharp — Wed, 04 May 2022 09:32:36 GMT

Do you have SAS/IML ?

Maybe @Rick_SAS could give you a hand.

Re: Kronecker Product

JOADUTWUM — Wed, 04 May 2022 09:36:39 GMT

Yes I do.

Re: Kronecker Product

Rick_SAS — Wed, 04 May 2022 10:11:33 GMT

To create an AR(1) matrix from the estimate, rho, see the first section in "Fast simulation of multivariate normal data with an AR(1) correlation structure."

Let's suppose both matrices are 3x3, although they can be any square dimensions. The following program computes the Kronecker product of an AR(1) correlation matrix with rho=0.8 and a unstructured matrix with estimates UN[1,2]=0.63, UN[1,3]=0.12, and UN[2,3]=0.34:

proc iml;
/* return p x p matrix whose (i,j)th element is rho^|i - j| 
See https://blogs.sas.com/content/iml/2018/10/03/ar1-cholesky-root-simulation.html */
start AR1Corr(rho, p); 
   return rho##distance(T(1:p), T(1:p), "L1");
finish;
 
/* test on a matrix with rho = 0.8 */
rho = 0.8;  p = 3;
AR1 = AR1Corr(rho, p);

UN = {1    0.63 0.12,
      0.63 1    0.34,
      0.12 0.34 1   };

K = UN@AR1;
print UN, AR1, K;