How to specify arh(1) covariance structure in PROC IML simulation code for repeated measures?

[Lyson]

Helo family!

I am looking for a code to specify arh(1) covariance structure in the PROC IML simulation procedure. I saw the IML code for specifying the R-side ar(1)  covariance structure by Gibbs & Kiernan (2020), but I am struggling to figure out how to capture arh(1) instead and generate my simulation samples. Here is the code below by Gibbs & Kiernan (2020), when the R-side covariance structure is ar(1):

proc iml worksize=100;
  resid=2;
  rho=0.8;
  m=5; /* m = number of time points  */
  cov=j(m,m,1);
  do i=1 to m;
      do j=1 to m;
        cov=[i, j]=resid*rho**abs(i-j);
    end;
  end;
mean={0 0 0};
n=200;
call randseed(61345);
z=RandNormal(n, mean, cov);

create AR1Errors from z;
append from z;
quit;

Kindly assist with the parameters (code) required for arh(1) structure in a similar code as given above. Alternative approaches (codes) are most welcome.

Thank you for your assistance.

[Rick_SAS · Accepted solution]

The ARH(1) structure is shown in the PROC MIXED documentation.

The structure is the Hadamard product of an outer-product matrix sigma*sigma` and an AR1 matrix (which is a Toeplitz matrix). I think the following IML program will sample normal errors from an ARH1 covariance matrix:

proc iml;
sigma = {8, 16, 4, 8, 1};
rho=0.5;
dim = nrow(sigma);

/* AR1 correlation structure:
   https://blogs.sas.com/content/iml/2012/11/05/constructing-common-covariance-structures.html
*/
start AR1Corr(dim, rho);
   u = cuprod(j(1,dim-1,rho)); /* cumulative product */
   return( toeplitz(1 || u) );
finish;

/* ARH(1) is Hadamard product of outer product sigma*sigma` 
   and the AR(1) correlation matrix */
OutProd = sigma * sigma`;
AR1 = AR1Corr(dim, rho);
ARH1Cov = OutProd # AR1;
print OutProd, AR1, ARH1Cov;

mean = j(1, dim, 0);
n = 200;
call randseed(61345);
z = RandNormal(n, mean, ARH1Cov);

create ARH1Errors from z[c=('x1':'x5')];
append from z;
quit;

/* visualize */
proc sgscatter data=ARH1Errors;
matrix x1-x5;
run;

[Lyson]

Hi Rick,

Thank you so much. You are very right. I got a chance to go through the suggested code and additional reading material. It is quite clear. I thank you.

[Rick_SAS]

For a general discussion about how to generate heterogeneous covariance matrices, see "Construct heterogeneous structured covariance matrices in SAS."

[Lyson]

Dear Rick

I will definitely follow the discussion.

Thank you so much.