glmm-with-spatial-structure-described-in-terms-of-covariance-function

Poisson GLMM

Filed under: Spatial Models

GLMM with spatial correlation, where the locations do not lie on a grid. Illustrates how you can parameterize a large correlation matrix in terms of an isotropic correlation function r(d), where "d" is the distiance between two locations.

Model description

Our data are 100 Poisson counts (y), each with parameter lambda. The datapoints are index by i and j (i,j=1,...,10). It is assumed that 

where  X_i,jb is a linear predictor and e_i,j are Gaussian random variables with covariance

Here d is the Euclidean distance between the two positions.

Orthogonalization

This example shows a mathematical trick that is useful in all sorts of regression analysis: make the columns of the design matrix X orthogonal. This makes the model more stable, but when you later shall interpret the output (b vector) you must "transform back".

DATA_SECTION

matrix dd(1,n,1,n); // Distance matrix

LOC_CALCS

int i, j;

dmatrix tX=trans(X);

ncol1=norm(tX(1));

tX(1)/=ncol1;

tX(2)-= tX(1)*tX(2)*tX(1);

cout << tX(1)*tX(2) << endl;

ncol2=norm(tX(2));

tX(2)/=ncol2;

X=trans(tX);

END_CALCS

###Files:

• spatial.dat
• spatial.tpl
• spatial.par
• spatial.pin