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Poisson GLMM

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 

 

log(lambdai,j) = Xi,jb + ei,j.

 

where  Xi,jb is a linear predictor and ei,j are Gaussian random variables with covariance

cov(ei1,j1,ei2,j2) = s2 exp(a-1 d),

 

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