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<h1 id="parent-fieldname-title" class="documentFirstHeading">
Poisson GLMM
</h1>
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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.
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<h3><strong>Model description</strong></h3>
Our data are 100 Poisson counts (y), each with parameter lambda. The datapoints are index by i and j (<em>i,j</em>=1,...,10). It is assumed that
<p> </p>
<div align="center">log(lambda<sub>i,j</sub>) = X<sub><em>i,j</em></sub>b + e<sub><em>i,j</em></sub>.</div>
<p> </p>
where <em>X<sub>i,j</sub></em>b is a linear predictor and e<sub>i,j</sub> are Gaussian random variables with covariance
<div align="center"><br />cov(e<sub><em>i1,j1</em></sub>,e<sub><em>i2,j2</em></sub>) = s<sup>2</sup> exp(a<sup>-1</sup> <em>d</em>),</div>
<p> </p>
Here <em>d</em> is the Euclidean distance between the two positions.
<p> </p>
<base />
<h3>Orthogonalization</h3>
<p>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".</p>
<pre>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
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