Generalized linear array model

Generalized linear array model

In statistics, the generalized linear array model(GLAM) is used for analyzing the data sets with array structure. It based on the generalized linear model with the regression matrix written as a Kronecker product.

Overview

In the article published in the Journal of the Royal Statistical Society series B, 2006, Currie, Durban and Eilers introduced the generalized linear array model or GLAM. GLAMs provide a structure and a computational procedure for fitting generalized linear models or GLMs whose model matrix can be written as a Kronecker product and whose data can be written as an array. In a large GLM, the GLAM approach gives very substantial savings in both storage and computational time over the usual GLM algorithm.

Suppose the data mathbf Y is arranged in a d-dimensional array with size n_1 imes n_2 imesldots imes n_d; thus,the corresponding data vector mathbf y = extbf{vec}(mathbf Y) has size n_1n_2n_3cdots n_d. Suppose also that the regression matrix mathbf X = mathbf X_dotimesmathbf X_{d-1}otimesldotsotimesmathbf X_1.

The standard analysis of a GLM with data vector mathbf y and regression matrix mathbf X proceeds by repeated evaluation of the scoring algorithm

mathbf X' ilde{mathbf W}_deltamathbf Xhat{oldsymbol heta} = mathbf X' ilde{mathbf W}_delta ilde{mathbf z}

where ilde{oldsymbol heta} represents the approximate solution of oldsymbol heta, and hat{oldsymbol heta} is the improved value of it; mathbf W_delta is the diagonal weight matrix with elements

w_{ii}^{-1} = left(frac{partialeta_i}{partialmu_i} ight)^2 ext{var}(y_i),

and mathbf z = oldsymboleta + mathbf W_delta^{-1}(mathbf y - oldsymbolmu) is the working variable.

Computationally, GLAM provides array algorithms to calculate the linear predictor, oldsymboleta = mathbf X oldsymbol heta and the weighted inner product mathbf X' ilde{mathbf W}_deltamathbf X without evaluation of the model matrix mathbf X .

Example: In 2 dimensions, let mathbf X = mathbf X_2otimesmathbf X_1 then the linear predictor is written mathbf X_1 oldsymbolTheta mathbf X_2' where oldsymbolTheta is the matrix of coefficients; the weighted inner product is obtained from G(mathbf X_1)' mathbf W G(mathbf X_2) and mathbf W is the matrix of weights; here G(mathbf M) is the row tensor function of the r imes c matrix mathbf M given by

G(mathbf M) = (mathbf M otimes mathbf 1') * (mathbf 1' otimes mathbf M) where * means element by element multiplcation and mathbf 1 is a vector of 1's of length c.

These low storage high speed formulae extend to d-dimensions.

Applications: GLAM is designed to be used in d-dimensional smoothing problems where the data are arranged in an array and the smoothing matrix is constructed as a Kronecker product of d one-dimensional smoothing matrices.

References

* I.D Currie, M. Durban and P. H. C. Eilers (2006) Generalized linear array models with applications to multidimensional smoothing,"Journal of Royal Statistical Society - Series B", 68, part 2, 259-280.


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