Generalized linear array model

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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.

[edit] 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 $n_1\times n_2\times\ldots\times n_d$ ; thus,the corresponding data vector $\mathbf y = \textbf{vec}(\mathbf Y)$ has size $n_1n_2n_3\cdots n_d$ . Suppose also that the regression matrix $\mathbf X = \mathbf X_d\otimes\mathbf X_{d-1}\otimes\ldots\otimes\mathbf X_1$ .

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

$\mathbf X'\tilde{\mathbf W}_\delta\mathbf X\hat{\boldsymbol\theta} = \mathbf X'\tilde{\mathbf W}_\delta\tilde{\mathbf z}$

where $\tilde{\boldsymbol\theta}$ represents the approximate solution of $\boldsymbol\theta$ , and $\hat{\boldsymbol\theta}$ is the improved value of it; $\mathbf W_\delta$ is the diagonal weight matrix with elements

$w_{ii}^{-1} = \left(\frac{\partial\eta_i}{\partial\mu_i}\right)^2\text{var}(y_i),$

and $\mathbf z = \boldsymbol\eta + \mathbf W_\delta^{-1}(\mathbf y - \boldsymbol\mu)$ is the working variable.

Computationally, GLAM provides array algorithms to calculate the linear predictor, $\boldsymbol\eta = \mathbf X \boldsymbol\theta$ and the weighted inner product $\mathbf X'\tilde{\mathbf W}_\delta\mathbf X$ without evaluation of the model matrix $\mathbf X$ .

Example: In 2 dimensions, let $\mathbf X = \mathbf X_2\otimes\mathbf X_1$ then the linear predictor is written $\mathbf X_1 \boldsymbol\Theta \mathbf X_2'$ where $\boldsymbol\Theta$ 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 $r \times c$ 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.

[edit] 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.