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] Reference

  • 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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