In statistics, the generalized linear array model (GLAM) is used for analyzing data sets with array structures. It based on the generalized linear model with the design matrix written as a Kronecker product.
Overview.
The generalized linear array model or GLAM was introduced in 2006. Such models 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 that the data formula_1 is arranged in a formula_2-dimensional array with size formula_3; thus, the corresponding data vector formula_4 has size formula_5. Suppose also that the design matrix is of the form
formula_6
The standard analysis of a GLM with data vector formula_7 and design matrix formula_8 proceeds by repeated evaluation of the scoring algorithm
formula_9
where formula_10 represents the approximate solution of formula_11, and formula_12 is the improved value of it; formula_13 is the diagonal weight matrix with elements
formula_14
and 
formula_15 
is the working variable.
Computationally, GLAM provides array algorithms to calculate the linear predictor, 
formula_16 
and the weighted inner product 
formula_17
without evaluation of the model matrix formula_18
Example.
In 2 dimensions, let formula_19, then the linear predictor is written formula_20 where formula_21 is the matrix of coefficients; the weighted inner product is obtained from formula_22 and formula_23 is the matrix of weights; here formula_24 is the row tensor function of the formula_25 matrix formula_26 given by
formula_27
where formula_28 means element by element multiplication and formula_29 is a vector of 1's of length formula_30.
On the other hand, the row tensor function formula_24 of the formula_25 matrix formula_26 is the example of Face-splitting product of matrices, which was proposed by Vadym Slyusar in 1996:
formula_34
where formula_35 means Face-splitting product.
These low storage high speed formulae extend to formula_2-dimensions.
Applications.
GLAM is designed to be used in formula_2-dimensional smoothing problems where the data are arranged in an array and the smoothing matrix is constructed as a Kronecker product of formula_2 one-dimensional smoothing matrices.
References.
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