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.
The generalized linear array model or GLAM was introduced in 2006.
[1] 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.
-dimensional array with size
; thus, the corresponding data vector
Suppose also that the design matrix is of the form The standard analysis of a GLM with data vector
and design matrix
proceeds by repeated evaluation of the scoring algorithm where
represents the approximate solution of
is the diagonal weight matrix with elements and is the working variable.
Computationally, GLAM provides array algorithms to calculate the linear predictor, and the weighted inner product without evaluation of the model matrix
, then the linear predictor is written
is the matrix of coefficients; the weighted inner product is obtained from
is the matrix of weights; here
is the row tensor function of the
On the other hand, the row tensor function
is the example of Face-splitting product of matrices, which was proposed by Vadym Slyusar in 1996:[2][3][4][5] where
means Face-splitting product.
These low storage high speed formulae extend to
GLAM is designed to be used in
-dimensional smoothing problems where the data are arranged in an array and the smoothing matrix is constructed as a Kronecker product of
one-dimensional smoothing matrices.