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.

Suppose that the data ${\displaystyle \mathbf {Y} }$ is arranged in a ${\displaystyle d}$-dimensional array with size ${\displaystyle n_{1}\times n_{2}\times \dots \times n_{d}}$; thus, the corresponding data vector ${\displaystyle \mathbf {y} =\operatorname {vec} (\mathbf {Y} )}$ has size ${\displaystyle n_{1}n_{2}n_{3}\cdots n_{d}}$. Suppose also that the design matrix is of the form

${\displaystyle \mathbf {X} =\mathbf {X} _{d}\otimes \mathbf {X} _{d-1}\otimes \dots \otimes \mathbf {X} _{1}.}$

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

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

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

${\displaystyle w_{ii}^{-1}=\left({\frac {\partial \eta _{i}}{\partial \mu _{i}}}\right)^{2}\mathrm {var} (y_{i}),}$

and

${\displaystyle \mathbf {z} ={\boldsymbol {\eta }}+\mathbf {W} _{\delta }^{-1}(\mathbf {y} -{\boldsymbol {\mu }})}$

is the working variable.

Computationally, GLAM provides array algorithms to calculate the linear predictor,

${\displaystyle {\boldsymbol {\eta }}=\mathbf {X} {\boldsymbol {\theta }}}$

and the weighted inner product

${\displaystyle \mathbf {X} '{\tilde {\mathbf {W} }}_{\delta }\mathbf {X} }$

without evaluation of the model matrix ${\displaystyle \mathbf {X} .}$

In 2 dimensions, let ${\displaystyle \mathbf {X} =\mathbf {X} _{2}\otimes \mathbf {X} _{1}}$, then the linear predictor is written ${\displaystyle \mathbf {X} _{1}{\boldsymbol {\Theta }}\mathbf {X} _{2}'}$ where ${\displaystyle {\boldsymbol {\Theta }}}$ is the matrix of coefficients; the weighted inner product is obtained from ${\displaystyle G(\mathbf {X} _{1})'\mathbf {W} G(\mathbf {X} _{2})}$ and ${\displaystyle \mathbf {W} }$ is the matrix of weights; here ${\displaystyle G(\mathbf {M} )}$ is the row tensor function of the ${\displaystyle r\times c}$ matrix ${\displaystyle \mathbf {M} }$ given by^[1]

${\displaystyle G(\mathbf {M} )=(\mathbf {M} \otimes \mathbf {1} ')\circ (\mathbf {1} '\otimes \mathbf {M} )}$

where ${\displaystyle \circ }$ means element by element multiplication and ${\displaystyle \mathbf {1} }$ is a vector of 1's of length ${\displaystyle c}$.

On the other hand, the row tensor function ${\displaystyle G(\mathbf {M} )}$ of the ${\displaystyle r\times c}$ matrix ${\displaystyle \mathbf {M} }$ is the example of Face-splitting product of matrices, which was proposed by Vadym Slyusar in 1996:^[2][3][4][5]

${\displaystyle \mathbf {M} \bullet \mathbf {M} =\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)\circ \left(\mathbf {1} ^{\textsf {T}}\otimes \mathbf {M} \right),}$

where ${\displaystyle \bullet }$ means Face-splitting product.

These low storage high speed formulae extend to ${\displaystyle d}$-dimensions.

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