Generalized linear array model

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.

Overview

In the artcile published on Journal of Royal Statistical Society series B, 2006, Currie, Durban and Eiler introduced the generalized linear array model. Suppose the data set $\mathbf {Y}$ is arranged in a $d$-dimensional array with size $n_{1}\times n_{2}\times \ldots \times n_{d}$, thus, the corresponding data vector $\mathbf {y} ={\textbf {vec}}(\mathbf {Y} )$ with size $n_{1}n_{2}n_{3}\cdots n_{d}$, which can be analyzed with regression matrix $\mathbf {X} =\mathbf {X} _{d}\otimes \mathbf {X} _{d-1}\otimes \ldots \otimes \mathbf {X} _{1}$.

With this method, this model can be fitted as a standard Generalized linear model with data vector $\mathbf {y}$ and regression matrix $\mathbf {X}$ by repeat 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; $\mathbf {W} _{d}elta$ 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.