Generalized linear array model

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 ${\displaystyle \mathbf {X} .}$

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.