Generalized linear array model

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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#edit

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

Example#edit

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.

Applications#edit

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.

References#edit

^ a b Currie, I. D.; Durban, M.; Eilers, P. H. C. (2006). "Generalized linear array models with applications to multidimensional smoothing". Journal of the Royal Statistical Society. 68 (2): 259–280. doi:10.1111/j.1467-9868.2006.00543.x. S2CID 10261944.
^ Slyusar, V. I. (December 27, 1996). "End products in matrices in radar applications" (PDF). Radioelectronics and Communications Systems. 41 (3): 50–53.
^ Slyusar, V. I. (1997-05-20). "Analytical model of the digital antenna array on a basis of face-splitting matrix products" (PDF). Proc. ICATT-97, Kyiv: 108–109.
^ Slyusar, V. I. (1997-09-15). "New operations of matrices product for applications of radars" (PDF). Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv.: 73–74.
^ Slyusar, V. I. (March 13, 1998). "A Family of Face Products of Matrices and its Properties" (PDF). Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz. 1999. 35 (3): 379–384. doi:10.1007/BF02733426. S2CID 119661450.

Categories Regression models Generalized linear models

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