Generalized linear array model

The generalized linear array model or GLAM was introduced in 2006.{{cite journal |last1=Currie |first1=I. D. |last2=Durban |first2=M. |last3=Eilers |first3=P. H. C. |year=2006 |title=Generalized linear array models with applications to multidimensional smoothing |journal=Journal of the Royal Statistical Society |volume=68 |issue=2 |pages=259–280 |doi=10.1111/j.1467-9868.2006.00543.x |s2cid=10261944 }} 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 $\backslash mathbf\; Y$ is arranged in a $d$-dimensional array with size $n\_1\backslash times\; n\_2\backslash times\backslash dots\backslash times\; n\_d$; thus, the corresponding data vector $\backslash mathbf\; y\; =\; \backslash operatorname\{vec\}(\backslash mathbf\; Y)$ has size $n\_1n\_2n\_3\backslash cdots\; n\_d$. Suppose also that the design matrix is of the form

:$\backslash mathbf\; X\; =\; \backslash mathbf\; X\_d\backslash otimes\backslash mathbf\; X\_\{d-1\}\backslash otimes\backslash dots\backslash otimes\backslash mathbf\; X\_1.$

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

:$\backslash mathbf\; X\text{'}\backslash tilde\{\backslash mathbf\; W\}\_\backslash delta\backslash mathbf\; X\backslash hat\{\backslash boldsymbol\backslash theta\}\; =\; \backslash mathbf\; X\text{'}\backslash tilde\{\backslash mathbf\; W\}\_\backslash delta\backslash tilde\{\backslash boldsymbol\backslash theta\}\; ,$

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

:$w\_\{ii\}^\{-1\}\; =\; \backslash left(\backslash frac\{\backslash partial\backslash eta\_i\}\{\backslash partial\backslash mu\_i\}\backslash right)^2\backslash mathrm\{var\}(y\_i),$

and

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

is the working variable.

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

:$\backslash boldsymbol\backslash eta\; =\; \backslash mathbf\; X\; \backslash boldsymbol\backslash theta$

and the weighted inner product

:$\backslash mathbf\; X\text{'}\backslash tilde\{\backslash mathbf\; W\}\_\backslash delta\backslash mathbf\; X$

without evaluation of the model matrix $\backslash mathbf\; X\; .$

In 2 dimensions, let $\backslash mathbf\; X\; =\; \backslash mathbf\; X\_2\backslash otimes\backslash mathbf\; X\_1$, then the linear predictor is written $\backslash mathbf\; X\_1\; \backslash boldsymbol\backslash Theta\; \backslash mathbf\; X\_2\text{'}$ where $\backslash boldsymbol\backslash Theta$ is the matrix of coefficients; the weighted inner product is obtained from $G(\backslash mathbf\; X\_1)\text{'}\; \backslash mathbf\; W\; G(\backslash mathbf\; X\_2)$ and $\backslash mathbf\; W$ is the matrix of weights; here $G(\backslash mathbf\; M)$ is the row tensor function of the $r\; \backslash times\; c$ matrix $\backslash mathbf\; M$ given by

:$G(\backslash mathbf\; M)\; =\; (\backslash mathbf\; M\; \backslash otimes\; \backslash mathbf\; 1\text{'})\; \backslash circ\; (\backslash mathbf\; 1\text{'}\; \backslash otimes\; \backslash mathbf\; M)$

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

On the other hand, the row tensor function $G(\backslash mathbf\; M)$ of the $r\; \backslash times\; c$ matrix $\backslash mathbf\; M$ is the example of Face-splitting product of matrices, which was proposed by Vadym Slyusar in 1996:{{Cite journal|last=Slyusar|first=V. I.|date= December 27, 1996|title=End products in matrices in radar applications. |url=http://slyusar.kiev.ua/en/IZV_1998_3.pdf|journal=Radioelectronics and Communications Systems |volume=41 |issue=3|pages=50–53}}{{Cite journal|last=Slyusar|first=V. I.|date=1997-05-20|title=Analytical model of the digital antenna array on a basis of face-splitting matrix products. |url=http://slyusar.kiev.ua/ICATT97.pdf|journal=Proc. ICATT-97, Kyiv|pages=108–109}}{{Cite journal|last=Slyusar|first=V. I.|date=1997-09-15|title=New operations of matrices product for applications of radars|url=http://slyusar.kiev.ua/DIPED_1997.pdf|journal=Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv.|pages=73–74}}{{Cite journal|last=Slyusar|first=V. I.|date=March 13, 1998|title=A Family of Face Products of Matrices and its Properties|url=http://slyusar.kiev.ua/FACE.pdf|journal=Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz. 1999.|volume=35|issue=3|pages=379–384|doi=10.1007/BF02733426|s2cid=119661450 }}

:$\backslash mathbf\{M\}\; \backslash bull\; \backslash mathbf\{M\}\; =\; \backslash left(\backslash mathbf\; \{M\}\; \backslash otimes\; \backslash mathbf\; \{1\}^\backslash textsf\{T\}\backslash right)\; \backslash circ\; \backslash left(\backslash mathbf\; \{1\}^\backslash textsf\{T\}\; \backslash otimes\; \backslash mathbf\; \{M\}\backslash right)\; ,$

where $\backslash bull$ means Face-splitting product.

These low storage high speed formulae extend to $d$-dimensions.

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

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Category:Regression models

Array model