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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 ==
The generalized linear array model or GLAM was introduced in 2006.<ref name="GLAM">{{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 }}</ref> Such models provide a structure and a computational procedure for fitting [[generalized linear model]]s 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 <math>\mathbf Y</math> is arranged in a <math>d</math>-dimensional array with size <math>n_1\times n_2\times\dots\times n_d</math>; thus, the corresponding data vector <math>\mathbf y = \operatorname{vec}(\mathbf Y)</math> has size <math>n_1n_2n_3\cdots n_d</math>. Suppose also that the [[design matrix]] is of the form
:<math>\mathbf X = \mathbf X_d\otimes\mathbf X_{d-1}\otimes\dots\otimes\mathbf X_1.</math>

The standard analysis of a GLM with data vector <math>\mathbf y</math> and design matrix <math>\mathbf X</math> proceeds by repeated evaluation of the scoring algorithm

:<math> \mathbf X'\tilde{\mathbf W}_\delta\mathbf X\hat{\boldsymbol\theta} = \mathbf X'\tilde{\mathbf W}_\delta\tilde{\boldsymbol\theta} ,</math>

where <math>\tilde{\boldsymbol\theta}</math> represents the approximate solution of <math>\boldsymbol\theta</math>, and <math>\hat{\boldsymbol\theta}</math> is the improved value of it; <math>\mathbf W_\delta</math> is the diagonal weight matrix with elements

:<math> w_{ii}^{-1} = \left(\frac{\partial\eta_i}{\partial\mu_i}\right)^2\mathrm{var}(y_i),</math>

and
:<math>\mathbf z = \boldsymbol\eta + \mathbf W_\delta^{-1}(\mathbf y - \boldsymbol\mu)</math>
is the working variable.

Computationally, GLAM provides array algorithms to calculate the linear predictor,
:<math> \boldsymbol\eta = \mathbf X \boldsymbol\theta </math>
and the weighted inner product
:<math> \mathbf X'\tilde{\mathbf W}_\delta\mathbf X </math>
without evaluation of the model matrix <math> \mathbf X .</math>

===Example===

In 2 dimensions, let <math>\mathbf X = \mathbf X_2\otimes\mathbf X_1</math>, then the linear predictor is written <math>\mathbf X_1 \boldsymbol\Theta \mathbf X_2' </math> where <math>\boldsymbol\Theta </math> is the matrix of coefficients; the weighted inner product is obtained from <math>G(\mathbf X_1)' \mathbf W G(\mathbf X_2)</math> and <math> \mathbf W </math> is the matrix of weights; here <math>G(\mathbf M) </math> is the row tensor function of the <math> r \times c</math> matrix <math> \mathbf M </math> given by<ref name="GLAM" />

:<math>G(\mathbf M) = (\mathbf M \otimes \mathbf 1') \circ (\mathbf 1' \otimes \mathbf M)</math>
where <math>\circ</math> means element by element multiplication and <math>\mathbf 1</math> is a vector of 1's of length <math> c</math>.

On the other hand, the row tensor function <math>G(\mathbf M) </math> of the <math> r \times c</math> matrix <math> \mathbf M </math> is the example of [[Khatri–Rao product#Face-splitting product|Face-splitting product]] of matrices, which was proposed by [[Vadym Slyusar]] in 1996:<ref name=slyusar>{{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}}</ref><ref name=slyusar1>{{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}}</ref><ref name="DIPED">{{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}}</ref><ref name=slyusar2>{{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 }}</ref>

:<math> \mathbf{M} \bull \mathbf{M} = \left(\mathbf {M} \otimes \mathbf {1}^\textsf{T}\right) \circ \left(\mathbf {1}^\textsf{T} \otimes \mathbf {M}\right) ,</math>
where <math>\bull</math> means [[Khatri–Rao product#Face-splitting product|Face-splitting product]].

These low storage high speed formulae extend to <math>d</math>-dimensions.

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

==References==
{{reflist}}

[[Category:Regression models]]
[[Category:Generalized linear models|Array model]]

Generalized linear array model - Revision history

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