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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]].
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| == Overview ==
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| The generalized linear array model or GLAM was introduced in 2006.<ref>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.</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.
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| 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\ldots\times n_d</math>; thus,the corresponding data vector <math>\mathbf y = \textbf{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
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| :<math>\mathbf X = \mathbf X_d\otimes\mathbf X_{d-1}\otimes\ldots\otimes\mathbf X_1.</math>
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| 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
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| :<math> \mathbf X'\tilde{\mathbf W}_\delta\mathbf X\hat{\boldsymbol\theta} = \mathbf X'\tilde{\mathbf W}_\delta\tilde{\mathbf z} ,</math>
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| 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
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| :<math> w_{ii}^{-1} = \left(\frac{\partial\eta_i}{\partial\mu_i}\right)^2\text{var}(y_i),</math>
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| and
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| :<math>\mathbf z = \boldsymbol\eta + \mathbf W_\delta^{-1}(\mathbf y - \boldsymbol\mu)</math>
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| is the working variable.
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| Computationally, GLAM provides array algorithms to calculate the linear predictor,
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| :<math> \boldsymbol\eta = \mathbf X \boldsymbol\theta </math> | |
| and the weighted inner product
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| :<math> \mathbf X'\tilde{\mathbf W}_\delta\mathbf X </math>
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| without evaluation of the model matrix <math> \mathbf X .</math>
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| ===Example===
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| 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
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| :<math>G(\mathbf M) = (\mathbf M \otimes \mathbf 1') * (\mathbf 1' \otimes \mathbf M)</math> | |
| where <math>*</math> means element by element multiplication and <math>\mathbf 1</math> is a vector of 1's of length <math> c</math>.
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| These low storage high speed formulae extend to <math>d</math>-dimensions.
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| ==Applications==
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| 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.
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| ==References== | |
| {{reflist}}
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| [[Category:Multivariate statistics]]
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| [[Category:Generalized linear models]]
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