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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

The generalized linear array model or GLAM was introduced in 2006. 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 Y {\displaystyle \mathbf {Y} } is arranged in a d {\displaystyle d} -dimensional array with size n 1 × n 2 × ⋯ × n d {\displaystyle n_{1}\times n_{2}\times \dots \times n_{d}} ; thus, the corresponding data vector y = vec ⁡ ( Y ) {\displaystyle \mathbf {y} =\operatorname {vec} (\mathbf {Y} )} has size n 1 n 2 n 3 ⋯ n d {\displaystyle n_{1}n_{2}n_{3}\cdots n_{d}} . Suppose also that the design matrix is of the form

X = X d ⊗ X d − 1 ⊗ ⋯ ⊗ X 1 . {\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 y {\displaystyle \mathbf {y} } and design matrix X {\displaystyle \mathbf {X} } proceeds by repeated evaluation of the scoring algorithm

X ′ W ~ δ X θ ^ = X ′ W ~ δ θ ~ , {\displaystyle \mathbf {X} '{\tilde {\mathbf {W} }}_{\delta }\mathbf {X} {\hat {\boldsymbol {\theta }}}=\mathbf {X} '{\tilde {\mathbf {W} }}_{\delta }{\tilde {\boldsymbol {\theta }}},}

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

w i i − 1 = ( ∂ η i ∂ μ i ) 2 v a r ( y i ) , {\displaystyle w_{ii}^{-1}=\left({\frac {\partial \eta _{i}}{\partial \mu _{i}}}\right)^{2}\mathrm {var} (y_{i}),}

and

z = η + W δ − 1 ( y − μ ) {\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,

η = X θ {\displaystyle {\boldsymbol {\eta }}=\mathbf {X} {\boldsymbol {\theta }}}

and the weighted inner product

X ′ W ~ δ X {\displaystyle \mathbf {X} '{\tilde {\mathbf {W} }}_{\delta }\mathbf {X} }

without evaluation of the model matrix X . {\displaystyle \mathbf {X} .}

Example

In 2 dimensions, let X = X 2 ⊗ X 1 {\displaystyle \mathbf {X} =\mathbf {X} _{2}\otimes \mathbf {X} _{1}} , then the linear predictor is written X 1 Θ X 2 ′ {\displaystyle \mathbf {X} _{1}{\boldsymbol {\Theta }}\mathbf {X} _{2}'} where Θ {\displaystyle {\boldsymbol {\Theta }}} is the matrix of coefficients; the weighted inner product is obtained from G ( X 1 ) ′ W G ( X 2 ) {\displaystyle G(\mathbf {X} _{1})'\mathbf {W} G(\mathbf {X} _{2})} and W {\displaystyle \mathbf {W} } is the matrix of weights; here G ( M ) {\displaystyle G(\mathbf {M} )} is the row tensor function of the r × c {\displaystyle r\times c} matrix M {\displaystyle \mathbf {M} } given by

G ( M ) = ( M ⊗ 1 ′ ) ∘ ( 1 ′ ⊗ M ) {\displaystyle G(\mathbf {M} )=(\mathbf {M} \otimes \mathbf {1} ')\circ (\mathbf {1} '\otimes \mathbf {M} )}

where ∘ {\displaystyle \circ } means element by element multiplication and 1 {\displaystyle \mathbf {1} } is a vector of 1's of length c {\displaystyle c} .

On the other hand, the row tensor function G ( M ) {\displaystyle G(\mathbf {M} )} of the r × c {\displaystyle r\times c} matrix M {\displaystyle \mathbf {M} } is the example of Face-splitting product of matrices, which was proposed by Vadym Slyusar in 1996:

M ∙ M = ( M ⊗ 1 T ) ∘ ( 1 T ⊗ M ) , {\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 d {\displaystyle d} -dimensions.

Applications

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

References

  1. ^ 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.
  2. Slyusar, V. I. (December 27, 1996). "End products in matrices in radar applications" (PDF). Radioelectronics and Communications Systems. 41 (3): 50–53.
  3. 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.
  4. 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.
  5. 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.
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