Generalized linear array model

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 \mathbf Y is arranged in a d-dimensional array with size n_1\times n_2\times\dots\times n_d; thus, the corresponding data vector \mathbf y = \operatorname{vec}(\mathbf Y) has size n_1n_2n_3\cdots n_d. Suppose also that the design matrix is of the form :\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 \mathbf y and design matrix \mathbf X proceeds by repeated evaluation of the scoring algorithm

: \mathbf X'\tilde{\mathbf W}\delta\mathbf X\hat{\boldsymbol\theta} = \mathbf X'\tilde{\mathbf W}\delta\tilde{\boldsymbol\theta} ,

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

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

and :\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, : \boldsymbol\eta = \mathbf X \boldsymbol\theta and the weighted inner product : \mathbf X'\tilde{\mathbf W}_\delta\mathbf X without evaluation of the model matrix \mathbf X .

Example

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

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

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

: \mathbf{M} \bull \mathbf{M} = \left(\mathbf {M} \otimes \mathbf {1}^\textsf{T}\right) \circ \left(\mathbf {1}^\textsf{T} \otimes \mathbf {M}\right) , where \bull means Face-splitting product.

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

Applications

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.

References

References

1. (2006). "Generalized linear array models with applications to multidimensional smoothing". [[Journal of the Royal Statistical Society]].
2. Slyusar, V. I.. (December 27, 1996). "End products in matrices in radar applications.". Radioelectronics and Communications Systems.
3. Slyusar, V. I.. (1997-05-20). "Analytical model of the digital antenna array on a basis of face-splitting matrix products.". Proc. ICATT-97, Kyiv.
4. Slyusar, V. I.. (1997-09-15). "New operations of matrices product for applications of radars". Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv..
5. Slyusar, V. I.. (March 13, 1998). "A Family of Face Products of Matrices and its Properties". Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz. 1999..