Multi-way covariates regression models

The algorithm for multi-way covariates regression is explained using the example of regressing y (/ x 1) on X (/ x J x K), where a Tucker3 structure for X is assumed, with P, Q and R components in the I, J and K direction, respectively. Generalizations to multivariate (and multi-way) Y and other three-way structures for X can be inferred from this example. 

5 In practice a is estimated, e.g. through cross-validation, together with other meta-parameters such as the number of components. 

An algorithm for solving Equation (6.48) can be based on alternating least squares. Starting values are chosen and then iterations are performed to minimize the loss function between brackets in Equation (6.48). The full algorithm is described in Algorithm 6.8. Several parts of the algorithm are explained below. 

Assuming that all other parameters are given, W can be solved by rewriting the loss function of Equation (6.48) as 

The vector py only appears in the second term of the loss function. Given the other parameters, py can be estimated as py = (XW)+y. This is a simple least squares step. The matrix Px consists of three Tucker-type matrices B, C and G. Given the other parameters, these matrices B, C and G can be updated by proper Tucker3 steps (see Algorithm 6.4) where the score matrix A = XW is kept constant. 

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Cross-validation of X shows that a (3,2,3) Tucker3 model is a reasonable model for X [Louwerse et al. 1999], Hence, a multi-way covariates regression model relating y to X, and assuming a (3,2,3) Tucker3 structure for X was calculated. The optimal a was found by cross-validation to be 0.9. This indicates that stabilizing the predictions by... 

Multi-way covariates regression can be extended to multi-way arrays X and Y of arbitrary numbers of modes. For each array a specific multi-way structure can be assumed. For example, if both X and Y are three-way arrays and a Tucker3 structure is assumed for both arrays, then the multi-way covariates regression model is... 

The most significant difference between Tuckerl-PLS and /V-PLS on the one hand and multi-way covariates regression models on the other hand is that Tuckerl-PLS and /V-PLS models are calculated sequential and the multi-way covariates regression model is derived in a simultaneous fashion. This has several consequences ... 

The solution of a Tuckerl-PLS, /V-PLS and a multi-way covariates regression model for a given data set will be different. 

The solutions of Tuckerl-PLS models with an increasing number of components are nested this also holds for /V-PLS models but not for multi-way covariates regression models. 

When data from new samples are to be evaluated with an existing multi-way covariates regression model, the principle follows almost immediately from the procedures used in other decomposition models. The scores of size P x 1 of a new sample can be found by from the preprocessed data, Xnew (J x K) and the weights, W (JK x R)... 

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