Multiway Regression

Gurden SP, Westerhuis JA, Bro R, Smilde AK, A comparison of multiway regression and scaling methods, Chemometrics and Intelligent Laboratory Systems, 2001, 59, 121-136. 

Methods for simultaneous Af-way regression can be based on the decomposition of the X array by multiway methods introduced in Section 5.2 (parallel factor analysis (PARAFAC) or Tucker models) and regressing the dependent variable on the components of those models. A drawback with this approach is that the separately estimated components are not necessarily predictive for Y. This caused the development of improved algorithms for multiway regression analysis of that kind. 

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The background for the extension of two-way partial least squares regression to multiway data (fV-PLS) was provided by Bro [1996] and further elaborated on by Smilde [1997] and de Jong [1998], An improved model of X was later introduced which, however, maintains the same predictions as the original model [Bro et ol. 2001], Only the three-way version of A-PLS is considered here. It differs from the N-way (N > 3) partial least squares regression algorithm in that it has a closed-form solution for the situation with only one dependent variable. 

Boque R, Smilde AK, Monitoring and diagnosing batch processes with multiway covariates regression models, AIChE Journal, 1999,45, 1504-1520. 

Smilde AK, Kiers HAL, Multiway covariates regression models, Journal of Chemometrics, 1999,13, 31—48. 

Smilde AK, Westerhuis JA, Boque R. Multiway multiblock component and covariates regression models. J Chemometr 2000 14 301-31. 

Preprocessing of multivariate and multiway data sets prior to regression and discriminant analysis follow the general principles outlined earher with few exceptions. In general, the response matrix (i.e., the data that are to be predicted) should be mean centered because this serves an additional purpose in regression and classification models. By centering both the dependent and independent variables, any possible differences in offsets are ranoved. Row normalization can be implemented if the priority is to establish a relationship between variables, rather than estimate the magnitude of the response, or to stabilize the impact of differently concentrated samples on models, as previously described. For example, if the calibration model is intended to predict a concentration from data that follow the Beer-Lambert law (e.g., fluorescence), then it is crucial not to normalize as this would cause the loss of concentration information. If, on the other hand, the model is intended to classify samples, then normahzation may help the model focus on patterns rather than on concentration-induced variations. 

Principal components regression (PCR), partial least squares regression (PLS), and their multiway versions (N-PCR and N-PLS)... 

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