Multilinear PLS

De Jong S, Regression coefficients in multilinear PLS, Journal of Chemometrics, 1998,12, 77-81. 

As stated in Section 6.1, multilinear PLS, as its two-way analogue, aims at finding components which, sequentially, maximize the product of the explained variance of the dependent block and the variance of the component itself. Therefore, if one looks at a factor representation which at the same time accounts for as much as possible of the information in the two blocks, it can happen that this condition is not completely fulfilled by the PLS criterion, which involves the maximization of the variance of Xw, instead of the maximization of the amount of X variance explained by the component model. Accordingly, with the aim of building a factor model whose components had the property of explaining both the variance of the independent and... 

Another problem is to determine the optimal number of descriptors for the objects (patterns), such as for the structure of the molecule. A widespread observation is that one has to keep the number of descriptors as low as 20 % of the number of the objects in the dataset. However, this is correct only in case of ordinary Multilinear Regression Analysis. Some more advanced methods, such as Projection of Latent Structures (or. Partial Least Squares, PLS), use so-called latent variables to achieve both modeling and predictions. 

Perform experimental tests on this subset of compounds and then use some form of modelling to relate the desired activity to structural data. Note that this modelling does not have to be multilinear modelling as discussed in this section, but could also be PLS (partial least squares) as introduced in Chapter 5. 

It is clear that for an unsymmetrical data matrix that contains more variables (the field descriptors at each point of the grid for each probe used for calculation) than observables (the biological activity values), classical correlation analysis as multilinear regression analysis would fail. All 3D QSAR methods benefit from the development of PLS analysis, a statistical technique that aims to find the multidimensional direction in the X space that explains the maximum multidimensional variance direction in the F space. PLS is related to principal component analysis (PCA)." ° However, instead of finding the hyperplanes of maximum variance, it finds a linear model describing some predicted variables in terms of other observable variables and therefore can be used directly for prediction. Complexity reduction and data... 

This is also discussed elsewhere [Smilde 1997], V-PLS can be generalized for higher-order arrays X and Y and in all cases a PARAFAC-like multilinear model is assumed for all the multi-way arrays involved [Bro 1996a],... 

A frequently used sequential N-way method that predicts y and decomposes X in a PARAFAC-like mode is multilinear orAf-way partial least squares (Af-PLS). Denoting the component vector t and the weighing vectors for mode K, w, and for mode /, m, the following criterion has to be maximized (cf. Figure 6.12) ... 

Multilinear regression analysis (MLRA) Principal component analysis (PCA) Partial least squares regression (PLS) Principal component regression (PCR). 

NIR spectroscopic theory does not have to assume a linear relationship between the optical data and constituent concentration, as data transformations or pretreatments are used to linearize the reflectance data. The most used linear transforms include log(l/i ) and K-M as math pretreatments. Calibration equations can be developed that compensate to some extent for the nonlinear relationship between analyte concentrations and log(l/R) or K-M-transformed data. PCR, PLS, and multilinear regression can be used to compensate for the nonlinearity. 

If gas selectivity cannot be achieved by improving the sensor setup itself, it is possible to use several nonselective sensors and predict the concentration by model based, such as multilinear regression (MLR), principle component analysis (PCA), principle component regression (PCR), partial least squares (PLS), and multivariate adaptive regression splines (MARS), or data-based algorithms, such as cluster analysis (CA) and artificial neural networks (ANN) (for details see Reference 10) (Figure 22.5). For common applications of pattern recognition and multi component analysis of gas mixtures, arrays of sensors are usually chosen... 

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