Partial least squares weight vectors

A number of variable selection techniques were also suggested for the Partial Least Squares (PLS) regression method [Lindgren et al, 1994]. The different strategies for PLS-based variable selection are usually based on a rotation of the standard solution by a manipulation of the PLS weight vector w or of the regression coefficient vector b of the PLS closed form. 

Intermediate Least Squares regression (ILS) is an extension of the Partial Least Squares (PLS) algorithm where the optimal variable subset model is calculated as intermediate to PLS and stepwise regression, by two parameters whose values are estimated by cross-validation [Frank, 1987]. The first parameter is the number of optimal latent variables and the second is the number of elements in the weight vector w set to zero. This last parameter (ALIM) controls the number of selected variables by acting on the weight vector of each mth latent variable as the following ... 

Numerous software data treatments authorize the elucidation of mixture composition from spectra. One of the best-known methods is the Kalman s least squares filter algorithm, which operates through successive approximations based upon calculations using weighted coefficients (additivity law of absorbances) of the individual spectra of each components contained in the spectral library. Other software for determining the concentration of two or more components within a mixture uses vector quantification mathematics. These are automated methods better known by their initials PLS (partial least square), PCR (principal component regression), or MLS (multiple least squares) (Figure 9.26). 

Unlike regression, PLS is not based on the assumption of independent and precise X variables but rather on the more realistic assumption that X contains more or less collinear and noisy parameters. Partial least squares summarizes these X variables by means of a few orthogonal score vectors (taeT), and the matrix Y is also resumed in a few score vectors ( aeU) which are not orthogonal. Plots of columns from T and U provide a visual representation of the configuration of the observations in the X or Y space, respectively. The PLS procedure allows one to derive a number of factors and weights, which are used to describe the desired properties. Quantitative structure-property relationships models are built up from these factors and weights. 

The PLS approach was developed around 1975 by Herman Wold and co-workers for the modeling of complicated data sets in terms of chains of matrices (blocks), so-called path models . Herman Wold developed a simple but efficient way to estimate the parameters in these models called NIPALS (nonlinear iterative partial least squares). This led, in turn, to the acronym PLS for these models, where PLS stood for partial least squares . This term describes the central part of the estimation, namely that each model parameter is iteratively estimated as the slope of a simple bivariate regression (least squares) between a matrix column or row as the y variable, and another parameter vector as the x variable. So, for instance, in each iteration the PLS weights w are re-estimated as u X/(u u). Here denotes u transpose, i.e., the transpose of the current u vector. The partial in PLS indicates that this is a partial regression, since the second parameter vector (u in the... 

Two factor analysis methods that are used are the principal components regression (PCR) and the partial least squares (PLS). In the PRC method, the concentrations are expressed as functions of the principal components (PC) instead of absorbances as in ILS. The PC are orthogonal vectors that are linear combinations of the original spectral data of the standards. Here, PCI accounts for the maximum variability in the data, and PC2 accounts for the maximum variability not accounted for by PCI, etc. The other method PLS, is similar to the PCR method except that the PCs are weighted. The weighting is based on the correlation of the PCs with concentration. These are full-spectrum methods like CLS, but like ILS, one can analyze one component at a time. These methods are most often used for quantitative analysis in the near infrared region because of the broadness and overlapping nature of the bands here. 

---