PLS weights

In PCA, the weights W and the normalized loadings P are identical cf. Chapters 31 and 35). For any other data decomposition, e.g PLS, weights and loadings differ. 

It should be noted that there are other multivariate variable selection methods that one could consider for their application. For example, the interactive variable selection (IVS) method71 is an actual modification of the PLS method itself, where different sets of X-variables are removed from the PLS weights (W, see Equation 8.37) of each latent variable in order to assess the usefulness at each X-variable in the final PLS model. 

More precisely, let x p and n denote the mean-centered data matrices. The normalized PLS weight vectors ra and qa (with rj = q,J = 1) are then defined as the vectors that maximize... 

The cross-covariance matrix I., is then estimated by, and the PLS weight vectors ra are computed as in the SIMPLS algorithm, but now starting with instead of S. In analogy with Equation 6.32, the x-loadings py are defined as p = / (vf X/ ) Then the deflation of the scatter matrix is performed as... 

To determine which properties of the reaction systems contribute to the observed selectivity, plots of the PLS weights of the x variables were used in combination with a statistical criterion, modelling power [43]. We will omit these details, full accounts are given in Refs. [1,80]. 

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. 

To determine which variables in the X block contributed to the description of the systematic variation of the regioselectivity, the plots of the PLS weights were used. These plots are shown in Fig. 17.8. Variables with a high degree of contribution are projected at the periphery of the plots. Variables with no or minor influence are projected close to the origin. 

Fig. 17.8 Plots of the PLS weights. Significant variables are projected at the periphery of the plots, while insignificant variables are projected close to the origin.

JoosTE PL, Weight MJ and Lombard CJ (2001) Iodine concentration in household salt in South Africa. Bulletin of the World Health Organization 79 534- 540. 

Figure 8 Statistical analysis of bread wheat RP-HPLC data (A) chromatograms of 12 hard red spring wheats (B) PCA weights for data from (A) and (C) PLS weights for loaf volume. (From Ref. 87.)...

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... 

The variable importance for the projection (VIP) parameter represents a condensed summary of the importance of the X variables. It is a squared function of the PLS weights, w, weighted with the amount of explained Y variance for each PLS component. X variables having a VIP value larger than... 

Figure 12 PLS weight ( loading ) plot (notation as in Table 1)...

FIGURE 19 Sugar data set. N-PLS weights for the six factors plotted as landscapes. 

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