Regression loading weights

In PLS regression the weight vectors are normally defined to be of length one. Under this constraint, it is easy to show that for the two-way case with one dependent variable, the first loading vector is given by the vector... 

A detailed explanation on the derivation of PCA is available in a review paper by Svante (63). In many ways, PLS is similar to PCA except that it looks for correlations between matrices or a matrix and a vector. Robust calibrations can be created using PLS becau,se the correlation is multivariate in nature, making it less susceptible to random noise. PL/S captures the highest variance in the data set and correlates both data blocks simultaneously. Unlike PCA where independent score. sets for each data block is calculated, a common link or weight loading vector (w) is calculated. A regression coefficient, , is used to predict independent variables. 

This approach is closely related to PCR hut actually uses the concentration data generated for the calibration set in the process of determining the principal components and their score and loading vectors. This results in principal components that are quite different from those in PCR since there is an intrinsic weighting for concentration, so the first few loading vectors contain the majority of the concentration information. This results in two sets of vectors one for the spectral data and one for the concentration data. These will be interrelated through a regression, which may be a simple linear relation (as in PCR), but this is not assumed and the decomposition of the two sets of data is performed simultaneously. 

Excel has three built-in facilities for least-squares calculations, which provide the same (and, if you wish, much more) information. The first, LINEST, is a simple function. The second is the Regression macro in the Analysis Toolpak, which is part of Excel but must be loaded if this was not already done at the time the software was installed. The third (and often simplest) method is to use the Trendline feature, which is only available once the data appear in a graph. Later we will encounter yet another option, by using the weighted least squares macro described in chapter 10. Truly an embarrassment of riches Below we will illustrate how to use the first three of these tools. Table 2.6-1 lists their main attributes, so that you can make an informed choice of which one of them to use. 

A drawback of the procedure above of regressing on the loadings obtained from a separate decomposition of the three-way array is that these components are not necessarily predictive for y. To overcome this, a weighted criterion has been proposed [Smilde 1997, Smilde Kiers 1999] that balances the selection of components between fitting X and predicting y. This can be stated as selecting components A = XW in the column-space of X (I x JK), that simultaneously fit X and predict y ... 

In two recent reports Rosas-Romero and co-workers (Rosas-Romero, Herrera and Muccini, 1996 Rosa-Romero et al, 1994) contributed to new TLC-FID procedures for the quantitative analysis of lipid classes of interest to the fat and oil industries. Concerned with the lack of linearity (particularly noticeable at high loads in the older latroscan systems), they suggested there is a need to establish an arithmetic transformation of the data. In order to improve quantitation of lipid classes, they proposed the data be transformed by establishing a regression of the log of the peak-area ratios against the log of the weight ratio (1996). Improvements in the design of the ion collector and the FID itself introduced for the Mark V will certainly improve the linearity of the untransformed data. 

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