Principal components regression block

Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). 

In the above examples there is a natural way to order the complete data set in two blocks, where both blocks have one mode in common. In Chapter 3 the methods of multiple linear regression, principal component regression, and partial least squares regression will be discussed on an introductory level. 

In the previous chapter, it was commented on that the ordinary least-sqnares approach applied to multivariate data (multivariate linear regression, MLR) suffered from serious uncertainty problems when the independent variables were collinear. Principal components regression (PCR) can solve the collin-earity problem and provide additional benefits of factor-based regression methods, such as noise filtering. Recall that PCR compresses the original X-block e.g. matrix of absorbances) into a new block of scores T, containing fewer variables (the so-called factors, latent variables, or principal components), and then regression is performed between T and the property of... 

Because of the enormous number of x variables that are generated in the field calculations, regression analysis cannot be applied. In the very beginning of 3D QSAR studies, principal components were derived from the X block (i.e., the table of field values) and then correlated with the biological activity values. In 1986, Svante Wold proposed to use PLS analysis. PLS analysis resembles principal component regression analysis in its derivation of vectors from the Y and the X blocks. However, there is a fundamental difference in PLS analysis, the orientation of the so-called u and t vectors does not exactly correspond to the orientation of the principal components. They are slightly skewed within their confidence hyperboxes, in order to achieve a maximum intercorrelation (Figure 8). 

In this paper the PLS method was introduced as a new tool in calculating statistical receptor models. It was compared with the two most popular methods currently applied to aerosol data Chemical Mass Balance Model and Target Transformation Factor Analysis. The characteristics of the PLS solution were discussed and its advantages over the other methods were pointed out. PLS is especially useful, when both the predictor and response variables are measured with noise and there is high correlation in both blocks. It has been proved in several other chemical applications, that its performance is equal to or better than multiple, stepwise, principal component and ridge regression. Our goal was to create a basis for its environmental chemical application. 

Figure 33 Representation of a PLS regression through the inner relation u = b.t. The solid lines in X- and Y-space are the principal components and the dashed lines are the PLS vectors. These are slightly skewed to account for the correlation between the two data blocks (redrawn from Figure 9 of ref [487] with permission from Pergamon Press Ltd., Headington Hill Hall, Oxford 0X3 OBW, UK).

The principal additions to this Edition are a substantial enlargement of Chapter I and two new chapters, Chapter XIII on balanced incomplete blocks amd Chapter XIV on confounding. Further additions are the components of variance for unequal column size in Chapter VII (d), the exact formula for the residual variance about a regression line in Chapter IX (h), the Doolittle method of computation in multiple regression in Chapter X (d), and the partitioning of sums of squares in Chapter XII (c). 

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