Principal component regression solving

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). 

The prediction of Y-data of unknown samples is based on a regression method where the X-data are correlated to the Y-data. The multivariate methods, usually used for such a calibration, are principal component regression (PCR) and partial least squares regression (PLS). Both methods are based on the assumption of linearity and can deal with co-linear data. The problem of co-linearity is solved in the same way as the formation of a PCA plot. The X-variables are added together into latent variables, score vectors. These vectors are independent since they are orthogonal to each other and they can therefore be used to create a calibration model. 

The problem dealt with by principal component regression is regressing y (/ x 1) on a possibly ill-conditioned X (/ x J). Hence, principal component regression tries to solve Equation (3.27) and Equation (3.29) for ill-conditioned X. Principal component regression approximates X by a few, say R, components (its principal components) and regresses y on these R components. Principal component regression can be written as... 

Thus, the name for this type of model is principal components regression it combines principal components analysis and inverse least squares regression to solve the calibration equation for the model. All that remains is to come up with a single unified equation that represents the PCR model. Therefore, rearranging the previous matrix model equation to represent the scores as a function of the spectral absorbances and the eigenvectors produces... 

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... 

In regression work, if some of the features used are linearly dependent of each other, then collinearity problem happens. In order to solve the collinearity problem, principal component regression (PCR) and partial least squares regression (PLSR) methods are proposed. 

A crucial decision in PLS is the choice of the number of principal components used for the regression. A good approach to solve this problem is the application of cross-validation (see Section 4.4). 

Note that the lipophilicity parameter log P is defined as a decimal logarithm. The parabolic equation is only non-linear in the variable log P, but is linear in the coefficients. Hence, it can be solved by multiple linear regression (see Section 10.8). The bilinear equation, however, is non-linear in both the variable P and the coefficients, and can only be solved by means of non-linear regression techniques (see Chapter 11). It is approximately linear with a positive slope (/ ,) for small values of log P, while it is also approximately linear with a negative slope b + b for large values of log P. The term bilinear is used in this context to indicate that the QSAR model can be resolved into two linear relations for small and for large values of P, respectively. This definition differs from the one which has been introduced in the context of principal components analysis in Chapter 17. 

In the past few years, PLS, a multiblock, multivariate regression model solved by partial least squares found its application in various fields of chemistry (1-7). This method can be viewed as an extension and generalization of other commonly used multivariate statistical techniques, like regression solved by least squares and principal component analysis. PLS has several advantages over the ordinary least squares solution therefore, it becomes more and more popular in solving regression models in chemical problems. 

Chemometrics is an essential part of NIR and Vis/NIR spectroscopy in food sector. NIR and Vis/NIR instrumentation in fact must always be complemented with chemometiic analysis to enable to extract useful information present in the sp>ectra separating it both from not useful information to solve the problem and from sp>ectral noise. Chemometric techniques most used are the princip)al component analysis (PCA) as a technique of qualitative analysis of the data and PLS regression analysis as a technique to obtain quantitative prediction of the parameters of interest (Naes et al., 2002 Wold et al., 2001 Nicolai et al., 2007 Cen He, 200 . ... 

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