Discriminant analysis nonlinearity model

A number of PLS variants have been deployed, for instance, for developing nonlinear models and for predicting together several response variables (PLS-2). Furthermore, when category indices are taken as response variables, PLS may work as a classification method which is usually called PLS discriminant analysis (PLS-DA). 

Current methods for supervised pattern recognition are numerous. Typical linear methods are linear discriminant analysis (LDA) based on distance calculation, soft independent modeling of class analogy (SIMCA), which emphasizes similarities within a class, and PLS discriminant analysis (PLS-DA), which performs regression between spectra and class memberships. More advanced methods are based on nonlinear techniques, such as neural networks. Parametric versus nonparametric computations is a further distinction. In parametric techniques such as LDA, statistical parameters of normal sample distribution are used in the decision rules. Such restrictions do not influence nonparametric methods such as SIMCA, which perform more efficiently on NIR data collections. 

Linear or nonlinear multiple regression analysis is used as a statistical tool to derive quantitative models, to check the significance of these models and of each individual term in the regression equation. Other statistical methods, such as discriminant analysis, principal component analysis (PCA), or partial least squares (PLS) analysis (see Partial Least Squares Projections to Latent Structures (PLS) in Chemistry) are alternatives to regression analysis (see Che mo me tries Multivariate View on Chemical Problems)Newer approaches compare the similarity of molecules with respect to different physicochemical or other properties with their biological activities. 

Two examples are provided here to illustrate nonlinear parameter estimation, model discrimination, and analysis of variance. 

In nonlinear regression analysis, we search for those parameter t alues that minimize the sum of the squares of the differences beiw een the measured values and the calculated values for all the data points.- Not only can nonlinear regression find the best estimates of parameter values, it can al,so be used to discriminate between different rate law models, such as the Langmutr-Hin-shelw ood models discussed in Chapter 10. Many software programs are available to find these parameter values so that all one has to do is enter the data, The Polymath software will be used to illustrate this technique. In order to carry out the search efficiently, in some cases one has to enter initial estimates of the parameter -alues close to the actual values. These estimates can be obtained using Ihe linear-least-squares technique discussed on the CD-ROM Professional Reference Shelf. 

The t test on parameters, described in Sec. 7.3.2, is useful in establishing whether a model contains an insignificant parameter. This information can be used to make small adjustments to models and thus discriminate between models that vary from each other by one or two parameters. This test, however, does not give a criterion for testing the adequacy of this model. The residual sum of squares, calculated by Eq. (7.160), contains two components. One is due to the scatter in the experimental data and the other is due to the lack of fit of the model. In order to test the adequacy of the fit of a model, the sum of squares must be partitioned into its components. This procedure is called analysis of variance, which is summarized in Table 7.2. To maintain generality, we examine a set of nonlinear data and assume the availability of multiple values of the dependent variable y j at each point of the independent variable jc, (see Fig. 7.12). 

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