Statistical discrimination

Linear, polynomial, or statistical discriminant functions (Fukunaga, 1990 Kramer, 1991 MacGregor et al., 1991), or adaptive connectionist networks (Rumelhart et al, 1986 Funahashi, 1989 Vaidyanathan and Venkatasub-ramanian, 1990 Bakshi and Stephanopoulos, 1993 third chapter of this volume, Koulouris et al), combine tasks 1 and 2 into one and solve the corresponding problems simultaneously. These methodologies utilize a priori defined general functional relationships between the operating data and process conditions, and as such they are not inductive. Nearest-neigh-... 

In Sect. 4.1 wc will discuss the method of linear statistical discriminant analysis. Here, however, some comments are given in advance. 

Although the characterization of coal macerals on the basis of their fluorescence spectra is a recent innovation, it has already proven to be an excellent fingerprinting tool for the various macerals. In some cases, it is even more sensitive than normal petrographic analysis. The initial results of fluorescence spectral studies show that the various fluorescent macerals in single coals can be statistically discriminated on the basis of their spectral parameters and that even varieties of a single maceral type can be distinguished. Although the spectra obtained at this time are rather broad and not suitable for chemical structure analysis, the potential for structural analysis exists and may be realized with improvements in instrumentation. 

Friedman and Frank [75] have shown that SIMCA is similar in form to quadratic discriminant analysis. The maximum-likelihood estimate of the inverse of the covariance matrix, which conveys information about the size, shape, and orientation of the data cloud for each class, is replaced by a principal component estimate. Because of the success of SIMCA, statisticians have recently investigated methods other than maximum likelihood to estimate the inverse of the covariance matrix, e.g., regularized discriminant analysis [76], For this reason, SIMCA is often viewed as the first successful attempt by scientists to develop robust procedures for carrying out statistical discriminant analysis on data sets where maximum-likelihood estimates fail because there are more features than samples in the data set. 

Various types of plots are available for testing special kinetic hypotheses. Some of these are used in problems at the end of this chapter. The reader is urged to st2irt with simple plots to get a feel for the data and judge what kinds of rate expressions might be suitable. Graphical schemes are useful for preliminary selection of models, but the statistical discrimination methods of Chapters 6 and 7 are recommended for the later stages. 

Here Kj is the adsorption equilibrium constant of species j. which can be a function of potential. In this case estimation of the best fit of kinetic parameters Qj, m, a, k°, Kj, Ef, requires the use of nonlinear regression techniques (84a). Although experimental data can fit an equation similar to Eq. (16), mechanistic deductions from such information alone should be restrained. It can be shown that more than one mechanism can be devised, the rate expressions of which cannot be statistically discriminated (84a). 

Contribution plots presented in Section 7.4 provide an indirect approach to fault diagnosis by first determining process variables that have inflated the detection statistics. These variables are then related to equipment and disturbances. A direct approach would associate the trends in process data to faults explicitly. HMMs discussed in the first three sections of this chapter is one way of implementing this approach. Use of statistical discriminant analysis and classification techniques discussed in this section and in Section 7.6 provides alternative methods for implementing direct fault diagnosis. 

The models proposed in the literature are usually developed for the prediction of the final constant rate of coke formation [16-18], However, the present model predicts the evolution of the coke deposition from the start to the end of the reaction. The simplest possible equation has been developed, based on previous models [22], A comparison and statistical discrimination of the different kinetic models will be presented in a future paper [25],... 

Pitts SJ and Kratochvil B (1991) Statistical discrimination of flat glass fragments by instrumental neutron activation analysis methods for forensic science applications. Journal of Forensic Sciences 36 122-137. 

Two methods for the pattern recognition of evaluation profiles have been used (a), statistical discriminant analysis and (b) pattern clustering and nearest neighbor pattern classi-fication. Fourier and Hadamard transform, coefficients have been assumed for compact representation of profile shapes.. 

Usually a proper statistical discrimination between different reactor models cannot be based on experiments alone, since too few measurements are available. The discrimination is then done by proposing a model and testing both the leliabihty of the model and whether the found parameters are within acceptable physical boundaries. 

---