Fisher’s discriminant analysis

Fisher suggested to transform the multivariate observations x to another coordinate system that enhances the separation of the samples belonging to each class tt [74]. Fisher s discriminant analysis (FDA) is optimal in terms of maximizing the separation among the set of classes. Suppose that there is a set of n = ni + U2 + + rig) m-dimensional (number of process variables) samples xi, , x belonging to classes tt, i = 1, , g. The total scatter of data points (St) consists of two types of scatter, within-class scatter Sw and hetween-class scatter Sb- The objective of the transformation proposed by Fisher is to maximize S while minimizing Sw Fisher s approach does not require that the populations have Normal distributions, but it implicitly assumes that the population covariance matrices are equal, because a pooled estimate of the common covariance matrix (S ) is used (Eq. 3.45). 

If one does not wish to bias the boundaries of the NO region of a system, kernel density estimation (KDE) can be used to find the contours underneath the joint probability density of the PC pair, starting from the one that captures most of the information. Below, a brief review of KDE is presented first that will be used as part of the robust monitoring technique discussed in Section 7.7. Then, the use of kernel-based methods for formulating nonlinear Fisher s discriminant analysis (FDA) is discussed. 

LH Chiang, ME Kotanchek, and AK Kordon. Fault diagnosis based on Fisher s discriminant analysis and support vector machines. Corn-put Chem. Engg., 28(8) 1389-1401, 2004. 

Fisher suggested to transform the multivariate observations x to another coordinate system that enhances the separation of the samples belonging to each class tt [74]. Fisher s discriminant analysis (FDA) is optimal in terms of maximizing the separation among the set of classes. Suppose that there is a set of n(= ni + U2 H-+ rig) m-dimensional (number of process... 

Woods JH, Winger GD, France CP (1987) Reinforcing and discriminative stimulus effects of cocaine analysis of pharmacological mechanisms. In Fisher S, Raskin A, Uhlenhuth EH (Eds), Cocaine Clinical and Biobehavioral Aspects, pp. 21-65. Oxford UP, Oxford. 

Figure 15-5 Fisher s linear discriminant analysis determines the line, plane, or hyperplane that best separates two populations, based on their mean values and the variance, in the graph, a line bisecting the two curves In the lower left corner provides this best discriminator.

Discriminant plots were obtained for the adaptive wavelet coefficients which produced the results in Table 2. Although the classifier used in the AWA was BLDA, it was decided to supply the coefficients available upon termination of the AWA to Fisher s linear discriminant analysis, so we could visualize the spatial separation between the classes. The discriminant plots are produced using the testing data only. There is a good deal of separation for the seagrass data (Fig. 5), while for the paraxylene data (Fig. 6) there is some overlap between the objects of class I and 3. Quite clearly, the butanol data (Fig. 7) post a challenge in discriminating between the two classes. 

Fig. 5 Discriminant plots for the seagrass data produced by supplying the coefficients resulting from the AW A to Fisher s linear discriminant analysis.

V(j)g, i.e., the ratio of the E Z and Z —> E quantum yields at the two irradiation wavelengths should be equal, which is only partly true for azobenzene (see Section 1.3.2.1). Fisher s article is not easy to read one must be careful to discriminate strictly between irradiation and analysis wavelengths. 

There are a number of classification methods for analyzing data, including artificial neural (ANNs see Beale and Jackson, 1990) networks, -nearest-neighbor (fe-NN) methods, decision trees, support vector machines (SVMs), and Fisher s linear discriminant analysis (LDA). Among these methods, a decision tree is a flow-chart-like tree stmcture. An intermediate node denotes a test on a predictive attribute, and a branch represents an outcome of the test. A terminal node denotes class distribution. 

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