Discriminant functions

The discriminant functions, dpXx) and d x), for each sample in the training set of Table 5.1 can now be calculated. 

With the assumption of equal covariance matrices, the rule defined by Equation 5.11 becomes  

Sample dA d]i Assigned group Sample dA da Assigned group 

Once again, if the prior probabilities are equal, (G(A Ag(b i the classification rule is simplified  

Equations 5.25 are linear with respect to x and this classification technique is referred to as linear discriminant analysis, with the discriminant function obtained by least-squares analysis, analogous to multiple regression analysis. 

Sample Assigned group Sample d Assigned group 

Turning to our spectroscopic data of Table 1, we can evaluate the performance of this linear discriminant analyser. 

---

Fig. 12.37 Discriminant analysis defines a discriminant function (dotted line) and a discriminant surface (solid) line.

Discriminant emalysis is a supervised learning technique which uses classified dependent data. Here, the dependent data (y values) are not on a continuous scale but are divided into distinct classes. There are often just two classes (e.g. active/inactive soluble/not soluble yes/no), but more than two is also possible (e.g. high/medium/low 1/2/3/4). The simplest situation involves two variables and two classes, and the aim is to find a straight line that best separates the data into its classes (Figure 12.37). With more than two variables, the line becomes a hyperplane in the multidimensional variable space. Discriminant analysis is characterised by a discriminant function, which in the particular case of hnear discriminant analysis (the most popular variant) is written as a linear combination of the independent variables ... 

The surface that actually separates the classes is orthogonal to this discriminant function, as shown in Figure 12.37, and is chosen to maximise the number of compounds correctly classified. To use the results of a discriminant analysis, one simply calculates the appropriate value of the discriminant function, from which the class can be determined. 

Fig. 11. Liaear discriminant function defines the boundary between two categories.

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-... 

This mapping from to the classes C, is determined by the discriminant functions that define the boundaries of regions I = 1,2,...,K in 5. Let d, p) be the discriminant function associated with the 7th class of operating situations, where /= 1,2,..., K. Then, a pattern of measurements p, implies the operating situation C, iff... 

The inductive learning process determines the discriminant functions, using prior examples of (p,C,) associations. 

No geographic structure was revealed by this analysis, with trees from Arch Cape, Oregon, Whitetish, Montana, and sites from northern British Columbia, including Queen Charlotte Islands, being closely associated. The authors remarked on the lack of differentiation between coastal and interior populations. A reinvestigation of red cedar from 55 sites (3-6 trees per site) provided a new data set that was analyzed by numerical and discriminant-function analyses (von Rudloff et al., 1988). These analyses confirmed the low intra- and interpopulational variation seen in the earlier study, but did reveal small differences between coastal and interior populations. No correlations between northern and southern populations emerged from the analyses likewise, elevation had no effect on terpene composition. 

In the method of linear discriminant analysis, one therefore seeks a linear function of the variables, D, which maximizes the ratio between both variances. Geometrically, this means that we look for a line through the cloud of points, such that the projections of the points of the two groups are separated as much as possible. The approach is comparable to principal components, where one seeks a line that explains best the variation in the data (see Chapter 17). The principal component line and the discriminant function often more or less coincide (as is the case in Fig. 33.8a) but this is not necessarily so, as shown in Fig. 33.8b. 

Fig. 33.8. Situation where principal component (PC) and linear discriminant function (DF) are essentially the same (a) and very different (b).

Because a hyperplane corresponds to a boundary between pattern classes, such a discriminant function naturally forms a decision rule. The global nature of this approach is apparent in Fig. 19. An infinitely long decision line is drawn based on the given data. Regardless of how closely or distantly related an arbitrary pattern is to the data used to generate the discriminant, the pattern will be classified as either o>i or <02. When the arbitrary pattern is far removed from the data used to generate the discriminant, the approach is extremely prone to extrapolation errors. 

Once determined, these parameters can be used to create linear discriminant functions of the form... 

Figure 11.3 Positive ion FIESMS spectra of crude cell extracts from Escherichia coli HB101 (A), Bacillus sphaericus DSM 28 (B), and Bacillus licheniformis NTCC 10341 (C). (D) A pseudo-3D plot of the first three discriminant functions (DF1-3) obtained from positive ion whole-cell DIESMS spectra of seven Bacillus subtilis strains (a-g) (E) the corresponding abridged dendrogram obtained from the same information as in D. (Adopted from Vaidyanathan et al.57)...

Distance (in multivariate data) Discriminant function Discriminant variable... 

Fig. 8.5. Discrimination of object quality (+/—) by means of one variable Xi (a), of two variables X and x2 (b) which are represented as biplot in (c) from which the discriminant variable dv (discriminant function df) (d) can be derived...

Fig. 8.11. Representation of classification of 88 German wines by discriminant analysis (1st vs 2nd discriminant function) o Bad Diirkheim (Rhinelande-Palatinate)... 

Fig. 8.12. Representation of the both first discriminant functions df vs df2 obtained by discriminant analysis (MVDA) of 65 German wines from five wine-growing regions according to six different grape varieties by means of 58 features (bouquet components)...

Sawle, G. V., Playford, E. D., Burn, D. J. et al. Separating Parkinson s disease from normality. Discriminant function analysis of fluorodopa F-18 positron emission tomography data. Arch. Neurol. 51 237-243,1994. 

Total specimens for the seven reference units are given, by site provenience, in Table 3. The units are represented in five dimensional discriminant space, the coordinates for the first two dimensions being shown in figure 4. These two vectors account for 72 percent of the total discriminant power. For convenience, approximate territorial lines have been added to the plot. Apparently representing the compositional pattern of locally made Palenque pottery, units 1 and 2 are enclosed within a single territorial boundary. Less well represented but, as will be seen, of considerable interest for the incesario problem, is unit 7, which relates primarily to the site of Xupa. Observable separation of units 3, 4, and 5 is possible only when utilizing additional discriminant functions beyond the first and second. In... 

Table 4. Standardized Linear Discriminant Function Coefficients... 

---