Discriminant score

Maximizing the posterior probabilities in case of multivariate normal densities will result in quadratic or linear discriminant rules. However, the mles are linear if we use the additional assumption that the covariance matrices of all groups are equal, i.e., X = = Xk=X- In this case, the classification rule is based on linear discriminant scores dj for groups j... 

FIGURE 5.4 Linear discriminant scores dj for group j by the Bayesian classification rule based on (Equation 5.2). mj, mean vector of all objects in group j Sp1, inverse of the pooled covariance matrix (Equation 5.3) x, object vector (to be classified) defined by m variables Pj, prior probability of group j. 

The calculation of the discriminant scores in Equation 5.2 is schematically shown in Figure 5.4 for the estimated quantities. 

Thus, a new object x, is classified by calculating the discriminant score v, (projection on the direction defined by the decision vector)... 

The discriminant score v, is compared with a classification threshold... 

The higher it is, the higher is the so-called discriminant score (also known as the discriminating function)... 

The lines of equal probability (and of equal discriminant score) appear as hyperellipsoids in the space of the variables, equal for each class (see Fig. 21) the locus, where... 

Wenker et al. used dispersion polygons to show the differences in the dispersion of some classes of French brandies. The number of classification errors on the basis of these dispersion polygons appears to be very small in comparison with that based on the discriminating scores (i.e., when the canonical discriminating fimctions are used, and the classification made on the basis of the distance from category centroids). 

The terms classification ability and classification error used above refer to a procedure in which we have N objects of G categories, and all objects are used to compute class means and a pooled intraclass covariance matrix. The objects are then classified according to their highest discriminant score. This procedure is that commonly used. However, the classification ability (percentage of correctly classifiai objects) is an overestimate of the real utility of information, which must be considered as the ability to classify correctly unknown samples predictive ability. 

Two methods are used to evaluate the predictive ability for LDA and for all other classification techniques. One method consists of dividing the objects of the whole data set into two subsets, the training and the prediction or evaluation set. The objects of the training set are used to obtain the covariance matrix and the discriminant scores. Then, the objects of the training set are classified, so obtaining the apparent error rate and the classification ability, and the objects of the evaluation set are classified to obtain the actual error rate and the predictive ability. The subdivision into the training and prediction sets can be randomly repeated many times, and with different percentages of the objects in the two sets, to obtain a better estimate of the predictive ability. 

Stepwise discriminant analysis was used to determine how tree chemical, phenologlcal, and physical parameters differed between sites (Table VII). Only seven of the 18 variables used were needed to completely differentiate the trees at the 2 sites (F/y ] 93) = 210.36 p < 0.001). The magnitudes of the standardized discriminant function coefficients for the Included variables Indicated that the differences between sites were largely due to terpene chemistry (Table VIII). The discriminant function contrasts primarily the relative concentration of alpha-plnene versus the concentration of several terpenes, particularly bornyl acetate and beta-plnene. Examination of the discriminant scores showed that the stressed trees loaded negatively on the function (x discriminant score = -2.23), while the non-stressed trees loaded positively (x discriminant score = 3.38). In other wards, trees from the stressed site were higher In alpha-plnene vdille the non-stressed trees contained more bornyl acetate, beta-plnene, and other terpenes In their young needles. 

Fig. 13.3. Scatter plot of linear discriminant scores for each spectrum in the biopsy targeting model when tested against all the others, colour coded for consensus pathology...

With the ndf discriminant functions the ndf discriminant scores of each object may be calculated. As an example the first discriminant score of an object i is given by ... 

In the case of two groups one describes the groups by the discriminant scores of one discriminant function. The coordinates of the two class means with their variances are known and the new object may easily be discriminated, in other words classified to one of the groups. This is achieved simply by attributing the unknown object to that class into which confidence region it falls. This may be tested using ... 

The maximum-likelihood rule classifies an observation x IRp into na if ln(pja(x)) is the maximum of the set In(pfjix)) j = 1,. .., Z. If we assume that the density f for each group is Gaussian with mean pj and covariance matrix Xy, then it can be seen that the maximum-likelihood rule is equivalent to maximizing the discriminant scores df(x) with... 

Sillanaukee, R The diagnostic value of a discriminant score in the detection of alcohol abuse. Arch. Path. Lab. Med. 1992 116 924-929... 

Calculation of the left-hand side of this equation results in a value for each object which is a function of x, the pattern vector, and which is referred to as the discriminant score. 

Tables Discriminant scores using the quadratic discriminant function as classifier (a), and the resulting confusion matrix (b)...

Decision limit 32, 33 Degrees of freedom, 8 Dendrogram, 97, 105 Detection limit, 32, 33 Determination limit, 32, 33 Differentiation, 55 Savitsky-Golay, 57 Discriminant function, 124, 130 Discriminant score, 130 Discrimination, 123 Dispersion matrix, 82 Distance measures, 99 Dixon s Q-test, 13... 

The generalized variance Si, the prior probability pi and the Mahalanobis distance contribute to the quadratic score dj (x). Using the discriminant scores, the minimum total probability of misclassification rule for Normal populations and unequal covariance matrices becomes [126] ... 

Since the second and third terms are independent of i, they are the same for all dj (x) and can be ignored in classification. Since the remaining terms consist of a constant for each i (Inp — f/2/LtJ S/Xj) and a linear combination of the components of x, a linear discriminant score is defined as... 

FDA is used to diagnose faults by modifying the quadratic discrimination score for the ith population defined in Eq. 3.41 in the FDA framework such that... 

The Chemometricks analysis developed discriminant functions that distinguished each sample or sample class from every other based on the principal mass components of the spectra. Applied to individual spectra, the discriminant functions yielded a function score that reflected the abundance of those chemical components in the sample mixture that distinguished the sample or class from the others. Experimentally measured static elasticities for the samples were then compared with the discriminant scores to determine the degree of correlation between the spectral patterns and surface elasticity. 

Table 5.3 Discriminant scores using the quadratic discriminant function as...

Maximizing the posterior probabihty is related to minimizing the discriminant scores obtained by... 

If the discriminant score, C, exceeds a certain threshold value (discriminatory value or cutoff value), the credit quality of a corporation will be classified as good, otherwise as poor. The estimation of the discriminant function requires a set of corporations that are already classified as solvent or insolvent (good or poor). Thus it is possible to assess the reliability of the function via the number of misclassified corporations. 

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