WILKS lambda

Another approach requires the use of Wilks lambda. This is a measure of the quality of the separation, computed as the determinant of the pooled within-class covariance matrix divided by the determinant of the covariance matrix for the whole set of samples. The smaller this is, the better and one selects variables in a stepwise way by including those that achieve the highest decrease of the criterion. 

Despite similarity between the secretions of the two genital glands, differences between the chemical profiles of all three types of secretions emerged in the LDA (Fig. 8.4a). Likewise, reliable seasonal variation appeared in the chemical profiles derived from all three sources (labial Fig. 8.4b scrotal Wilks lambda = 0.018, P <0.01 brachial Wilks lambda = 0.136, P <0.05). We also found the anticipated individual-specific signatures in scent secretions derived from the genital glands (labial Wilks lambda = 0.000, P <0.01 scrotal Fig. 8.4c), but not from... 

Fig. 8.4 Discriminant analyses of the principal chemical components in L. catta scent secretions by (a) gland, (b) season, and (c) individual, (a) Accurate classification of 97.5% of labial, scrotal, and brachial samples in = 77) by gland of origin (Wilks lambda = 0.003 P < 0.001). (b) Reliable differentiation of 100% of labial samples (n = 26) into prebreeding, breeding, and nonbreeding seasons (Wilks lambda = 0.018, P < 0.01). (c) Individual scent signatures in the scrotal secretions from seven males. LDA performed on 17 principal components correctly classified 100% of these samples to the individuals from which they were collected (Wilks lambda = 0.000, P < 0.002)...

Instead of the imivariate Fisher ratio, SLDA considers the ratio between the generalized within-category dispersion (the determinant of the pooled within-category covariance matrix) and total dispersion (the determinant of the generalized covariance matrix). This ratio is called Wilks lambda, and the smaller it is, the better the separation between categories. The selected variable is that that produces the maximum decrease of Wilks lambda, tested by a suitable F statistic for the input of a new variable or for the deletion of a previously selected one. 

It is, however, very common to use the so-called partial lambda which enables evaluation of the process of selection of original variables for calculation of an optimum set of features. This partial lambda is the ratio of two WILKS lambda values, Asetl and Aset 2, where Asetl holds for the smaller set of features and Aset2 is computed after adding a feature to the former set ... 

In Tab. 5-13 we report the results of both mentioned strategies for the selection process. In both procedures the WILKS lambda varies monotonously and each set has a significant meaning. We may, therefore, stop the selection process following the misclassification rate. In the forward strategy the first zero error rate appears with the feature set Ti, Mg, Ca in step 3 (Fig. 5-25) whereas in the backward strategy the zero error rate is obtained with the remaining elements Si, Ca, Al, Mg in step 3. Now it is up to the expert to decide which feature set to retain in the future. 

If the filter coefficients are to be used for discriminatory purposes, then the criterion function should strive to reflect differences among classes. In this section three suitable discriminant criterion functions are described. These discriminant criterion functions are Wilk s lambda (3a), entropy (3e), and the cross-validated quadratic probability measure (3cvqpm)-... 

For the Wilk s Lambda criterion, optimization was based on band(3,3), while the entropy criterion optimized over band(3,2). The CVQPM criterion optimized over the scaling band(3,0). Some features which we might expect from the adaptive wavelet algorithm, is that at termination, the band on which optimization was based would outperform the other bands, at least in... 

Table 3. The percentage of correctly classified spectra, using the coefficients X (t) for T = 0,..., 3 at initialization and at termination of the adaptive wavelet algorithm. The discriminant criterion functions were Wilk s Lambda, symmetric entropy and the CVQPM.

One reason why the CVQPM, maybe outperforming the other criterion functions could be due to the fact that optimization and hence classification is based on scaling coefficients. So that a fair comparison could be made, the optimization routine using the Wilk s Lambda, and symmetric criterion functions was repeated, this time forcing optimization over the scaling band. These results are summarized in Table 4. 

Optimization over the scaling band did improve the results slightly for the Wilk s Lambda and symmetric entropy criterion, but these criterion functions were not able to improve upon the results previously obtained with the CVQPM criterion function. 

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