Elaboración de compost

Rand indices computed for standard averaging based ADWPT decompositions are shown in the Tables 4.1 and 4.2. The results are quite encouraging as relatively high values for Rand indices are obtained.

Table 4.1: Rand Indices for various features of Intersection decomposi- tion obtained using Standard averaging and various distance functions (HLD=Hellinger Distance, K-LD=Kullback-Leibler Distance, FD=Fishers Lin- ear Discriminant, J-SD=Jensen-Shannon Distance, BD=Bhattacharya Dis- tance, MD=Mahalanobis Distance, MCMD=Multi-Class Mahalanobis Distance, EneD=Energy Distance, REneD=Relative Energy Distance). Rµ_{=Average Rand} Index µ σ ν Γ ξ ε Gς Gυ Gε Gτ Rµ HLD 0.63 0.62 0.63 0.63 0.62 0.78 0.74 0.81 0.74 0.79 0.70 K-LD 0.58 0.62 0.63 0.63 0.63 0.77 0.74 0.81 0.73 0.79 0.69 FD 0.62 0.63 0.63 0.61 0.63 0.72 0.72 0.75 0.71 0.75 0.68 J-SD 0.58 0.62 0.63 0.63 0.63 0.77 0.77 0.81 0.73 0.79 0.70 BD 0.63 0.62 0.63 0.63 0.62 0.75 0.73 0.82 0.71 0.76 0.69 MD 0.62 0.59 0.63 0.61 0.59 0.68 0.70 0.76 0.72 0.81 0.67 MCMD 0.62 0.55 0.50 0.60 0.55 0.65 0.70 0.78 0.67 0.73 0.64 EneD 0.63 0.63 0.61 0.64 0.63 0.68 0.72 0.75 0.68 0.74 0.67 ReneD 0.58 0.47 0.62 0.64 0.60 0.73 0.72 0.82 0.67 0.77 0.66 Avg 0.61 0.60 0.61 0.63 0.61 0.72 0.73 0.79 0.70 0.77 0.68

4.4 Evaluation of Distance Functions

Table 4.2: Rand Indices for various features of Union decomposition obtained usingStandard averagingand various distance functions (HLD=Hellinger Dis- tance, K-LD=Kullback-Leibler Distance, FD=Fishers Linear Discriminant, J- SD=Jensen-Shannon Distance, BD=Bhattacharya Distance, MD=Mahalanobis Distance, MCMD=Multi-Class Mahalanobis Distance, EneD=Energy Distance, REneD=Relative Energy Distance). Rµ_{=Average Rand Index}

µ σ ν Γ ξ ε Gς Gυ Gε Gτ Rµ HLD 0.62 0.62 0.62 0.64 0.62 0.67 0.73 0.8 0.72 0.76 0.68 K-LD 0.58 0.64 0.61 0.64 0.64 0.67 0.72 0.81 0.72 0.76 0.68 FD 0.63 0.63 0.61 0.64 0.63 0.68 0.72 0.75 0.68 0.74 0.67 JS 0.56 0.64 0.6 0.64 0.64 0.67 0.72 0.81 0.72 0.76 0.68 BD 0.62 0.64 0.6 0.65 0.65 0.73 0.72 0.75 0.68 0.75 0.68 MD 0.63 0.63 0.61 0.64 0.63 0.68 0.72 0.75 0.68 0.74 0.67 MCMD 0.62 0.63 0.61 0.64 0.63 0.68 0.72 0.75 0.68 0.75 0.67 EneD 0.62 0.63 0.61 0.64 0.63 0.68 0.72 0.75 0.68 0.74 0.67 ReneD 0.63 0.62 0.61 0.65 0.62 0.68 0.72 0.75 0.71 0.75 0.67 Avg 0.61 0.63 0.61 0.64 0.63 0.69 0.72 0.77 0.70 0.74 0.67

4.4 Evaluation of Distance Functions

As seen from the tables the intersection of ADWPT decompositionsB_T∗ _pro-

duces slightly better results than union of ADWPT decompositions B_S∗_{. The}

main advantage of the intersection of ADWPT decompositions is that it reduces the length of the feature set greatly. We will also see in the next chapter that

B_T∗ _{is a superior method for selection of subbands. The highest ARI is produced}

by Hellinger and Jensen-Shannon distance which is 0.70 while the highest Rand value of 0.82 is obtained for the GLCM correlation feature for the Bhattacharyya and Relative Energy distance based decompositions. Kullback-Leibler also pro- duces relatively good Rand indices. Comparative analysis was made with LBP features. Two LBP based feature sets are acquired with neighbours 8 and radius 1 and neighbours 16 and radius 2. The Rand indices obtained after k-means clustering were 0.69 and 0.65 respectively.

On the other hand, in the case ofB_S∗_{, the better performing distance functions}

are Hellinger, Kullback-Leibler, Jensen-Shannon and Bhattacharyya (as shown in Table 4.2). The highest ARI produced for B∗_{S is 0.68 which is lower than}

the highest ARI of 0.70 produced for intersection of decompositions. From the results, for union and intersection, it can be seen that some distance functions perform better than others. After careful analysis of the Tables 4.1 and 4.2, it can be seen that the Energy distance and Mahalanobis distance measures (in both modes i.e. union and intersection) consistently perform worse than Hellinger, Kullback-Leibler, Jensen-Shannon and Bhattacharyya distance functions.

It may also be observed that the difference between the various ARI’s overall remains low but the difference between HRI’s is high. It would be important here to discuss the various decompositions and how subband selection affects the overall clustering efficiency. It must be noted here that we would be referring only to the intersection decompositions in our discussion as they produce better Rand indices. Figure 4.3 shows the intersection decomposition obtained for the various distance functions. It is clear from Figure 4.3 (a-e) that the decompositions with a high value of Rµ _{decompose subbands in the approximation, horizontal and}

vertical details i.e. the descendants of subbands W1,0,0, W1,1,0 and W1,0,1. The

more discriminant subbands are found usually amongst the descendants ofW1,1,0

andW1,0,1. The only exception is the Relative Energy distance for which the most

4.4 Evaluation of Distance Functions

a. b. c.

d. e.

f. g. h.

Figure 4.3: Standard averaging based intersection decompositions obtained for the distance functions a. Hellinger, b. Kullback-Leibler, c. Jensen-Shannon, d. Bhattacharyya, e. Relative Energy, f. Energy, g. Mahalanobis and h. Multiple- class Mahalanobis

W1,0,0. This is due to the fact that high energy is found in the descendants

of W1,0,0 and since relative energy compares energies of two subbands, therefore

more discriminant subbands are selected in the descendants of the more energetic subband. On the other hand, we can see that Energy distance does not do as well because it decomposes all subbands rendering the feature set too long and hence, adds features to the feature set which may not be the best for differentiating between Meningioma subtypes.

Mahalanobis distance fails to decompose subbands beyond 2 levels and hence fails to capture the intrinsic textural characteristics. Multi-class Mahalanobis is even worse as it fails to go beyond the first level. Multi-class Mahalanobis yields a feature set which is quite small and as the results indicate does not produce

4.4 Evaluation of Distance Functions

desirable results. Hence, it can safely be deduced that subband selection in the descendants of the approximation subband W1,0,0, the vertical W1,1,0 and the

horizontal detail subbands W1,0,1, is good for clustering as it keeps the feature

set at a sufficient length and also captures the most useful discriminant textural characteristics. A FWPT decomposition as seen in the case of Energy distance is undesirable as it increases the feature set length and does not capture the more discriminant information. The Mahalanobis distance selects fewer subbands, in fact, the decomposition obtained is only up to the 2nd level which is not good for capturing textural characteristics as the Rand index indicates. The same applies to multi-class Mahalanobis which decomposes subbands only up to the 1st level. Next we discuss the results for the pseudo-averaging case.