Entropy, purity and V-measure

Since complete cluster (all objects from a single class are assigned to a single cluster) and homogeneous cluster (each cluster contains only objects from a single class) are hardly achieved, we aim to reach a satisfactory balance between these two approaches. Therefore, we apply five well-known clustering criteria in order to evaluate the performance of partition, which are purity, entropy H, V-measure, RAND index and F-measure. This page expose the three first one. The others are exposed in another page.

Entropy measure is used to show how the clusters of sentences are partitioned within each cluster, and it is known as the average of weighted values in each cluster entropy over all clusters C={c_1, …, c_n}:

The purity of a cluster is the fraction of the cluster size that the largest class of sentences assigned to that cluster represents, that is:

Overall purity is the weighted sum of the individual cluster purities is given by:

While purity and entropy are useful for comparing clusterings with the same number of clusters, they are not reliable when comparing clusterings with different numbers of clusters. This is because entropy et purity perform on how the sets of sentences are partitioned within each cluster, and the will lead to homogeneity case. Highest scores however, of purity and lowest scores of entropy are usually obtained whhen the total number of clusters is too big, where this step will lead to being lowest in the completeness. The next measure considers both completeness and homogeneity approaches.

The V-measure is known as the homogeneity and completness harmonic mean; thatis, V=homogeneity*completeness/(homogeneity+completeness), where where homogeneity and completeness are defined as homogeneity=1-H(C|L)/H(C) and completeness=1-H(L|C)/H(L) where: