Loevinger's measures of rule quality for assessing cluster stability

A method is developed for measuring clustering stability under the removal of a few objects from a set of objects to be partitioned. Measures of stability of an individual cluster are defined as Loevinger's measures of rule quality. The stability of an individual cluster can be interpreted as a weighted mean of the inherent stabilities in the isolation and cohesion, respectively, of the examined cluster. The design of the method also enables us to measure the stability of a partition, that can be viewed as a weighted mean of the stability measures of all clusters in the partition. As a consequence, an approach is derived for determining the optimal number of clusters of a partition. Furthermore, using a Monte Carlo test, a significance probability is computed in order to assess how likely any stability measure is, under a null model that specifies the absence of cluster stability. In order to illustrate the potential of the method, stability measures that were obtained by using the batch K-Means algorithm on artificial data sets and on Iris Data are presented.