Measuring the component overlapping in the Gaussian mixture model

The ability of a clustering algorithm to deal with overlapping clusters is a major indicator of its efficiency. However, the phenomenon of cluster overlapping is still not mathematically well characterized, especially in multivariate cases. In this paper, we are interested in the overlap phenomenon between Gaussian clusters, since the Gaussian mixture is a fundamental data distribution model suitable for many clustering algorithms. We introduce the novel concept of the ridge curve and establish a theory on the degree of overlap between two components. Based on this theory, we develop an algorithm for calculating the overlap rate. As an example, we use this algorithm to calculate the overlap rates between the classes in the IRIS data set and clear up some of the confusion as to the true number of classes in the data set. We investigate factors that affect the value of the overlap rate, and show how the theory can be used to generate “truthed data” as well as to measure the overlap rate between a given pair of clusters or components in a mixture. Finally, we show an example of application of the theory to evaluate the well known clustering algorithms.