Controlling Mixture Component Overlap for Clustering Algorithms Evaluation

This paper presents two algorithms for generating artificial mixtures of univariate normal densities with controlled component overlap. Such mixtures are used basically as test benchmarks for the evaluation of clustering algorithms. Although the algorithms are different, they are based on the same definition. Both algorithms use formal methods to ensure the generation of “non totally overlapped” mixtures, in contrast to algorithms in the literature which generally use ad-hoc methods to control the overlap. For this purpose, the overlap is assumed to affect only two adjacent components of the mixture. Three configurations are defined: the “maximum overlap” beyond which two components are considered to be totally merged, the “minimum overlap” beyond which two components are considered “nonoverlapped”, and the “rate of overlap”, denoted by λ (0 < λ ≤ 1), which defines a given overlap between the maximum and the minimum. Both algorithms presented in this paper are, then, designed to generate mixtures that control these overlap configurations. The first algorithm uses the width of components to control the overlap, while the second uses the mean. The first algorithm allows a better control over the boundaries of the generated mixture, but involves complex relations. It has been used to generate unidimensional mixtures to approximate image histograms. The second algorithm involves less complex relations, and allows one to control the rate of overlap. The reason for controlling the rate of overlap is to generate artificial mixtures constituting a given degree of difficulty. Indeed, the greater the rate of overlap, the more complex the mixture. Both algorithms allow the generation of mixtures with any number of components. Received November 2, 2001 1