Multiobjective Simulated Annealing for Fuzzy Clustering With Stability and Validity

The present article deals with an application of multiobjective optimization (MOO) technique for fuzzy clustering. The problem of automatic clustering can be posed as one of the searching for a suitable set of cluster centers such that some measure of cluster validity is optimized. However, no single validity measure works equally well for different kinds of data sets. Moreover, it is extremely difficult to combine the different measures into one, since the modality for such combination may not be possible to ascertain, while sometimes the measures may be incompatible. Thus, it appears natural to keep a number of such measures separate, and to optimize them simultaneously by applying an MOO technique. Here, the mean value of a cluster validity index, computed over the partitionings obtained for different bootstrapped samples of the data, and a measure of its stability, that is defined newly in terms of the validity index, are used for optimization. The search capability of a recently proposed archived multiobjective simulated annealing algorithm called AMOSA is utilized for this purpose. The characteristic features of AMOSA are its concepts of amount of domination and archive in simulated annealing and situation-specific acceptance probabilities. The final solutions are stored in an archive, from which a single solution is chosen in a semisupervised manner that assumes the existence of some labeled points. Results are demonstrated for several artificial and real-life data sets. Comparison is made with another recent multiobjective evolutionary clustering scheme, in addition to a single objective approach and two classical methods.