A validity measure for hard and fuzzy clustering derived from Fisher's linear discriminant

Abstract

Cluster analysis has a growing importance in many research areas, especially those involving problems of pattern recognition. Generally, in real world problems, the number of classes is unknown in advance, being necessary to have criterions to Identify the best choice of clusters. Here we propose an extension to Fisher Linear Discriminant, the EFLD that does not impose limits on the minimum number of samples, can be applied to fuzzy and crisp partitions and can be calculated more efficiently. We also propose a nem fast and efficient validity method based in the EFLD that measures the compactness and separation of partitions produced by any fuzzy or crisp clustering algorithm. The simulations performed indicate that it's a efficient and fast measure even when the overlapping between clusters is high. Finally, we propose an algorithm that applies the new validity measure to the problem of finding the patterns for the fuzzy K-NN classifier. This algorithm is applied to the problem of cursive digits recognition.