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

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, criteria to identify the best choice of clusters being necessary. In this paper, we propose an extension to the Fisher linear discriminant (EFLD) that does not impose limits on the minimum number of samples, that can be applied to fuzzy and crisp partitions and that can be calculated more efficiently. We also propose a new 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 is 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 patterns for a fuzzy k-NN (k-nearest neighbors) classifier. This algorithm is applied to the problem of cursive digit recognition