A fuzzy extension of the silhouette width criterion for cluster analysis

The present paper proposes a new cluster validity measure as an additional criterion to help the decision making process in fuzzy cluster analysis. This measure, named Fuzzy Silhouette, is a generalization to the fuzzy case of the Average Silhouette Width Criterion, originally conceived to assess crisp (non-fuzzy) data partitions. The Fuzzy Silhouette is more appealing than its crisp counterpart in the context of fuzzy cluster analysis since it makes explicit use of the fuzzy partition matrix provided by the clustering algorithm. In addition, it has been designed to improve performance of the original silhouette criterion in detecting regions with higher data density when the data set involves overlapping clusters. The performance of the Fuzzy Silhouette is evaluated and compared to that of five well-known cluster validity measures. Six data sets are used to illustrate different scenarios in which the proposed Fuzzy Silhouette performs similar to or better than these other criteria, thus becoming eligible to join a pool of measures to be used all together in fuzzy cluster analysis.