Validation of fuzzy and crisp c-partitions

We propose a comparative study of crisp and fuzzy cluster validity measures; we focus mainly on the generalized definitions of Dunn´s indices (GDI). We propose three fuzzy versions of GDI indices: plain fuzzy GDI indices using directly the fuzzy membership degrees, (“semi-fuzzy”) GDI indices using fuzzy membership degrees only of points that are found (after defuzzification) to belong crisply to a cluster, and fuzzy GDI indices combined with a measure of cluster fuzziness. Two types of cluster fuzziness measure are proposed: (i) classical fuzziness measures (H) based on fuzzy membership grades of all data points in a cluster, and (ii) fuzziness measures (H^Semi) that use only fuzzy membership values of data points found to belong crisply to the cluster. Numerical experiments are conducted on nine data sets to compare the performance of the crisp and fuzzy GDI indices. The best among tested indices were found to be the crisp GDI indices, the “semi-fuzzy” version of GDI indices. These two indices resulted in 8 out of 9; correct prescriptions of the right number of clusters. Further, a fuzzy GDI combined with measures of fuzziness H^Semi produces 7 out of 9 correct prescriptions of the number of clusters