Evaluation measures for cluster ensembles based on a fuzzy generalized Rand index

Abstract

Cluster ensemble has become a general technique for combining multiple clustering partitions. There are various cluster ensemble methods to be used in real applications. Recently, Zhang et al. (2012) considered a generalized adjusted Rand index (ARI) for cluster ensembles by using a consensus matrix to evaluate ARI values. However, Zhang’s method for cluster ensembles cannot treat the cases in fuzzy partitions and fuzzy cluster ensembles. In this paper we propose evaluation measures for cluster ensembles based on the proposed fuzzy generalized Rand index (FGRI). We first use a graph and relation matrices to convert a membership matrix into a sign relation matrix, and have the trace of matrix multiplication to calculate similarity measures. We then use the FGRI to broaden the scope of the RI for considering other scenarios so that it can treat the following situations: (1) between a fuzzy cluster ensemble and a crisp partition, (2) between a fuzzy cluster ensemble and a cluster ensemble, (3) between a fuzzy cluster ensemble and a fuzzy partition, (4) between two fuzzy cluster ensembles, and (5) between two different object data sets with the same cardinal number and the same partition method. Finally, numerical comparisons and experimental results are used to demonstrate the key properties, rationality, and practicality of the proposed method.