Fuzzy Generalization and Comparisons for the Rand Index

Abstract

To generalize the Rand index (RI) from crisp partitions to fuzzy partitions, we first propose a graph method in which color edges in the graph for crisp partitions are used to determine the relation matrix between objects such that the matrix trace can be employed to calculate the RI. This approach is then introduced into fuzzy partitions to generalize the RI to the fuzzy RI (FRI). Compared with previous fuzzy generalizations, the most unique aspect of our method has the following important characteristics that for any two partition matrices M(1) and M(2), the result with M(1)=M(2) is the necessary and sufficient condition for the result that the FRI is equal to 1. This important characteristic renders our fuzzy generalization of the RI is not only able to determine the similarities between fuzzy partitions and crisp reference partitions, but also to identify the similarity between fuzzy partitions and fuzzy reference partitions. The method can even be used to explore and compare the similarities between various data sets and the same fuzzy reference partition. Finally, we use synthetic data and real data to give more demonstrations, and further perform comparisons of our method with those existing fuzzy extensions of the RI.

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

“…We note that there are several fuzzy extensions of RI in the literature (see [1,7,8,19]). However, our recently proposed fuzzy generalized Rand index [42] is well used as an evaluation measure for cluster ensembles. We first use a membership matrix to find a similarity measure for cluster ensembles and then extend it to a sign relation matrix for replacing the consensus matrix.…”

“…Therefore, extending evaluation measure from partitions and cluster ensembles to fuzzy partitions and fuzzy cluster ensembles is also one of our main purposes in this paper, and it is supposed to be important and also the first work in the literature. We follow our recent work in fuzzy generalized Rand index (Yang and Yeh [42]), and then propose evaluation measures for cluster ensembles based on the proposed fuzzy generalized Rand index to broaden the evaluation scope to fuzzy situations such as: between a fuzzy cluster ensemble and a crisp partition, between a fuzzy cluster ensemble and a cluster ensemble, between a fuzzy cluster ensemble and a fuzzy partition, between two fuzzy cluster ensembles, and so forth.…”