Fuzzy Generalization and Comparisons for the Rand Index

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.

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