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Adjusted Concordance Index, an extension of the Adjusted Rand index to fuzzy partitions

In comparing clustering partitions, Rand index (RI) and Adjusted Rand index (ARI) are commonly used for measuring the agreement between the partitions. Both these external validation indexes aim to analyze how close is a cluster to a reference (or to prior knowledge about the data) by counting corrected classified pairs of elements. When the aim is to evaluate the solution of a fuzzy clustering algorithm, the computation of these measures require converting the soft partitions into hard ones. It is known that different fuzzy partitions describing very different structures in the data can lead to the same crisp partition and consequently to the same values of these measures. We compare the existing approaches to evaluate the external validation criteria in fuzzy clustering and we propose an extension of the ARI for fuzzy partitions based on the normalized degree of concordance. Through use of real and simulated data, we analyze and evaluate the performance of our proposal.