The comparison of crisp partitions is a common operation in clustering and classification tasks. Partitions are induced from the values taken by object domains' attributes. When these values are nominal, the partitions are crisp and there are several measures (such as the Rand or the Jaccard index) that allow the comparison of partitions. When the attributes take continuous values, the usual way to deal with them is with discretization methods, but these techniques have some considerable drawbacks. An alternative way to handle continuous values is using fuzzy sets, but this generates fuzzy partitions and then appropriate measures to deal with them are needed. Several proposes have been introduced by Campello (2007) and by Hüllermeier and Rifqi (2009) with interesting formal properties.
In this talk we introduce FLM, a fuzzy extension of the López de Mántaras distance (LM). This measure was presented in the PhD dissertation of López de Mántaras (1977) as a tool useful to analyze the results of classification algorithms, since it allows to compute the distance between the obtained classification and the real classification. The distance LM is defined using the notions of entropy of a partition, conditional entropy, and joint entropy. In order to define the fuzzy version of LM, we use the notion of fuzzy partition in the sense of Ruspini (1969). For the notion of entropy relative to fuzzy partitions we follow the definition given by Kuriyama (1983) which uses the product operation in order to define a notion of intersection of fuzzy partitions. In the talk we discus the formal properties of FLM, we present some open problems, and we draw some lines of future work.