A probabilistic approach to comparing the distances between partitions of a set

This article describes and compares a number of classical metrics to compare different approaches to partition a given set, such as the Rand index, the Larsen and Aone coefficient, among others. We developed a probabilistic framework to compare these metrics and unified representation of distances that uses a common set of parameters. This is done by taking all possible values of similarity measurements between different possible partitions and graduating them by using quantiles of a distribution function. Let ${\lambda }_{\alpha }$ be a quantile with $\alpha $ level for distribution function $F_{\rho }\left(t\right)=P\left(\rho <t\right)$. Then if the proximity measurement $\rho $ is not less than ${\lambda }_{\alpha }$, we can conclude that $\alpha \cdot 100\%$ of randomly chosen pairs of partitions have a proximity measurement less than $\rho $. This means that these partitions can neither be considered close nor similar. This paper identifies the general case of distribution functions that describe similarity measurements, with a special focus on uniform distributions. The comparison results are presented in tables for quantiles of probability distributions, using computer simulations over our selected set of similarity metrics. Refs 9. Table 1.