Random models for adjusting fuzzy rand index extensions

The adjusted Rand index (ARI) is a widely used method for comparing hard clusterings, but requires a choice of random model that is often left implicit. Several recent works have extended the Rand index to fuzzy clusterings and adjusted for chance agreement with the permutation model, but the assumptions of this random model are difficult to justify for fuzzy clusterings. Previous work on random models for hard clusterings has shown that different random models can impact similarity rankings, so matching the assumptions of the random model to the algorithm is essential. We propose a single framework computing the ARI with three new random models that are intuitive and explainable for both hard and fuzzy clusterings. The theory and assumptions of the proposed models are contrasted with the existing permutation model, and computations on synthetic and benchmark data show that each model has distinct behaviour, meaning accurate model selection is important for the reliability of results.