Power fuzzy clustering: flexible distance metrics and inclusion of covariates

Clustering is a fundamental task in statistics and machine learning aimed at identifying homogeneous groups within data. Among the available approaches, fuzzy clustering (FC) is particularly suited to poorly separated boundaries, as it allows partial memberships and captures uncertainty and overlapping structures. We introduce Power Fuzzy Clustering (PFC), a general framework that unifies several FC methods, including fuzzy K-means and probabilistic distance clustering. By complementing the classical fuzzifier with a power parameter on distances, PFC enhances model flexibility while retaining interpretability. To accommodate clusters with different shapes and scales, we incorporate a volume parameter and alternative distance metrics (e.g., Mahalanobis and Minkowski). We further propose the Power Fuzzy Cluster-wise Regression (PFCR) model, a generalization of PFC that reduces to it when only intercepts are included. Simulation studies investigate the impact of the power and fuzzification parameters, and a real-world application illustrates the method's practical relevance.