Novel distance measures for probabilistic hesitant Pythagorean fuzzy information and applications in clustering and pattern recognition

Abstract Probabilistic hesitant Pythagorean fuzzy sets enable robust modeling of complex uncertain information, but distance measure, which underpins similarity assessment, clustering, and pattern recognition, remains underexplored in the literature. This study addresses the gap by introducing two novel distance measures: (1) probabilistic hesitant Pythagorean fuzzy element distance measures that integrate probability weighting with hesitant set operations to simultaneously capture uncertainty and hesitation, and (2) a probabilistic hesitant Pythagorean fuzzy vector distance measure that enables comprehensive similarity evaluation. Key contributions include: theoretical validation of the measure properties (non-negativity, identity, symmetry, and triangle inequality); development of a probabilistic hesitant Pythagorean fuzzy clustering algorithm based on element-level distances; and design of a pattern recognition method utilizing vector-level distances. Numerical experiments and comparative analysis demonstrate that clustering and pattern recognition methods based on the proposed distance measures achieve superior accuracy and robustness. By establishing a unified distance measure framework, this research advances probabilistic hesitant Pythagorean fuzzy set theory and highlights its practical utility in data-driven decision-making.