A generalized Rényi entropy to measure the uncertainty of a random permutation set

Abstract

Random permutation set (RPS) introduces a novel set that considers all subsets with ordered elements from a given set. Each subset with ordered elements represents a permutation event within the permutation event space (PES). The permutation mass function (PMF) represents the chance of occurrence of events in the PES. PES and PMF make up RPS, which contains ordered information and also provides a new insight to consider the uncertainty. This characteristic aligns more closely with the occurrence of ordered events in the real world. However, existing entropies cannot measure the uncertainty with ordered information. To address this issue, a generalized Rényi entropy is proposed, it degenerates into different entropies with the changing of scenarios and parameters, in other words, it is compatible with these entropies. When the events in permutation event space are not ordered, Rényi-RPS entropy degenerates into Deng entropy. In addition, Rényi-RPS entropy further degenerates into Rényi entropy under the probability distribution. In a further way, when the parameter α→1, Rényi-RPS entropy evolves into Shannon entropy. Several numerical examples will illustrate the characteristics of the presented Rényi-RPS entropy.