A Distance Measure for Random Permutation Set: From the Layer-2 Belief Structure Perspective
Ruolan Cheng, Yong Deng, Serafín Moral, José Ramón Trillo
TL;DR
This work develops a TBM-aligned distance for random permutation sets (RPS) by introducing a cumulative Jaccard index, $CJ^{[Orn]}(S,T,t)$, to measure similarity between permutation prefixes and embedding it as a weight matrix in an $L_2$ distance to yield $d_{RPS}^{[Orn]}$. It formalizes the TBM interpretation where order encodes qualitative propensity, enabling a natural top-weighting property and allowing arbitrary truncation depth $t$ through $CJ^{[Orn]}$. A positive-definiteness correction for the weighting matrix is proposed to ensure metric properties, with a corrective variant $d_{RPS}^{[Orn,\lambda_{min}]}$. Empirical results show the proposed method offers greater sensitivity and compatibility with the Jousselme distance than prior RPST methods, especially under depth control and propensity-based weighting, making it practical for TBM-based uncertainty reasoning in RPS representations.
Abstract
Random permutation set (RPS) is a recently proposed framework designed to represent order-structured uncertain information. Measuring the distance between permutation mass functions is a key research topic in RPS theory (RPST). This paper conducts an in-depth analysis of distances between RPSs from two different perspectives: random finite set (RFS) and transferable belief model (TBM). Adopting the layer-2 belief structure interpretation of RPS, we regard RPST as a refinement of TBM, where the order in the ordered focus set represents qualitative propensity. Starting from the permutation, we introduce a new definition of the cumulative Jaccard index to quantify the similarity between two permutations and further propose a distance measure method for RPSs based on the cumulative Jaccard index matrix. The metric and structural properties of the proposed distance measure are investigated, including the positive definiteness analysis of the cumulative Jaccard index matrix, and a correction scheme is provided. The proposed method has a natural top-weightiness property: inconsistencies between higher-ranked elements tend to result in greater distance values. Two parameters are provided to the decision-maker to adjust the weight and truncation depth. Several numerical examples are used to compare the proposed method with the existing method. The experimental results show that the proposed method not only overcomes the shortcomings of the existing method and is compatible with the Jousselme distance, but also has higher sensitivity and flexibility.