A Distance Measure for Random Permutation Set: From the Layer-2 Belief Structure Perspective

2025-10-12

AI Summary

This paper addresses the critical problem of distance measurement between random permutation sets (RPSs). We propose a novel distance metric grounded in the layer-2 belief structure, interpreting ordered focal elements as qualitative propensity expressions and constructing the metric via a cumulative Jaccard index matrix. The approach inherently assigns higher weights to top-ranked permutations and incorporates an adjustable truncation depth with a correction mechanism, significantly improving sensitivity and flexibility. Theoretically, it integrates random finite set theory, the transferable belief model, and positive-definiteness analysis. Practically, it generalizes the classical Jousselme distance and demonstrates superior discriminability and robustness across multiple numerical experiments. This work provides a more interpretable and adaptive distance tool for uncertainty modeling and fusion of RPSs.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

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.

Problem

• Proposes a distance measure for random permutation sets using cumulative Jaccard index
• Analyzes permutation mass functions from set theory and belief structure perspectives
• Overcomes existing method limitations with enhanced sensitivity and flexibility

Innovation

• Proposes cumulative Jaccard index for permutation similarity
• Introduces distance measure using cumulative Jaccard matrix
• Provides adjustable weight and truncation depth parameters