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随机置换集的距离度量：基于第二层信念结构视角

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

Institute of Fundamental and Frontier Science（基础与前沿科学研究院）；University of Electronic Science and Technology of China（电子科学与技术大学）；Andalusian Research Institute in Data Science and Computational Intelligence（数据科学与计算智能安达卢西亚研究所）；Dept. of Computer Science and AI, University of Granada（计算机科学与人工智能系，格拉纳达大学）；School of Medicine（医学院）；Vanderbilt University（范德比大学）；Computer Science and Artificial Intelligence, University of Granada（计算机科学与人工智能，格拉纳达大学）

Ruolan Cheng, Yong Deng, Serafín Moral, José Ramón Trillo

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AI总结：

本文基于第二层信念结构视角，提出了一种基于累积Jaccard指数矩阵的随机置换集距离度量方法，克服了现有方法缺点，兼容Jousselme距离，并具备更高灵敏度与灵活性。

摘要：

随机置换集（RPS）是最近提出的一个框架，旨在表示有序结构的不确定信息。测量置换质量函数之间的距离是RPS理论（RPST）中的一个关键研究课题。本文从随机有限集（RFS）和可转移信念模型（TBM）两个不同视角对RPS之间的距离进行了深入分析。采用RPS的第二层信念结构解释，我们将RPST视为TBM的一种细化，其中有序焦点集中的顺序代表定性倾向。从置换出发，我们引入了累积Jaccard指数的新定义来量化两个置换之间的相似性，并进一步提出了一种基于累积Jaccard指数矩阵的RPS距离度量方法。我们研究了所提出距离度量的度量与结构特性，包括累积Jaccard指数矩阵的正定性分析，并提供了一种修正方案。所提出的方法具有自然的顶部加权特性：较高排名元素之间的不一致性往往会导致更大的距离值。该方法为决策者提供了两个参数，以调整权重和截断深度。文中使用几个数值示例将所提方法与现有方法进行了比较。实验结果表明，所提方法不仅克服了现有方法的缺点并与Jousselme距离兼容，而且具有更高的灵敏度和灵活性。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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