Study And Application Of Random Permutation Set Theory

Uncertainty theory is a discipline that aims to deal with data and knowledge effectively in the presence of multiple types of uncertainty like imprecision,incompleteness,vagueness,randomness,etc.It is also a fundamental theory for the development of generalized artificial intelligence models.Existing uncertainty theories mainly model uncertain information in the basic event space and power set space,and the representative theories are probability theory and evidence theory,respectively.In exploring the physical meaning of power sets,the research found that there is a close relationship between the power set and combinatorial numbers: the power set is the set of all event combinations in the basic event space.Naturally,some simple and intuitive questions arise: What is the set space composed of all event permutations? How to deal with uncertain information containing ordinal information? To address these issues,Random Permutation Set Theory(RPS theory)defined the permutation mass function(PMF)and Deng’s combination rule to explore the reasoning and decision-making of order information in an uncertain environment.RPS theory represents a seminal exploration in modeling uncertainty within the permutation structure space.Theoretically,it offers richer modeling of ordinal information compared to uncertainty theories based on the power set structure,leveraging the information distribution of permutation sets.From the perspective of evidence theory,ordinal information is conceptualized as a symbol sequence representing a tendency towards different elements during belief transfer.Thus,RPS theory enables the simultaneous use of quantitative and qualitative descriptions to model the uncertainty of data and knowledge,thereby possessing a more comprehensive knowledge representation framework.Furthermore,from the viewpoint of the Transferable Belief Model(TBM),RPS theory provides richer information for the decision layer,which is conducive to enhancing the flexibility of decision-making.Although the RPS theory has unique advantages,research on RPS theory is currently in its initial stages,necessitating further exploration of how its theoretical advantages can be transformed into practical benefits in information processing.From the perspective of evidence theory,this paper explores the specific interpretation and application methods of RPS theory in practical information processing,including the following aspects:(1)Research on the generation of the permutation mass function.Existing literature on PMF information assumes prior knowledge or relies on direct input from domain experts,thus exhibiting considerable subjectivity.Therefore,this dissertation proposes a data-driven approach for PMF generation within a classification context: firstly,Basic Probability Assignment(BPA)is generated based on statistical characteristics such as the mean and standard deviation of the dataset.Subsequently,an analysis of the distances between samples and the means of different classes is conducted to generate a tendency degree towards ordered propositions for sample pairs.PMFs are then generated by combining BPA and the tendency degree.The proposed method provides an effective approach for generating PMFs from the perspective of supervised learning,and the experimental results of classification show that the proposed method can effectively improve the classification accuracy compared with the classification method based on basic confidence assignment.The reason is that under the framework of RPS theory,permutation mass function can not only represent the quantitative membership of the samples to different classes,but also represent the qualitative information of the samples’ propensity to the class,and thus provide more effective information to participate in the decision-making,thus improving the classification performance.(2)Research on the uncertainty measurement of PMF.Deng entropy is a valid uncertainty measure of power set structural distributions and BPA,which has been widely used in evidence theory.This dissertation extends Deng entropy to the framework of RPS theory,introducing RPS entropy to measure the uncertainty of PMF characterized by the permutation structure.RPS entropy can be regarded as type-2 Deng entropy,and it is compatible with Deng entropy and Shannon entropy.Its properties under the five axiomatic system of evidence theory are analyzed,and finally,the properties and validity of RPS entropy are verified through numerical examples.(3)Research on the discrepancy measure of PMF.This dissertation proposes two different measures of RPS distance and RPS divergence for computing the discrepancy between PMFs from the perspectives of distance and divergence,respectively.For RPS distance,the dissertation begins with the classical Jousselme distance in evidence theory,then permutation distance and Jaccard similarity coefficient are combined to construct an order-structure matrix for calculating the distance between PMFs.Regarding RPS divergence,this dissertation extends the belief divergence of evidence theory to RPS theory,measuring the discrepancy between different PMFs from the perspective of random variables.The effectiveness and rationality of RPS distance and RPS divergence are analyzed through numerical examples.Finally,a reliability assessment algorithm based on the difference measure of permutation mass function is proposed.The experimental results of threat assessment show that the algorithm can reflect the conflict between different information sources by using RPS distance or RPS divergence,which effectively solves the problem of the existing algorithms that cannot determine the fusion order of information sources,so as to efficiently and accurately identify threat targets.(4)Research on the information fusion algorithm within the framework of RPS Theory.This paper proposes an information fusion algorithm based on the uncertainty and discrepancy of the permutation mass function,which integrally utilizes the RPS entropy and RPS distance(or RPS divergence)to calculate the weights of the information sources,aiming to enhance the impact of positive evidence and diminish that of negative evidence,thereby effectively determining the weights of information sources.The experimental results of fault diagnosis show that,compared with the information fusion method based on evidence theory,due to the advantage of the permutation mass function carrying propensity information,the proposed fusion model can fully utilize the propensity information to participate in decision-making,and then obtain diverse decision reasoning results,providing a more flexible decision-making framework.Particularly,when Orn = 0.5,the result of the proposed model is consistent with that of the fusion model based on evidence theory.(5)Research on clustering within the framework of RPS Theory.Considering that the credal partition results of evidence clustering methods cannot represent the tendency information of samples to different clusters,this dissertation put forward an Ordered Credal C-Means clustering based on Random Permutation Set theory,abbreviated as OCCMRPS.OCCM-RPS defines the concept of ordered credal partition to describe the clustering results for the first time.With the advantage of the ordered information characterization of RPS theory,an ordered credal partition can not only quantitatively represent the membership relationship between samples and clusters but also qualitatively represent the tendency information of samples towards clusters using symbolic information,reflecting the characteristics of the original data more comprehensively in a multi-level manner.Through dataset verification,it is found that the OCCM-RPS algorithm can effectively improve clustering validity metrics under hard partitioning and exhibits superior clustering performance compared to the compared algorithms.The OCCM-RPS algorithm not only contributes to the refinement of RPS theory to clustering,but also provides an interpretable and effective granulation method within the RPS theory environment from the perspective of granular computing.