Random Set Theory Research Articles

Random Permutation Set Reasoning.

In artificial intelligence, it is crucial for pattern recognition systems to process data with uncertain information, necessitating uncertainty reasoning approaches such as evidence theory. As an orderable extension of evidence theory, random permutation set (RPS) theory has received increasing attention. However, RPS theory lacks a suitable generation method for the element order of permutation mass function (PMF) and an efficient determination method for the fusion order of permutation orthogonal sum (POS). To solve these two issues, this paper proposes a reasoning model for RPS theory, called random permutation set reasoning (RPSR). RPSR consists of three techniques, including RPS generation method (RPSGM), RPSR rule of combination, and ordered probability transformation (OPT). Specifically, RPSGM can construct RPS based on Gaussian discriminant model and weight analysis; RPSR rule incorporates POS with reliability vector, which can combine RPS sources with reliability in fusion order; OPT is used to convert RPS into a probability distribution for the final decision. Besides, numerical examples are provided to illustrate the proposed RPSR. Moreover, the proposed RPSR is applied to classification problems. An RPSR-based classification algorithm (RPSRCA) and its hyperparameter tuning method are presented. The results demonstrate the efficiency and stability of RPSRCA compared to existing classifiers.

Multi-sample means comparisons for imprecise interval data

In recent years, interval data have become an increasingly popular tool to solving modern data problems. Intervals are now often used for dimensionality reduction, data aggregation, privacy censorship, and quantifying awareness of various uncertainties. Among many statistical methods that are being studied and developed for interval data, significance tests are of particular importance due to their fundamental value both in theory and practice. The difficulty in developing such tests mainly lies in the fact that the concept of normality does not extend naturally to intervals, making the exact tests hard to formulate. As a result, most existing works have relied on bootstrap methods to approximate null distributions. However, this is not always feasible given limited sample sizes or other intrinsic characteristics of the data. In this paper, we propose a novel asymptotic test for comparing multi-sample means with interval data as a generalization of the classic ANOVA. Based on the random sets theory, we construct the test statistic in the form of a ratio of between-group interval variance and within-group interval variance. The limiting null distribution is derived under usual assumptions and mild regularity conditions. Simulation studies with various data configurations validate the asymptotic result, and show promising small sample performances. Finally, a real interval data ANOVA analysis is presented that showcases the applicability of our method.

Random walk in random permutation set theory.

Random walk is an explainable approach for modeling natural processes at the molecular level. The random permutation set theory (RPST) serves as a framework for uncertainty reasoning, extending the applicability of Dempster-Shafer theory. Recent explorations indicate a promising link between RPST and random walk. In this study, we conduct an analysis and construct a random walk model based on the properties of RPST, with Monte Carlo simulations of such random walk. Our findings reveal that the random walk generated through RPST exhibits characteristics similar to those of a Gaussian random walk and can be transformed into a Wiener process through a specific limiting scaling procedure. This investigation establishes a novel connection between RPST and random walk theory, thereby not only expanding the applicability of RPST but also demonstrating the potential for combining the strengths of both approaches to improve problem-solving abilities.

Probability hypothesis density filter for parameter estimation of multiple hazardous sources

This study introduces an advanced methodology for estimating the source term of multiple, variable-number biochemical hazard releases, where the exact count of sources is not predetermined. Focusing on environments monitored via a network of sensors, we tackle this challenge through a multi-source Bayesian filtering paradigm, employing the theory of random finite sets (RFS). Our novel approach leverages a modified particle filter-based probability hypothesis density (PHD) filter within the RFS framework, enabling simultaneous estimation of critical source characteristics (such as location, emission rate, and effective release height) and the quantification of source numbers. This method not only accurately estimates pertinent source parameters but is also adept at identifying the emergence of new sources and the cessation of existing ones within the monitored area. The efficacy of our approach is validated through extensive simulations, which mimic a range of scenarios with varying and unknown source counts, highlighting the proposed method’s robustness and precision.

Negation of permutation mass function in random permutation sets theory for uncertain information modeling

Negation provides a novel perspective for the representation of information. However, current research seldom addresses the issue of negation within the random permutation set theory. Based on the concept of belief reassignment, this paper proposes a method for obtaining the negation of permutation mass function in the of random set theory. The convergence of proposed negation is verified, the trends of uncertainty and dissimilarity after each negation operation are investigated. Furthermore, this paper introduces a negation-based uncertainty measure, and designs a multi-source information fusion approach based on the proposed measure. Numerical examples are used to verify the rationality of proposed method.

Methodological issues of the fuzzy set theory (generalizing article)

The theory of fuzziness is an important area of modern theoretical and applied mathematics. The methodology of the theory of fuzziness is a doctrine of organizing activities in the field of development and application of the scientific results of this theory. We discuss some methodological issues of the theory of fuzziness, i.e., individual components of the methodology in the area under consideration. The theory of fuzziness is a science of pragmatic (fuzzy) numbers and sets. The ancient Greek philosopher Eubulides showed that the concepts «Heap» and «Bald» cannot be described using natural numbers. E. Borel proposed to define a fuzzy set using a membership function. A fundamentally important step was taken by L. A. Zadeh in 1965. He gave the basic definitions of the algebra of fuzzy sets and introduced the operations of intersection, product, union, sum, negation of fuzzy sets. The main thing he did was demonstration of the possibilities of expanding («doubling») mathematics: by replacing the numbers and sets used in mathematics with their fuzzy counterparts, we obtain new mathematical formulations. In the statistics of non-numerical data, methods of statistical analysis of fuzzy sets have been developed. Interval and triangular fuzzy numbers are often used specific types of membership functions. The theory of fuzzy sets in a certain sense is reduced to the theory of random sets. We think fuzzy and that is the only reason we understand each other. The paradox of the fuzzy theory is that it is impossible to consistently implement the thesis «Everything in the world is fuzzy». For ordinary fuzzy sets, the argument and values of the membership function are crisp. If they are replaced by fuzzy analogues, then their description will require their own clear arguments and membership functions, and so on ad infinitum. System fuzzy interval mathematics proceeds from the need to take into account the fuzziness of the initial data and the prerequisites of the mathematical model. One of the options for its practical implementation is an automated system-cognitive analysis and the intellectual system «Eidos».

The multiple pairwise Markov chain model-based labeled multi-Bernoulli filter

Most of existing multi-target tracking (MTT) algorithms, which are rooted in random finite set theory, generally rely on two hypotheses, i.e., the single dynamic model hypothesis and the hidden Markov chain (HMC) hypothesis, and the HMC hypothesis requires the target state to conform to a Markov process and the detection process to be independent. Unfortunately, these hypotheses may not always hold at the same time in many practical situations. Therefore, it is important to study the MTT algorithms in such scenarios when the HMC hypothesis and the single dynamic model hypothesis fail simultaneously. As a result, this paper presents a multiple model MTT algorithm, which is designed to tackle the MTT problem effectively in scenarios where both hypotheses are invalid. Firstly, when the HMC hypothesis is not satisfied, an MTT algorithm was presented based on pairwise Markov chain (PMC) and the labeled multi-Bernoulli filter (PMC-LMB). Secondly, in case that both hypotheses are not met, a multiple model MTT algorithm was proposed by extending the previously presented PMC-LMB filter to multiple PMC model case. Finally, extensive simulation was done to demonstrate the efficiency of the presented algorithms.

A Network-Group Target State and Network Topology Estimation Method Based on Signals Containing Delay, Doppler and Address

Network-group targets are a set of objectives that adhere to a shared communication protocol, perform common tasks, and exhibit relatively coordinated movements. Typically, network-group targets emit radar and communication signals. However, they often employ a silent mode to evade our perception. Despite this, they continue to exchange data through their communication networks. By intercepting the communication signals of these targets, this paper proposes a method for estimating the state and network topology of network-group targets based on random finite set (RFS) theory. This method is based on the modeling of network-group targets using a labeled multi-Bernoulli (LMB) RFS. In state estimation, the usual method involves extracting the localization parameters from the signals in the first step and use these parameters for target tracking in the second step. This study focused on estimating the kinematic states of network-group targets using communication signals containing delay and Doppler information received by multiple mobile sensor platforms. The proposed method considers the coherency between frequency-hopping pulses in the signals, resulting in an improved estimation performance. In addition, considering that network-group targets require network access for information exchange, graph theory concepts were utilized to estimate the network topology of network-group targets by address measurement. The simulation results validated the effectiveness of the proposed method and demonstrated its superior performance.

Distributed Large-Scale Swarm Control in Obstacle Environment Based on Random Finite Set Theory

Distributed Large-Scale Swarm Control in Obstacle Environment Based on Random Finite Set Theory

Bernoulli Filter for Extended Target Tracking in the Framework of Possibility Theory

Extended objects give rise to a varying number of noisy measurements from its reflection (scattering or feature) points. Due to imperfect detection, only some of the feature points are detected in each scan of input data, while false alarms can also be present. The optimal sequential Bayesian state estimator in the framework of random set theory is the Bernoulli filter for an extended target (BF-X). In this paper we formulate and derive the analog of the BF-X in the framework of possibility theory, where uncertainty is represented using <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">possibility</i> functions, rather than <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">probability</i> distributions. Possibility functions have the capacity to model partial (imprecise) probabilistic specifications and thus the main advantage of the proposed possibilistic BF-X, is enhanced robustness in the absence of precise measurements or dynamic models.

A Labeled RFS-Based Framework for Multiple Integrity Attackers Detection and Identification in Cyber–Physical Systems

The problem of multiple integrity attacks (attackers) detection and identification (MIADI) in cyber-physical systems (CPSs) is still a challenging problem to date. The goal of this paper is to develop a knowledge-based method capable of simultaneously detecting and identifying multiple integrity attacks aiming at different sensors in a CPS. In this paper, with the help of labeled random finite set (RFS) theory, a new solution to solve the MIADI problem is proposed. The main contributions of this paper, lie in the the following two aspects, the first is the novel formulation of the MIADI problem, in which labeled RFSs are used to model the behaviors of multiple integrity attackers for the first time, and the second is the proposed labeled RFS-based solution, which provides an elegant framework to cope with the MIADI problem. Numerical experiments are conducted and experimental results demonstrate the effectiveness of the proposed solution. This proposed solution further extends the feasibility of the labeled RFS theory in the context of CPSs cybersecurity.

Conjunctive and disjunctive combination rules in random permutation set theory: A layer-2 belief structure perspective

In uncertainty reasoning, conjunctive and disjunctive combination rules are the core tools for information updates. As an extension of evidential reasoning, random permutation set reasoning models uncertain information based on ordered focal sets. Existing combination rules in random permutation set theory, orthogonal sums, do not satisfy the commutativity of one of the original distributions and overly obey its order information. In this paper, the random permutation set theory is interpreted as an refined extension of Dempster–Shafer theory, and a layer-2 belief structure is proposed to describe the permutation event space. Compared with the traditional belief structure, the proposed structure can model both symbolic and numerical uncertainty. Based on the above, the conjunctive and disjunctive combination rules in Dempster–Shafer theory are extended to random permutation set theory. Through properties analysis and simulation demonstration, we demonstrate that the proposed methods can not only resolve the counter-intuitive results of orthogonal sums, but make full use of order information in distributions as well. In addition, we also extend the product space operations and discounting methods based on the proposed methods, and give a general framework of multi-source information fusion under the random permutation set theory.

Coordinated target localization and tracking with node scheduling for a network of autonomous underwater vehicles

Underwater mobile sensor networks (UMSNs) have the advantages of large area coverage, flexible network structure, and good local resolution. Particularly, spatial diversity can be exploited through node mobility to get more target information. To take those advantages in the context of regional cooperative target localization and tracking, a joint node scheduling and acoustic model fusion method is proposed for an UMSN consisting of multiple autonomous underwater vehicles (AUVs). First, a target tracking algorithm framework based on the random finite set theory is established. To take the acoustic environment into account, the a priori acoustic field and target distribution information are combined to evaluate the detection probability. A path-planning scheme with node scheduling is then developed based on the information gain to optimize the detection performance. To achieve effective node scheduling, three reward functions with different information gains respectively derived from the Rényi divergence and Fisher information matrix are proposed and compared. The rationale and the performance of the developed method are verified via simulations. A simplified version of the approach is also tested in field experiments with three AUVs, verifying the effectiveness of node scheduling in a realistic scenario.

Research on the development model of fusion of poetry communication and literary creation based on information fusion technology under the perspective of national fusion

Abstract Under the perspective of national integration, the fusion development of poetry communication and literary creation is flourishing. Through analyzing the characteristics of contemporary poetry dissemination and creation, this paper proposes an innovative model for the fusion development of poetry dissemination and literary creation based on information fusion technology. For poetry dissemination of multi-source information data, random set theory is used for representation, information fusion is performed on the representation results of random sets based on D-S evidence theory, and uncertainty is measured by introducing trust functions and multiple basic probability distribution functions synthesized into one basic trust function through orthogonal sum according to synthesis rules. From the viewpoint of dissemination information sources, news and information websites, video websites, podcasts or audiobooks are called the most important information sources for poetry dissemination, with a combined weight of more than 95%. In the case of poetry literature creation under the integration development mode, the periodicity of the works has significantly improved, with an improvement interval of (18.3% and 27.6%). The dissemination degree also increased effectiveness, with a maximum of 30.7% and a minimum of 21.5%. The development model based on information fusion can open up a new path for the fusion development of contemporary poetry communication and literary creation.

Parametric families of continuous belief functions based on generalized Gaussian random fuzzy numbers

The theory of epistemic random fuzzy sets is a general theory of uncertainty encompassing both possibility theory and the Dempster-Shafer theory of belief functions as special cases. Within this framework, Gaussian random fuzzy numbers have recently been introduced as a practical model of uncertainty about real variables. However, the limited flexibility of this model does not allow it to represent all kinds of beliefs encountered in applications. In this paper, it is extended in two ways. First, we study one-to-one transformations of random fuzzy sets and show that such transformations commute with combination. This property allows us to define parametric families of easily combinable random fuzzy numbers and vectors on different frames based on the Gaussian model. We then go one step further by studying mixtures of random fuzzy variables, which provide a very flexible model making it possible to construct belief functions on continuous frames with arbitrary complexity. To demonstrate the applicability and practical interest of these models, two applications are studied: the elicitation of expert beliefs about numerical quantities, and generalized Bayesian inference with weak prior information represented by random fuzzy numbers.

Cell Multi-Bernoulli (Cell-MB) Sensor Control for Multi-Object Search-While-Tracking (SWT).

Information-driven control can be used to develop intelligent sensors that can optimize their measurement value based on environmental feedback. In object tracking applications, sensor actions are chosen based on the expected reduction in uncertainty also known as information gain. Random finite set (RFS) theory provides a formalism for quantifying and estimating information gain in multi-object tracking problems. However, estimating information gain in these applications remains computationally challenging. This paper presents a new tractable approximation of the RFS expected information gain applicable to sensor control for multi-object search and tracking. Unlike existing RFS approaches, the information gain approximation presented in this paper considers the contributions of non-ideal noisy measurements, missed detections, false alarms, and object appearance/disappearance. The effectiveness of the information-driven sensor control is demonstrated through two multi-vehicle search-while-tracking experiments using real video data from remote terrestrial and satellite sensors.

Marginalization in random permutation set theory: from the cooperative game perspective

Marginalization in random permutation set theory: from the cooperative game perspective

Quantifying coal rib stability and support requirements using bonded block modeling and reliability analysis based on the random set theory

An apparent flaw in geomechanical analysis of rock structures is the application of single value parameters as input to numerical models. A known but often overlooked characteristic of the rock mass is its inherent strong property variability. In conventional reliability analysis of rock mechanics, subjective assumptions are often made on the probability density function of parameters because, in many cases, the results of geotechnical investigations are set-valued rather than being precise and point-valued. On the other hand, imprecise probability theories such as the Random Set Theory could be considered a possible attractive candidate to employ in these analyses. In this study, the Random Set Theory in combination with the Bonded Block Modeling (RS-BBM) approach is employed to demonstrate its practical applicability in determining coal rib stability and support requirements. Sensitivity analysis of the BBM input parameters is conducted. The uncertainties of the geomechanical parameters of the BBM contacts, which control model behavior, are considered in the reliability analysis model. The analyses were performed in UDEC, and the strength reduction method was used to evaluate rib stability. It is shown that RS-BBM is an efficient tool to determine the most likely bounds of coal rib performance. A method combining the RS-BBM approach with empirical methods is also described to illustrate its practical application.

Multiple extended target tracking based on distributed multi‐sensor fusion and shape estimation

AbstractDistributed multi‐sensor fusion based on the generalised covariance intersection (GCI) fusion has been widely integrated into the Random Finite Set theory, which is promising for multi‐target tracking with an unknown number of targets. However, it has not been widely investigated in the multiple extend target tracking (METT) field, and there is still an open problem on how to solve the inconsistency of label space among the sensors. For these problems, we first introduce the GCI fusion into the METT and proposed an association algorithm by considering the estimated shapes and the target positions to avoid the phenomenon of the label inconsistency as well as to reduce the computational burden. Simulation results show that the proposed algorithm has a better tracking performance than the traditional METT algorithms.

Optimization hierarchy for fair statistical decision problems

Data-driven decision making has drawn scrutiny from policy makers due to fears of potential discrimination, and a growing literature has begun to develop fair statistical techniques. However, these techniques are often specialized to one model context and based on ad hoc arguments, which makes it difficult to perform theoretical analysis. This paper develops an optimization hierarchy, which is a sequence of optimization problems with an increasing number of constraints, for fair statistical decision problems. Because our hierarchy is based on the framework of statistical decision problems, this means it provides a systematic approach for developing and studying fair versions of hypothesis testing, decision making, estimation, regression, and classification. We use the insight that qualitative definitions of fairness are equivalent to statistical independence between the output of a statistical technique and a random variable that measures attributes for which fairness is desired. We use this insight to construct an optimization hierarchy that lends itself to numerical computation, and we use tools from variational analysis and random set theory to prove that higher levels of this hierarchy lead to consistency in the sense that it asymptotically imposes this independence as a constraint in corresponding statistical decision problems. We demonstrate numerical effectiveness of our hierarchy using several data sets, and we use our hierarchy to fairly perform automated dosing of morphine.