Remarks on “Measuring Ambiguity in the Evidence Theory”

In a recent paper, a functional AM (ambiguity measure) is introduced and an attempt is made to show that this functional qualifies as a measure of total aggregated uncertainty in the Dempster-Shafer theory. We show that this attempt fails due to a particular error in the proof of one of the principal theorems in the paper. Some additional remarks are made regarding recent research pertaining to the subject of the discussed paper.     

UNCERTAINTY IN THE DEMPSTER-SHAFER THEORY: A CRITICAL RE-EXAMINATION

Measures of two types of uncertainty that coexist in the Dempster-Shafer theory are overviewed. A measure of one type of uncertainty, which expresses nonspecificity of evidential claims, is well justified on both intuitive and mathematical grounds. Proposed measures of the other types of uncertainty, which atlempt to capture conflicts among evidential claims, are shown to have some deficiencies. To alleviate these deficiencies, a new measure is proposed. This measure, which is called a measure of discord, is not only satisfactory on intuitive grounds, but has also desirable mathematical properties. A measure of total uncertainty, which is defined as the sum of nonspecificity and discord, is also discussed. The paper focuses on conceptual issues. Mathematical properties of the measure of discord are only slated; their proofs are given in a companion paper.¹⁵     

基于证据理论的不确定性推理方法及其应用

针对民航无线电管理中遇到的干扰问题以及客观世界中描述客观现象的知识和信息具有不确定性的特点,结合民航无线电干扰查处的实际情况,提出基于一种产生式规则的知识表示方法,以区间确定因子描述不确定性。提出基于Dempster-Shafer证据理论和区间确定因子的不确定性推理方法。进一步根据领域专家经验建立民航无线电干扰查处规则,以基于Dempster-Shafer证据理论和区间确定因子的不确定性推理方法为推理机,给出民航无线电干扰查处专家系统的结构及工作流程,采用C#语言结合MySQL数据库开发民航无线电干扰查处专家系统,并通过案例分析说明该系统的有效性和实用性。     

Monotonicity of entropy computations in belief functions

This article addresses the issue of quantitative information measurement within the Dempster-Shafer belief function formalism. Entropy computation in Dempster-Shafer depends on the way uncertainty measures are conceptualized. However, freed of most probability constraints, uncertainty measures in Dempster-Shafer theory can lead to further advances in optimization in information theory, which in turn may have a wide impact on decision and control. This article examines one form of current development regarding the entropy measure induced from the measure of dissonance. For a significant period, the measure of dissonance has been taken as a measure of entropy. We present in this article the entropy measure as a monotonically decreasing function, symmetrical to the measure of dissonance.     

A review of: “ANTICIPATORY SYSTEMS”, by Robert Rosen,Pergamon Press,Oxford, 1985, X + 436pp.

In rough set theory there exists a pair of approximation operators, the upper and lower approximations, whereas in Dempster-Shafer theory of evidence there exists a dual pair of uncertainty measures, the plausibility and belief functions. It seems that there is some kind of natural connection between the two theories. The purpose of this paper is to establish the relationship between rough set theory and Dempster-Shafer theory of evidence. Various generalizations of the Dempster-Shafer belief structure and their induced uncertainty measures, the plausibility and belief functions, are first reviewed and examined. Generalizations of Pawlak approximation space and their induced approximation operators, the upper and lower approximations, are then summarized. Concepts of random rough sets, which include the mechanisms of numeric and non-numeric aspects of uncertain knowledge, are then proposed. Notions of the Dempster-Shafer theory of evidence within the framework of rough set theory are subsequently formed and interpreted. It is demonstrated that various belief structures are associated with various rough approximation spaces such that different dual pairs of upper and lower approximation operators induced by the rough approximation spaces may be used to interpret the corresponding dual pairs of plausibility and belief functions induced by the belief structures.     

Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster-Shafer theory

There are many different methods for interval and fuzzy number comparison proposed in the literature which provide the results of comparison in the form of a real or Boolean value. In this paper, we use the Dempster-Shafer theory of evidence with its probabilistic interpretation to justify and construct the method which provides the result of comparison in the form of an interval or fuzzy number. The complete and consistent set of expressions for inequality and equality relations between intervals and fuzzy numbers has been obtained in the framework of the probabilistic approach. These relations make it possible to compare intervals and fuzzy numbers with real values and may be considered as an asymptotic limit of the results we obtain using the Dempster-Shafer theory. A natural fuzzy extension of the proposed approach is considered and discussed using some illustrative examples.     

On the fusion of imprecise uncertainty measures using belief structures

Our interest is in the fusion of information from multiple sources when the information provided by the individual sources is expressed in terms of an imprecise uncertainty measure. We observe that the Dempster-Shafer belief structure provides a framework for the representation of a wide class of imprecise uncertainty measures. We then discuss the fusion of multiple Dempster-Shafer belief structures using the Dempster rule and note the problems that can arise when using this fusion method because of the required normalization in the face of conflicting focal elements. We then suggest some alternative approaches fusing multiple belief structures that avoid the need for normalization.     

A new matrix game with payoffs of generalized Dempster-Shafer structures

How to deal with matrix game under uncertainty is an open issue. Compared with the Dempster-Shafer structure, the insignificant deviation caused by the subjectivity of the expert is effectively eliminated generalized Dempster-Shafer structures. In this paper, a new matrix game with payoffs of generalized Dempster-Shafer structures is presented. Generalized Dempster-Shafer structures presents probability nonspecificity and inaccuracy as interval values. Then payoff is determined by linear programming. A zero-sum matrix game is illustrated the efficiency of the proposed method.     

Performance of an extended certainty-weighted detection model

This paper introduces a certainty-weighted detection system (CWDS) based on distributed decision makers that can classify a binary phenomenon as true, false, or uncertain. The CWDS is composed of two main blocks: the definite decision block (DDB), which provides a decision regarding the presence or absence of the phenomenon, and the uncertainty measure block (UMB) that provides a measure of uncertainty. The final decision, which may be definite (true or false) or uncertain, depends on characteristic parameters that define the region of uncertainty (RU/sub i/ and /spl alpha/) used by piecewise linear certainty functions in the DDB and in the UMB. The Bayes cost analysis is extended to include the cost of uncertain classifications and cost of errors. A cost function is used to compare the CWDS to decision structures based on the Dempster-Shafer theory and fuzzy logic that also provide uncertain decisions. The CWDS performs similarly to a classical Bayes detection system when no uncertain classifications are provided. By changing the parameters RU/sub i/ and /spl alpha/, the CWDS can also be adjusted to perform similarly to the Dempster-Shafer and fuzzy structures. The differences between these approaches are mainly in their characterization of uncertainty, and they can reduce the total costs below that of the Bayesian model if the cost of uncertain classifications is sufficiently smaller than the cost of errors. The performance of the CWDS was less sensitive to changes in the ratio of the costs of uncertain decisions to the cost of incorrect certain decisions, showing the CWDS to be more robust to system parameters than the fuzzy and Dempster-Shafer systems.     

Uncertain reasoning using the Dempster-Shafer method: an application in forecasting and marketing management

Abstract: The intended purpose of this article is twofold: to study techniques for uncertainty management in expert systems, particularly the Dempster-Shafer theory of belief; and to use this method in the construction of an expert system for the field of forecasting and marketing management. Compared with the probabilistic approach, which can only deal with singleton possibilities, the Dempster-Shafer approach proves to be superior because it provides the ability to deal with sets of possibilities. Since market analysis and forecasting always have a strong element of uncertainty associated with them, and since, in general, we consider a combination of several different forecasting techniques for planning our marketing strategies, the Dempster-Shafer approach is particularly suitable. Here we present a short introduction to this theory, briefly discuss the domain of application (selection of forecasting techniques for marketing planning), discuss the interesting components of our expert system, and analyze our experiences in applying the theory to this domain.     

Uncertainty representation using fuzzy measures

We introduce the fuzzy measure and discuss its use as a unifying structure for modeling knowledge about an uncertain variable. We show that a large class of well-established types of uncertainty representations can be modeled within this framework. A view of the Dempster-Shafer (D-S) belief structure as an uncertainty representation corresponding to a set of possible fuzzy measures is discussed. A methodology for generating this set of fuzzy measures from a belief structure is described. A measure of entropy associated with a fuzzy measure is introduced and its manifestation for different fuzzy measures is described. The problem of uncertain decision making for the case in which the uncertainty represented by a fuzzy measure is considered. The Choquet integral is introduced as providing a generalization of the expected value to this environment.     

Uncertainty measure based on Tsallis entropy in evidence theory

Dempster-Shafer evidence theory has been widely used in many applications due to its advantages with weaker conditions than Bayes probability. How to measure the uncertainty of basic probability assignment (BPA) in Dempster-Shafer evidence theory is an open and essential issue. Tsallis entropy as nonextensive entropy proposed according to multifractals has been used in many fields. In this paper, a new uncertainty measure of BPA is presented based on Tsallis entropy. The key issue is to determine the value of $q$ in Tsallis entropy. In addition, this paper also analyzes the properties of proposed uncertainty measure. Some numerical examples are used to illustrate the efficiency of the proposed method. Finally, the paper also discusses the application of the proposed method in decision-making.     

Measuring ambiguity in the evidence theory

In the framework of evidence theory, ambiguity is a general term proposed by Klir and Yuan in 1995 to gather the two types of uncertainty coexisting in this theory: discord and nonspecificity. Respecting the five requirements of total measures of uncertainty in the evidence theory, different ways have been proposed to quantify the total uncertainty, i.e., the ambiguity of a belief function. Among them is a measure of aggregate uncertainty, called AU, that captures in an aggregate fashion both types of uncertainty. But some shortcomings of AU have been identified, which are that: 1) it is complicated to compute; 2) it is highly insensitive to changes in evidence; and 3) it hides the distinction between the two types of uncertainty that coexist in every theory of imprecise probabilities. To overcome the shortcomings, Klir and Smith defined the TU/sub 1/ measure that is a linear combination of the AU measure and the nonspecificity measure N. But the TU/sub 1/ measure cannot solve the problem of computing complexity, and brings a new problem with the choice of the linear parameter /spl delta/. In this paper, an alternative measure to AU for quantifying ambiguity of belief functions is proposed. This measure, called Ambiguity Measure (AM), besides satisfying all the requirements for general measures also overcomes some of the shortcomings of the AU measure. Indeed, AM overcomes the limitations of AU by: 1) minimizing complexity for minimum number of focal points; 2) allowing for sensitivity changes in evidence; and 3) better distinguishing discord and nonspecificity. Moreover, AM is a special case of TU/sub 1/ that does not need the parameter /spl delta/.     

A NOTE ON UNCERTAINTY,DEMPSTER RULE OF COMBINATION,AND CONFLICT

Abstract

The relationship between uncertainty and the Dempster rule of combination in Dempster-Shafer theory is discussed. In particular, a condition is presented that is sufficient to guarantee that uncertainty does not increase by using the rule. A new definition of conflict of pieces of evidence is proposed and discussed.     

Multi-sensor recognition of electronic components

The Dempster-Shafer theory and the convex Bayesian theory have recently been proposed as alternatives to the (strict) Bayesian theory in the field of reasoning with uncertainty. These relatively new formalisms claim that missing information in the probabilistic model of a process not necessarily disables uncertainty reasoning. However, this paper shows that this does not apply to processes where the reasoning is part of a decision-making process, such as object recognition. In these cases, a complete probabilistic model is required and can be obtained by estimating missing probabilistic information. An examplary approach towards the estimation of uncertain probabilistic information is described in this paper for a multi-sensor system for recognition of electronic components on printed circuit boards. Received: 21 June 1998 / Accepted: 23 May 2000     

基于证据距离和不确定度的冲突数据融合算法

Dempster-Shafer证据理论广泛应用于信息融合的许多领域。但是,当使用证据理论对高度冲突的数据进行融合时,此时会出现有违常理的结果。为了解决冲突数据融合的问题,提出了一种基于证据距离和不确定度的冲突数据融合方法。通过证据距离计算证据之间的相对距离,将证据分为两种类别：可信证据和不可信证据。再应用新提出的信度熵对证据不确定度进行度量,对每个证据分配相应的权重,再根据权重对每个证据的基本信度值进行修正,再运用Dempster融合规则对修正后的信度进行组合得到最终全局信度值。通过算例实验,与其他几种经典的数据融合算法进行对比,仿真结果证明算法能够有效地解决数据冲突的问题。     

基于证据理论的小波域多特征医学图像融合

针对多源医学图像融合过程中融合权值选择的不确定性,根据DS证据理论,采用证据理论中的基本概率分配函数来描述判决结果的不确定性。利用图像的区域方差、区域能量、区域信息熵三个特征,然后对特征进行归一化,将各个特征值作为基本概率分配的依据,在小波域内对高频分量采用基于DS证据理论的多特征融合规则进行图像融合。利用拉普拉斯能量,在小波域内对低频分量采用拉普拉斯能量自适应融合规则。实验结果表示：所提算法综合了多个特征的优势,降低了融合过程中的不确定性,较大程度地保留了图像信息。     

决策建模支持中的不确定性分析

分析决策建模的简化过程，认为模型的建立都是在一系列模型假设的基础上，模型假设的设定不同，模型的结构也就不同，将模型结构的不确定怀视为模型假设命题的不确定性。将Ｄｅｍｐｓｔｅｒ－Ｓｈａｆｅｒ证据理论用于描述建模中的不确定性，讨论了扩张与限制运算，提出一种不确定性传播方法。     

A comprehensive comparison between generalized incidence calculus and the Dempster-Shafer theory of evidence

Dealing with uncertainty problems in intelligent systems has attracted a lot of attention in the AI community. Quite a few techniques have been proposed. Among them, the Dempster-Shafer theory of evidence (DS theory) has been widely appreciated. In DS theory, Dempster's combination rule plays a major role. However, it has been pointed out that the application domains of the rule are rather limited and the application of the theory sometimes gives unexpected results. We have previously explored the problem with Dempster's combination rule and proposed an alternative combination mechanism in generalized incidence calculus. In this paper we give a comprehensive comparison between generalized incidence calculus and the Dempster-Shafer theory of evidence. We first prove that these two theories have the same ability in representing evidence and combining DS-independent evidence. We then show that the new approach can deal with some dependent situations while Dempster's combination rule cannot. Various examples in the paper show the ways of using generalized incidence calculus in expert systems.