Joint Propagation and Exploitation of Probabilistic and Possibilistic Information in Risk Assessment

Random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk. While random variability can be represented by probability distribution functions, imprecision (or partial ignorance) is better accounted for by possibility distributions (or families of probability distributions). Because practical situations of risk computation often involve both types of uncertainty, methods are needed to combine these two modes of uncertainty representation in the propagation step. A hybrid method is presented here, which jointly propagates probabilistic and possibilistic uncertainty. It produces results in the form of a random fuzzy interval. This paper focuses on how to properly summarize this kind of information; and how to address questions pertaining to the potential violation of some tolerance threshold. While exploitation procedures proposed previously entertain a confusion between variability and imprecision, thus yielding overly conservative results, a new approach is proposed, based on the theory of evidence, and is illustrated using synthetic examples.     

Practical representations of incomplete probabilistic knowledge

The compact representation of incomplete probabilistic knowledge which can be encountered in risk evaluation problems, for instance in environmental studies is considered. Various kinds of knowledge are considered such as expert opinions about characteristics of distributions or poor statistical information. The approach is based on probability families encoded by possibility distributions and belief functions. In each case, a technique for representing the available imprecise probabilistic information faithfully is proposed, using different uncertainty frameworks, such as possibility theory, probability theory, and belief functions, etc. Moreover the use of probability-possibility transformations enables confidence intervals to be encompassed by cuts of possibility distributions, thus making the representation stronger. The respective appropriateness of pairs of cumulative distributions, continuous possibility distributions or discrete random sets for representing information about the mean value, the mode, the median and other fractiles of ill-known probability distributions is discussed in detail.     

Stochastic dynamics simulation with generalized interval probability

Two types of uncertainties are generally recognized in modelling and simulation, including variability caused by inherent randomness and incertitude due to the lack of perfect knowledge. In this paper, a generalized interval-probability theory is used to model both uncertainty components simultaneously, where epistemic uncertainty is quantified by generalized interval in addition to probability measure. Conditioning, independence, and Markovian probabilities are uniquely defined in generalized interval probability such that its probabilistic calculus resembles that in the classical probability theory. A path-integral approach can be taken to solve the interval Fokker–Planck equation for diffusion processes. A Krylov subspace projection method is proposed to solve the interval master equation for jump processes. Thus, the time evolution of both uncertainty components can be simulated simultaneously, which provides the lower and upper bound information of evolving probability distributions as an alternative to the traditional sensitivity analysis.     

Fuzzy probabilistic approximation spaces and their information measures

Rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information. By introducing probability into fuzzy approximation space, a theory about fuzzy probabilistic approximation spaces is proposed in this paper, which combines three types of uncertainty: probability, fuzziness, and roughness into a rough set model. We introduce Shannon's entropy to measure information quantity implied in a Pawlak's approximation space, and then present a novel representation of Shannon's entropy with a relation matrix. Based on the modified formulas, some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space, respectively. As a result, uniform representations of approximation spaces and their information measures are formed with this work.     

Sensitivity analysis of structural response uncertainty propagation using evidence theory

Sensitivity analysis for the quantified uncertainty in evidence theory is developed. In reliability quantification, classical probabilistic analysis has been a popular approach in many engineering disciplines. However, when we cannot obtain sufficient data to construct probability distributions in a large-complex system, the classical probability methodology may not be appropriate to quantify the uncertainty. Evidence theory, also called Dempster–Shafer Theory, has the potential to quantify aleatory (random) and epistemic (subjective) uncertainties because it can directly handle insufficient data and incomplete knowledge situations. In this paper, interval information is assumed for the best representation of imprecise information, and the sensitivity analysis of plausibility in evidence theory is analytically derived with respect to expert opinions and structural parameters. The results from the sensitivity analysis are expected to be very useful in finding the major contributors for quantified uncertainty and also in redesigning the structural system for risk minimization.     

Conversion of probabilistic information into fuzzy sets for engineering decision analysis

To carry out seismic hazard analysis in the framework of fuzzy set theory, it may become necessary to convert probabilistic information regarding some of the variables into triangular or trapezoidal fuzzy sets. In this paper, three approaches for converting probabilistic information, represented by a probability distribution, into an equivalent triangular or trapezoidal fuzzy set are discussed. In all the three approaches, the probability distribution is first converted into a probabilistic fuzzy set, which is then converted into the equivalent triangular or trapezoidal fuzzy set. The first approach is based on the method of least-square curve fitting, the second approach is based on the conservation of uncertainty (represented by the entropy) associated with the probabilistic fuzzy set in a mean square sense, and the third approach is based on the minimisation of Hausdorff distance (HD) between the probabilistic and the equivalent fuzzy sets. The effectiveness of these approaches in preserving the entropy as well as in preserving the elements of the fuzzy set and their corresponding grades of membership are also discussed with the help of a numerical example of obtaining equivalent fuzzy set for peak ground acceleration. It is found that the approach based on minimisation of Hausdorff distance provides a simple and efficient way for converting the probabilistic information into an equivalent fuzzy set.     

FUZZY MEASURES AND FUZZY INTEGRALS: AN OVERVIEW

This paper provides an overview of fuzzy measures, fuzzy integration theories and Choquet's capacity theory. Belief, plausibility, and possibility measures are characterized as Choquet capacities and as fuzzy measures. The relationship between possibility measures, fuzzy sets, and approximate reasoning is established. Recent results on extensions of fuzzy measures, structural characteristics of fuzzy measures, and convergence of function sequences on fuzzy measure spaces are presented. Fuzzy measure integration concepts due to Sugeno and Choquet and their applications are discussed. An extensive list of references to the literature of fuzzy measures, Sugeno and Choquet integrals, fuzzy probabilities, fuzzy random variables, probabilistic sets, and random sets is provided. Applicalions discussed or referenced include information fusion, information retrieval, approximate reasoning, artificial intelligence, uncertainty theory, and control and decision theory.     

Diagnostic reasoning and medical decision-making with fuzzy influence diagrams

Influence diagrams have been widely used as knowledge bases in medical informatics and many applied domains. In conventional influence diagrams, the numerical models of uncertainty are probability distributions associated with chance nodes and value tables for value nodes. However, when incomplete knowledge or linguistic vagueness is involved in the reasoning systems, the suitability of probability distributions is questioned. This study intends to propose an alternative numerical model for influence diagrams, possibility distributions, which extend influence diagrams into fuzzy influence diagrams. In fuzzy influence diagrams, each chance node and value node is associated with a possibility distribution which expresses the uncertain features of the node. This study also develops a simulation algorithm and a fuzzy programming model for diagnosis and optimal decision in medical settings.     

Construction of families of probability boxes and corresponding membership functions at different fractiles

Uncertainty comes in many forms in the real world and is an unavoidable component of human life. Generally, two types of uncertainties arise, namely, aleatory and epistemic uncertainty. Probability is a well established mathematical tool to handle aleatory uncertainty and fuzzy set theory is a tool to handle epistemic uncertainty. However, in certain situations, parameters of probability distributions may be tainted with epistemic uncertainty; and so, representation of parameters of probability distributions may be treated as fuzzy numbers (may be of different shapes). A probability box (P‐box) can be constructed when parameters are not precisely known. In this paper, an attempt has been made to construct families of P‐boxes when parameters of probability distributions are bell shaped or normal fuzzy numbers; and from these families of P‐boxes, membership functions are generated at different fractiles for different alpha levels.     

Testing Preorders for Probabilistic Processes

We present a testing preorder for probabilistic processes based on a quantification of the probability with which processes pass tests. The theory enjoys close connections with the classical testing theory of De Nicola and Hennessy in that whenever a process passes a test with probability 1 (respectively some nonzero probability) in our setting, then the process must (respectively may) pass the test in the classical theory. We also develop an alternative characterization of the probabilistic testing preorders that takes the form of a mapping from probabilistic traces to the interval [0, 1], where a probabilistic trace is an alternating sequence of actions and probability distributions over actions. Finally, we give proof techniques, derived from the alternative characterizations, for establishing preorder relationships between probabilistic processes. The utility of these techniques is demonstrated by means of some simple examples.     

Elicitation, assessment, and pooling of expert judgments usingpossibility theory

The problem of modeling expert knowledge about numerical parameters in the field of reliability is reconsidered in the framework of possibility theory. Usually expert opinions about quantities such as failure rates are modeled, assessed, and pooled in the setting of probability theory. This approach does not seem to always be natural since probabilistic information looks too rich to be currently supplied by individuals. Indeed, information supplied by individuals is often incomplete, imprecise rather than tainted with randomness. Moreover, the probabilistic framework looks somewhat restrictive to express the variety of possible pooling modes. In this paper, the authors formulate a model of expert opinion by means of possibility distributions that are thought to better reflect the imprecision pervading expert judgments. They are weak substitutes to unreachable subjective probabilities. Assessment evaluation is carried out in terms of calibration and level of precision, respectively, measured by membership grades and fuzzy cardinality indexes. Finally, drawing from previous works on data fusion using possibility theory, the authors present various pooling modes with their formal model under various assumptions concerning the experts. A comparative experiment between two computerized systems for expert opinion analysis has been carried out, and its results are presented in this paper     

Probability‐Generated Aggregators

The paper addresses a relation between logical reasoning and probability and presents probability‐generated aggregators. The obtained aggregators implement probability distributions for specification of generator functions; as it was proven in the paper, such implementation is always possible. In the paper, the relation between neutral element of the probabilistic uninorm and parameters of the underlying probability distribution is demonstrated, and a method for specification of the probabilistic uninorm, and thus—of the probability distribution using t‐norm and t‐conorm—is constructed. In addition, the obtained probabilistic uninorm and probabilistic absorbing norm or nullnorm are briefly considered as algebraic operations on the open unit interval. In is demonstrated, that, in general, the obtained algebra is nondistributive and depends on the distributions, which are used for generating probabilistic uninorm and absorbing norm. The obtained results bridge several gaps between fuzzy and probabilistic logics and provide a basis both for theoretical studies in the field and for practical techniques of digital/analog schemes synthesis and analysis.     

A review of: “FUZZY SETS,UNCERTAINTY, AND INFORMATION,” by George J. Klir and Tina A. Folger. Prentice Hall,Englewood Cliffs,N.J., 1988, XI+ 355 pp.

Possibility distribution introduced by Zadeh ["Fuzzy sets as a basis for a theory of possibility theory", Fuzzy Sets Syst. 1 (1978) 3-28] in his introductory paper of possibility theory assumes a normal distribution, in the sense that it supposes the existence of at least one element s 0 of the universe of discourse U , for which the distribution ~ is fully compatible with the context of interest: ~ (s 0 )=1. However, when such element does no longer exist, it leads to a subnormal possibility distribution. This situation may arise from incomplete data, inconsistent statements, or contradictory beliefs. To deal with such case, many authors like Yager ["On the relationships of methods of aggregation evidence in expert systems", Cybern. Syst. , 16 (1985) 1-21; "A modification of the certainty measure to handle subnormal distributions", Fuzzy Sets Syst. , 20 (1986) 317-324], Dubois and Prade ["An alternative approach to the handling of subnormal possibility distributions--A critical comment on a proposal of Yager", Fuzzy Sets Syst. , 24 (1987) 123-126] have put forward some proposals in order to keep track of the consistency of the basic axioms attached to possibility and necessity measures. In this paper, the proposals are reviewed in the light of new results regarding some appealing criteria. Particularly, when subnormal distribution and normal distribution are encountered in the same level, intuitively, two approaches are possible: Either the subnormal distributions are risen up to a normal distribution level, or the normal ones are flatted down to agree with the normal ones. In both cases there is a sort of gaining or losing information. We review some of the proposal solutions. The flatting approach is mainly related to fuzzy arithmetic calculus while the rising effect is motivated by Dempster-Shafer theory of evidence and its normalization paradigm. The two approaches will also be investigated with respect to some appealing criteria like preference preservation, distance minimization, entropy, minimum/maximum specificity, and, further, particular interest is focused on information based uncertainty preservation. Later on, the proposals are discussed according to the f -certainty qualification where the greatest value h of the subnormal distribution is understood as a degree of certainty that must be attached to the resulting normal distribution.     

On weighted lower and upper possibilistic means and variances of fuzzy numbers and its application in decision

In this paper, we define the weighted lower and upper possibilistic variances and covariances of fuzzy numbers. We also obtain their many properties similar to variance and covariance in probability theory. On the basis of the weighted lower and upper possibilistic means and variances, we present two new possibilistic portfolio selection models with tolerated risk level and holdings of assets constraints. The conventional probabilistic mean–variance model can be transformed to a linear programming problem under possibility distributions. Finally, an estimation method of possibility distribution is offered and a real example for portfolio selection problem is given to illustrate the usability of the approach and the effectiveness of our methods.     

Delay-Distribution-Dependent Stability and Stabilization of T–S Fuzzy Systems With Probabilistic Interval Delay

In this paper, we are concerned with the problem of stability analysis and stabilization control design for Takagi–Sugeno (T–S) fuzzy systems with probabilistic interval delay. By employing the information of probability distribution of the time delay, the original system is transformed into a T–S fuzzy model with stochastic parameter matrices. Based on the new type of T–S fuzzy model, the delay-distribution-dependent criteria for the mean-square exponential stability of the considered systems are derived by using the Lyapunov–Krasovskii functional method, parallel distributed compensation approach, and the convexity of some matrix equations. The solvability of the derived criteria depends not only on the size of the delay but also on the probability distribution of the delay taking values in some intervals. The revisions of the main criteria in this paper can also be used to deal with the case when only the information of variation range of the delay is considered. It is shown by practical examples that our method can lead to very less conservative results than those by other existing methods.      

Some methods for strategic decision‐making problems with immediate probabilities in Pythagorean fuzzy environment

In this article, a new decision‐making model with probabilistic information and using the concept of immediate probabilities has been developed to aggregate the information under the Pythagorean fuzzy set environment. In it, the existing probabilities have been modified by introducing the attitudinal character of the decision maker by using an ordered weighted average operator. Based on it, we have developed some new probabilistic aggregation operator with Pythagorean fuzzy information, namely probabilistic Pythagorean fuzzy weighted average operator, immediate probability Pythagorean fuzzy ordered weighted average operator, probabilistic Pythagorean fuzzy ordered weighted average, probabilistic Pythagorean fuzzy weighted geometric operator, immediate probability Pythagorean fuzzy ordered weighted geometric operator, probabilistic Pythagorean fuzzy ordered weighted geometric, etc. Furthermore, we extended these operators by taking interval‐valued Pythagorean fuzzy information and developed their corresponding aggregation operators. Few properties of these operators have also been investigated. Finally, an illustrative example about the selection of the optimal production strategy has been given to show the utility of the developed method.     

基于调和犹豫模糊信息的多属性决策方法

实践中发现,犹豫模糊信息和概率犹豫模糊信息在计算过程中存在着计算繁琐、与数量运算规则不相容等问题.对此,提出一套基于调和犹豫模糊元的解决方法.通过定义调和犹豫模糊元为一组概率分布相同的概率犹豫模糊元,在犹豫模糊信息和概率犹豫模糊信息之间架起一座桥梁,将它们纳入统一处理框架.在此基础上,定义调和犹豫模糊信息的基本运算规则、信息集成算子、距离测度和混合熵测度,构建基于调和犹豫模糊信息的多属性决策方法,并将其应用于陆军合成旅指挥控制能力评估.数值实验表明:调和犹豫模糊决策理论克服了已有理论的缺陷,具有计算量可控、易于编程实现、与数量运算规则相容等优势.     

基于模糊集和DS证据理论的信息安全风险评估方法*

在信息安全风险评估过程中,存在着很多不确定和模糊的因素,针对专家评价意见的不确定性和主观性问题,提出了一种将模糊集理论与DS证据理论进行结合的的风险评估方法。首先,根据信息安全风险评估的流程和要素,建立风险评估指标体系,确定风险影响因素;其次,通过高斯隶属度函数,求出专家对各影响因素的评价意见隶属于各个不同评价等级的程度;再次,将其作为DS理论所需的基本概率分配,引入基于矩阵分析和权值分配的融合算法综合多位专家的评价意见;最后,结合贝叶斯网络模型的推理算法,得出被测信息系统所面临的风险大小,并对其进行分析。结果显示,将模糊集理论和DS证据理论应用到传统贝叶斯网络风险评估的方法,在一定程度上能够提高评估结果的客观性。     

概率犹豫模糊决策理论与方法综述

概率犹豫模糊集是在犹豫模糊集的基础上为每个隶属度添加与之相对应的概率值.与犹豫模糊集相比,它可以更加准确和全面地表达专家给出的初始决策信息,因此,基于概率犹豫模糊集的决策理论与方法更加可靠且符合实际.这里对概率犹豫模糊决策理论与方法进行综述.首先介绍其发展过程;然后分别对它的信息融合理论、偏好关系理论以及决策方法等进行阐述;最后展望了概率犹豫模糊决策理论与方法的未来研究方向.