规则基专家系统概率区间推理模式研究

知识不确定性的二值表示法已经成为专家系统中不确定推理模型设计的重要出发点,本文基于不确定信息的概率区间表示,给出了专家系统中不确定信息的组合、积累和传播过程中概率区间的计算模式,并结合一个实例进行了说明。最后,我们对概率区间计算理论的一般情况进行了讨论。     

Representation of analysis results involving aleatory and epistemic uncertainty

Procedures are described for the representation of results in analyses that involve both aleatory uncertainty and epistemic uncertainty, with aleatory uncertainty deriving from an inherent randomness in the behaviour of the system under study and epistemic uncertainty deriving from a lack of knowledge about the appropriate values to use for quantities that are assumed to have fixed but poorly known values in the context of a specific study. Aleatory uncertainty is usually represented with probability and leads to cumulative distribution functions (CDFs) or complementary CDFs (CCDFs) for analysis results of interest. Several mathematical structures are available for the representation of epistemic uncertainty, including interval analysis, possibility theory, evidence theory and probability theory. In the presence of epistemic uncertainty, there is not a single CDF or CCDF for a given analysis result. Rather, there is a family of CDFs and a corresponding family of CCDFs that derive from epistemic uncertainty and have an uncertainty structure that derives from the particular uncertainty structure (e.g. interval analysis, possibility theory, evidence theory or probability theory) used to represent epistemic uncertainty. Graphical formats for the representation of epistemic uncertainty in families of CDFs and CCDFs are investigated and presented for the indicated characterisations of epistemic uncertainty.     

Adaptive sampling for generalized probabilistic roadmaps

In this paper,an adaptive sampling strategy is presented for the generalized sampling-based motion planner,generalized probabilistic roadmap (GPRM).These planners are designed to account for stochastic...     

A note on generalized hold functions

In this note, we study the generalized hold functions for use of digital control of continuous-time systems. First we define and solve the problem of simultaneous stabilization and minimization. It is shown that every plant with a strictly proper transfer function is simultaneously stabilizable and minimizable, provided that the sampling interval is small enough. We also consider robust designs of generalized hold functions. A condition is given for the stability of a system using generalized hold functions in the presence of uncertainty.     

A PRINCIPLE OF UNCERTAINTY AND INFORMATION INVARIANCE*

The paper introduces a new principle, referred to as the principle of uncertainty and information invariance, for making transformations between different mathematical theories by which situations under uncertainty can be characterized. This principle requires that the amount of uncertainty (and related information) be preserved under these transformations. The principle is developed in sufficient details for transformations between probability theory and possibility theory under interval, log-interval and ordinal scales. Its broader use is discussed only in general terms and illustrated by an example.     

A total uncertainty measure for D numbers based on belief intervals

Uncertainty quantification is very important in many applications. As a generalization of Dempster-Shafer theory, the theory of D numbers is a new theoretical framework for uncertainty reasoning. Measuring the uncertainty of knowledge or information represented by D numbers is an unsolved issue in that theory. In this paper, inspired by distance-based uncertainty measures for Dempster-Shafer theory, a total uncertainty measure for a D number is proposed based on its belief intervals. The proposed total uncertainty measure can simultaneously capture the discord, and nonspecificity, and nonexclusiveness involved in D numbers. And some basic properties of this total uncertainty measure, including range, monotonicity, generalized set consistency, are also presented. At last, an illustrative application about feature evaluation is given to verify the effectiveness of the proposed uncertainty measure.     

Collision Avoidance by Using Space-Time Representations of Motion Processes

This paper handles the problem of collision avoidance in a multi-robot environment. To solve this problem, the motion processes of the mobile robots are modelled in space-time. Since the robots are autonomous and communication is non-deterministic, there is temporal uncertainty in addition to spatial uncertainty. The paper presents a method to model both uncertainty components in a homogeneous way. It is shown, that it is not sufficient to guarantee a spatial security distance between the robots. Distances in space-time and space-time vectors must be considered. The main result of this paper is a straightforward and efficient solution to the problem of collision avoidance between up to three mobile robots by applying a space-time displacement vector. The solution is based on space-time, which is a helpful view onto our world in relativity theory and quantum physics. Space-time methods are also very valuable in Robotics, especially for problems in dynamic environments and for motion coordination of mobile robots. Practical experiments with up to two robots, and simulations of up to three robots have been performed and are reported.     

RECONSTRUCTABILITY ANALYSIS USING PROBABILITY INTERVALS

It is proposed that probability intervals be used in reconstructability analysis. A probability interval is a subinterval of the real interval [0,1]. Regarded as “interval-valued probabilities”, these intervals generalize real-valued probabilities and arise naturally in many situations. They may represent confidence intervals resulting from sampling; imprecisely stated subjective probabilities; known linear equality or inequality constraints; etc. Thus, probability intervals are sometimes a more realistic characterization of uncertainty than are real-valued probabilities. Furthermore, the problem of inconsistency can often be avoided by their use

Although the utility of interval-valued probability distributions for the identification problem is emphasized, a reconstruction technique is also developed. This reconstruction method employs a metric distance for interval distributions that is monotonic with respect to model refinement.     

An extended probabilistic method for reliability analysis under mixed aleatory and epistemic uncertainties with flexible intervals

The reliability analysis approach based on combined probability and evidence theory is studied in this paper to address the reliability analysis problem involving both aleatory uncertainties and epistemic uncertainties with flexible intervals (the interval bounds are either fixed or variable as functions of other independent variables). In the standard mathematical formulation of reliability analysis under mixed uncertainties with combined probability and evidence theory, the key is to calculate the failure probability of the upper and lower limits of the system response function as the epistemic uncertainties vary in each focal element. Based on measure theory, in this paper it is proved that the aforementioned upper and lower limits of the system response function are measurable under certain circumstances (the system response function is continuous and the flexible interval bounds satisfy certain conditions), which accordingly can be treated as random variables. Thus the reliability analysis of the system response under mixed uncertainties can be directly treated as probability calculation problems and solved by existing well-developed and efficient probabilistic methods. In this paper the popular probabilistic reliability analysis method FORM (First Order Reliability Method) is taken as an example to illustrate how to extend it to solve the reliability analysis problem in the mixed uncertainty situation. The efficacy of the proposed method is demonstrated with two numerical examples and one practical satellite conceptual design problem.     

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.     

考虑认知不确定性的模型确认贝叶斯因子法*

针对认知不确定性条件下计算机建模仿真所面临的模型确认问题,提出一种结合了二阶概率法与区间数排序的改进贝叶斯模型确认方法。该方法首先采用二阶概率法对模型的不确定性进行量化,量化结果被做为先验模型输出,再基于实验数据对模型输出的先验概率密度进行贝叶斯更新,最后通过区间数排序的方式对比模型输出的后验和先验概率密度。由此所得的贝叶斯因子能够在模型存在认知不确定性的情况下为模型确认提供可信的结果。算例分析结果显示了本文方法的合理性。     

Risk-informed decision-making in the presence of epistemic uncertainty

An important issue in risk analysis is the distinction between epistemic and aleatory uncertainties. In this paper, the use of distinct representation formats for aleatory and epistemic uncertainties is advocated, the latter being modelled by sets of possible values. Modern uncertainty theories based on convex sets of probabilities are known to be instrumental for hybrid representations where aleatory and epistemic components of uncertainty remain distinct. Simple uncertainty representation techniques based on fuzzy intervals and p-boxes are used in practice. This paper outlines a risk analysis methodology from elicitation of knowledge about parameters to decision. It proposes an elicitation methodology where the chosen representation format depends on the nature and the amount of available information. Uncertainty propagation methods then blend Monte Carlo simulation and interval analysis techniques. Nevertheless, results provided by these techniques, often in terms of probability intervals, may be too complex to interpret for a decision-maker and we, therefore, propose to compute a unique indicator of the likelihood of risk, called confidence index. It explicitly accounts for the decision-maker's attitude in the face of ambiguity. This step takes place at the end of the risk analysis process, when no further collection of evidence is possible that might reduce the ambiguity due to epistemic uncertainty. This last feature stands in contrast with the Bayesian methodology, where epistemic uncertainties on input parameters are modelled by single subjective probabilities at the beginning of the risk analysis process.     

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.     

L2增益约束下跳变系统鲁棒控制

不确定跳变系统的L2增益条件与系统可镇定性和鲁棒性密切相关,本文阐述利用依赖于模态的状态反馈鲁棒控制律,以实现闭环系统输入输出L2增益约束的要求,一方面用凸多面体界定跳变系统各模态间的跳变转移率的变化,另一方面,用L2增益界定对象状态方程中时变不确定参数,在一定概率空间下,获得的反馈控制律对不确定跳变概率和时变参数具有鲁棒性.反馈控制律的设计利用线性矩阵不等式技术,通过凸优化计算易于实现,最后给出了数值计算示例.     

A new algorithm for importance analysis of the inputs with distribution parameter uncertainty

Importance analysis is aimed at finding the contributions by the inputs to the uncertainty in a model output. For structural systems involving inputs with distribution parameter uncertainty, the contributions by the inputs to the output uncertainty are governed by both the variability and parameter uncertainty in their probability distributions. A natural and consistent way to arrive at importance analysis results in such cases would be a three-loop nested Monte Carlo (MC) sampling strategy, in which the parameters are sampled in the outer loop and the inputs are sampled in the inner nested double-loop. However, the computational effort of this procedure is often prohibitive for engineering problem. This paper, therefore, proposes a newly efficient algorithm for importance analysis of the inputs in the presence of parameter uncertainty. By introducing a ‘surrogate sampling probability density function (SS-PDF)’ and incorporating the single-loop MC theory into the computation, the proposed algorithm can reduce the original three-loop nested MC computation into a single-loop one in terms of model evaluation, which requires substantially less computational effort. Methods for choosing proper SS-PDF are also discussed in the paper. The efficiency and robustness of the proposed algorithm have been demonstrated by results of several examples.     

基于混合布谷鸟算法的不确定手术调度模型求解

针对应急救援情境下手术调度中存在不确定性因素导致无法获得精确的手术时间和结束期的问题,设计了一种灰色调度模型和求解该问题的混合灰色布谷鸟算法。首先引入三参数和四参数区间灰数来描述不确定手术时间和不确定结束期,并定义了可能性测度和必然性测度,提出了拖期可信度指标用于度量手术发生拖期的概率;然后建立了以最小化手术平均拖期可信度为目标的灰色混合整数规划模型,提出了一种混合灰色布谷鸟算法的求解方法,并以规模为6（3）×3的经典算例为例进行仿真测试。实验表明该算法能很好地解决问题,比基本布谷鸟求解算法有更好的性能。     

A robust critical path in an environment with hybrid uncertainty

In this article, we consider the project critical path problem in an environment with hybrid uncertainty. In this environment, the duration of activities are considered as random fuzzy variables that have probability and fuzzy natures, simultaneously. To obtain a robust critical path with this kind of uncertainty a chance constraints programming model is used. This model is converted to a deterministic model in two stages. In the first stage, the uncertain model is converted to a model with interval parameters by alpha-cut method and distribution function concepts. In the second stage, the interval model is converted to a deterministic model by robust optimization and min-max regret criterion and ultimately a genetic algorithm with a proposed exact algorithm are applied to solve the final model. Finally, some numerical examples are given to show the efficiency of the solution procedure.     

Application of a generalized block pulse function to a scaled system

A generalized block pulse function (GBPF) is employed to solve a functional differential equation. The operational matrix for integration and the functional operational matrix of the GBPF are introduced to solve the state equation in order to simplify the calculation procedure. The greatest advantage of using a GBPF is that the time interval of calculation can be adjusted arbitrarily. A small time interval is chosen for a steep change of state variable with time and a large time interval is chosen for a flat change of state variable with time. Therefore, the number of expansion coefficients is greatly reduced and the computer time is also minimized when using GBPF compared with the conventional BPF. An illustrative example is given. It is shown that computational results are more accurate for a steep change of state variable using a GBPF rather than a BPF.     

A random set approach to the optimization of uncertain structures

The single-objective optimization of structures, whose parameters are assigned as fuzzy numbers or fuzzy relations, is presented in this paper as a particular case of the random set theory and evidence theory approach to uncertainty. Some basic concepts concerning these theories are reviewed and the relationships among interval analysis, convex modeling, possibility theory and probability theory are pointed out. In this context a frequentistic view of fuzzy sets makes sense and it is possible to calculate bounds on the probability that the solution satisfies the constraints. Some special but useful cases illustrate in detail the meaning of the approach proposed and its links with a recent formulation conceived within the context of convex modeling. Some theorems allow a very efficient computational procedure to be set up in many real design situations. Two numerical examples illustrate the model presented.     

Minimum cost consensus modelling under various linear uncertain-constrained scenarios

Group decision-making combined with uncertainty theory is verified as a more conclusive theory, by building a bridge between deterministic and indeterministic group decision-making in this paper. Due to the absence of sufficient historical data, reliability of decisions are mainly determined by experts rather than some prior probability distributions, easily leading to the problem of subjectivity. Thus, belief degree and uncertainty distribution are used in this paper to fit individual preferences, and five scenarios of uncertain chance-constrained minimum cost consensus models are further discussed from the perspectives of the moderator, individual decision-makers and non-cooperators. Through deduction, reaching conditions for consensus and analytic formulas of the minimum total cost are both theoretically given. Finally, with the application in carbon quota negotiation, the proposed models are demonstrated as a further extension of the crisp number or interval preference-based minimum cost consensus models. In other words, the basic conclusions of the traditional models are some special cases of the uncertain minimum cost consensus models under different belief degrees.