Dempster-Shafer Structures Research Articles

МЕТОД ВЕРОЯТНОСТНОГО АНАЛИЗА НАДЕЖНОСТИ УЗЛОВ СТАЛЬНЫХ ФЕРМ НА ОСНОВЕ Р-БЛОКОВ

The article presents a new approach to probabilistic analysis of reliability for steel trusses’ joints based on p-boxes. The p-box is an area formed by two boundary cumulative distribution functions (CDFs) inside which there is a real cumulative distribution function (which is unknown due to the epistemic uncertainty of statistical data). The research demonstrates an algorithm for arithmetic operations with p-boxes by discretizing them into the Dempster-Shafer structures. Information about the failure probability of trusses’ joints is necessary to consider the truss reliability as a mechanical serial system in terms of reliability theory. The use of p-boxes makes it possible to obtain a more cautious model of a random variable due to the absence of the need to adopt some statistical hypotheses about the type of probability distribution of various random variables in mathematical models.

A Novel Uncertainty Quantification Framework for PF and OPF Considering Nonlinear Correlated Power Injections With Limited Information

This paper proposes a new uncertainty quantification framework for power flow (PF) and optimal power flow (OPF) considering the nonlinear correlations of uncertain power injections with limited information. The D-Vine copula is leveraged to capture the nonlinear correlations among uncertain power injections from historical data. This is further integrated with the evidence theory and the reformulated quadratic affine form to obtain PF and OPF results. The D-Vine copula and evidence theory allow one to effectively eliminate the explosive-growth joint focal elements of large-scale power system with large numbers of partial known uncertain power injections, leading to significant reduction of computing time. The reformulated quadratic affine form aims at characterizing the PF and OPF outputs with partially known uncertain power injections in a simple form based on Dempster-Shafer structure, yielding further computational efficiency improvement without the loss of accuracy. Comparison results with other alternatives show that the proposed framework leads to more accurate PF and OPF outcomes while achieving high computational efficiency for large-scale power systems with large numbers of nonlinear correlated power injections in presence of limited information.

РАЗРАБОТКА УТОЧНЕННОГО Р-БЛОКА КАК МОДЕЛИ СЛУЧАЙНОЙ ВЕЛИЧИНЫ В ЗАДАЧА Х АНАЛИЗА НАДЕЖНОСТИ СТРОИТЕЛЬНЫХ КОНСТРУКЦИЙ

The article describes a new type of the p-box (probability box) for random variables modeling in problems of structural reliability analysis. The advantage of the presented random variable model is that it is based on confidence estimates of statistical parameters without the need to confirm the hypothesis about the type of probability distribution function. The numerical example shows that the introduction of additional information in the form of a median’s confidence interval to the p-boxes based on the Chebyshev’s inequality is leading to reducing the area between the boundary distribution functions of the random variable. This makes it possible to reduce the level of uncertainty during further arithmetical operations with p-boxes in problems of structural reliability analysis, which is clearly visible after the discretization of the p-boxes into the Dempster-Shafer structure.

Belief Functions on Distributive Lattices

The Dempster-Shafer theory of belief functions is an important approach to deal with uncertainty in AI.In the theory, belief functions are defined on Boolean algebras of events. In many applications of belief functions in real world problems, however, the objects that we manipulateis no more a Boolean algebra but a distributive lattice. In this paper, we extend the Dempster-Shafer theory to the setting of distributive lattices, which has a mathematical theory as attractive as in that of Boolean algebras.Moreover, we apply this more general theory to a simple epistemic logic the first-degree-entailment fragment of relevance logic R, provide a sound and complete axiomatization for reasoning about belief functions for this logic and show that the complexity of the satisfiability problem of a belief formula with respect to the class of the corresponding Dempster-Shafer structures is NP-complete.

Modeling heterogeneous multi-attribute emergency decision-making with Dempster-Shafer theory

Conventional multi-attribute decision-making (MADM) methods, with regard to different attributes, typically require decision-makers (DMs) to use a unified information expression to evaluate the performance of various alternatives. This does not properly reflect the heterogeneity among attributes. This paper presents a novel heterogeneous multi-attribute decision-making approach based on Dempster-Shafer theory (DST). A human-centric heterogeneous multi-attribute emergency decision-making framework is constructed based on Dempster-Shafer structures. The term “human-centric” is emphasized here as the proposed framework encompasses aspects specific to human judgments, specifications, or evaluations. Considering the differences in DM backgrounds and preferences, our framework incorporates a heterogeneous decision environment. DST allows DMs to express decision information in a manner closely consistent with human cognition. A series of parameters capture the subjective attitudes of DMs in our framework, the embodiments of which are manifold: (1) they reveal the sensitivity of DMs to alternative ranks in the process of basic probability assignment generation from decision information; (2) they reflect the tolerance of DMs to uncertainty in the process of weight calculation from belief entropy; and (3) they facilitate analysis of DMs in the decision process based on belief intervals. The proposed approach is demonstrated on a emergency decision-making application for medical supplier selection in the context of COVID-19.

Intuitionistic fuzzy decision‐making in the framework of Dempster–Shafer structures

Intuitionistic fuzzy decision‐making in the framework of Dempster–Shafer structures

Interval-valued belief entropies for Dempster-Shafer structures.

In practical application problems, the uncertainty of an unknown object is often very difficult to accurately determine, so Yager proposed the interval-valued entropies for Dempster–Shafer structures, which is based on Dempster–Shafer structures and classic Shannon entropy and is an interval entropy model. Based on Dempster–Shafer structures and classic Shannon entropy, the interval uncertainty of an unknown object is determined, which provides reference for theoretical research and provides help for industrial applications. Although the interval-valued entropies for Dempster–Shafer structures can solve the uncertainty interval of an object very efficiently, its application scope is only a traditional probability space. How to extend it to the evidential environment is still an open issue. This paper proposes interval-valued belief entropies for Dempster–Shafer structures, which is an extension of the interval-valued entropies for Dempster–Shafer structures. When the belief entropy degenerates to the classic Shannon entropy, the interval-valued belief entropies for Dempster–Shafer structures will degenerate into the interval-valued entropies for Dempster–Shafer structures. Numerical examples are applied to verify the validity of the interval-valued belief entropies for Dempster–Shafer structures. The experimental results demonstrate that the proposed entropy can obtain the interval uncertainty value of a given uncertain object successfully and make decision effectively.

Rolling element bearing diagnosis based on probability box theory

Feature extraction leads to the loss of statistical information of raw data and ignores the sampling uncertainty and the fluctuations in the signal over time in mechanical fault diagnosis. In this paper, novel modeling methods for mechanical signals based on probability box theory were proposed to solve the above problem. First, the type of random distribution of the bearing signals were analyzed. Then, a Dempster-Shafer structure was obtained to establish a probability box model. To address the identification difficulty of the type of random distribution for the bearing signals, a second probability box model was established based on a vector consisting of features from the bearing signals. If the data are not found to follow a random distribution, a third modeling method based on the definition of probability boxes was proposed. The effectiveness and applicability of the three proposed models were compared with experimental data from rolling element bearings. The combination of probability box theory and mechanical fault diagnosis theory can open up a new research direction for mechanical fault diagnosis.

On combination rule in Dempster–Shafer theory using OWA‐based soft likelihood functions and its applications in environmental impact assessment

Dempster–Shafer theory (DST) was presented as an effective mathematical tool to represent uncertainty. Its significant innovation is to allow the allocation of the belief of mass to sets or intervals, and it becomes a valuable method in the field of decision making and evaluation when accurate information is not available or when knowledge is expressed subjectively by humans. A crucial research issue in DST is the combination of multi-sources of evidence. In this paper, a novel combination rule for Dempster–Shafer structures is developed based on ordered weighted average (OWA)-based soft likelihood functions proposed by Yager. First, the belief intervals, including the belief measures and plausibility measures, of all the hypotheses in the frame of discernment (FOD) are calculated. Second, the representative value of belief interval is defined based on golden rule introduced by Yager. Third, the soft likelihood value of each hypothesis is calculated based on the proposed OWA-based soft likelihood function for belief interval, which can be considered as the combined evidence. The final evaluation results can be employed for practical applications, such as decision making and evaluation. In addition, the improved evidence combination rule is presented which takes into account the weight of evidence. Several illustrative examples are conducted to manifest the use of the developed methods. Finally, an application for environmental impact assessment is given to demonstrate the usefulness of the developed combination rule in DST.

Health Risk Assessment through Incorporation of Dempster–Shafer Structure and Fuzzy Numbers within the Same Framework

It is always most important to accrue knowledge on the features of each and every existing data, information and model parameters in risk assessment. It is observed that most frequently model parameters, data, information are fouled with uncertainty due to lack of precision, deficiency in data, diminutive sample sizes. In such situations, fuzzy set theory or Dempster–Shafer structure (DSS) can be explored to represent uncertainty. More often, both type of uncertainty representation theories coexist in health risk assessment and need to amalgamate. This paper presents two algorithms to combine DSS and fuzzy numbers (FNs) of different shapes.• The algorithms deal with the combination of DSS with fuzzy focal elements of different types and different fuzzy numbers of various types and shapes within the same framework. • Finally, non-cancer human health risk assessment is carried out under this setting. • Results are obtained in fuzzy numbers at different fractiles.The approaches presented here have the ability to be used in any mathematical model which represents real-world problems, in which model parameters are tainted with uncertainty, where representations of uncertain model parameters are DSS and fuzzy numbers with different shapes and types.

Generalized Dempster–Shafer Structures

We review the Dempster–Shafer (D–S) belief structure and discuss the measures of plausibility and belief and other measures that can be associated with this uncertainty modeling structure. We provide a view of the D–S structure as a two-step process, the first is the selection of one of the focal elements and the second is the selection of an object from the chosen focal element. The selection of the focal element is uncertain, the uncertainty is based on a probability distribution. Here, we provide a generalization of this model by allowing a more general type uncertainty, a monotonic set measure, to model the underlying uncertainty associated with the selection of the focal element. We look at form of the plausibility and belief in this more general case. We provide a procedure for uncertain decision making when our uncertainty is modeled using this more general D–S belief structure.

On operations of possibilistic belief structures

This paper studies operations on possibilistic belief structures, which are constructed by basic possibility assignments, or equivalently, maxitive belief structures, as a framework for modeling imprecise possibility distributions. Different from Dempster–Shafer structures, it is shown that there is no one-to-one correspondence between basic possibility assignments and their induced upper and lower possibilities. The operations on the set of basic possibility assignments might not induce the operations on their upper and lower possibilities. An operation on basic possibility assignments is called proper if it is also the operation on their upper and lower possibilities. We introduce set operations and point operations on basic possibility assignments. Set operations include intersection, union, complement, difference, projection, marginalization, cylindrical extension, vacuous extension and Cartesian product operations. Point operations include all binary and unary operations. We show that all point operations are proper. For set operations, it is shown that union, projection, marginalization, cylindrical extension, vacuous extension and Cartesian product operations are proper, while intersection, complement and difference operations are not proper. In addition, we study construction of compatible possibility distributions and show that the lower possibility of a basic possibility assignment can be represented by the infinimum of compatible possibility distributions. We also show that the local computation technique can be applied to the computation problem of possibilistic belief structures.

Interval valued entropies for Dempster–Shafer structures

We first describe the basic ideas associated with the Dempster–Shafer belief structure and note its role in modeling uncertainty that manifests both imprecision and randomness. The inherent imprecision requires the use of intervals in expressing the associated probabilities. The ideas of plausibility and belief are introduced to provide upper and lower bounds on these interval-valued probabilities. We turn to the question of formalizing the entropy for these belief structures. As was the case with the associated probabilities the imprecision mandates the use of an interval-valued formulations for entropy. We suggest both Shannon type and Gini type formulations of interval-valued entropy for D–S belief structures. Here we make use of the classic scalar formulations of the D–S entropy introduced by Hoehle and Yager to provide the bounds for this interval-valued formulation. We discuss a more general type of belief structure in which our underlying uncertainty, while still manifesting imprecision, is based on a measure instead of a probability distribution. We show how to formulate entropy for these measure-based belief structures.

Estimation of the lower and upper bounds on the probability of failure using subset simulation and random set theory

Abstract Random set theory is a general framework which comprises uncertainty in the form of probability boxes, possibility distributions, cumulative distribution functions, Dempster-Shafer structures or intervals; in addition, the dependence between the input variables can be expressed using copulas. In this paper, the lower and upper bounds on the probability of failure are calculated by means of random set theory. In order to accelerate the calculation, a well-known and efficient probability-based reliability method known as subset simulation is employed. This method is especially useful for finding small failure probabilities in both low- and high-dimensional spaces, disjoint failure domains and nonlinear limit state functions. The proposed methodology represents a drastic reduction of the computational labor implied by plain Monte Carlo simulation for problems defined with a mixture of representations for the input variables, while delivering similar results. Numerical examples illustrate the efficiency of the proposed approach.

Dempster–Shafer structure based fuzzy logic system for stochastic modeling

The Dempster–Shafer (D–S) theory of evidence is introduced to improve fuzzy inference under the complex stochastic environment. The Dempster–Shafer based fuzzy set (DFS) is first proposed, together with its union and intersection operations, to capture the principal stochastic uncertainties. Then, the fuzzy inference will be modified based on the extensional Dempster rule of combination. This new approach is able to capture the stochastic disturbance acting on fuzzy membership function, and provide a more effective inference under strong stochastic uncertainty. Finally, the numerical simulation and the experimental prediction of the wind speed are conducted to show the potential of the proposed method in stochastic modeling.

Dempster Shafer Structure-Fuzzy Number Based Uncertainty Modeling in Human Health Risk Assessment

In risk assessment, generally model parameters are affected by uncertainty arises due to vagueness, imprecision, lack of data, small sample sizes etc. Fuzzy set theory and Dempster-Shafer theory (In short DST) of evidence should be explored to handle this type of uncertainty. Representation of parameters of risk assessment models may be Dempster-Shafer structure (in short DSS) and fuzzy numbers. To deal with such situations, it is important to device new techniques. This paper presents two algorithms to combine Dempster-Shafer structure with generalized/normal fuzzy focal elements, generalized/normal fuzzy numbers within the same framework. Sampling technique for evidence theory and alpha-cut for fuzzy numbers are considered to execute the algorithms. Finally, results are obtained in the form of fuzzy numbers (normal/generalized) at different fractiles.

Uncertainty Modeling in Risk Assessment Based on Dempster–Shafer Theory of Evidence with Generalized Fuzzy Focal Elements

Dempster–Shafer theory of evidence is one of the important tools for decision making under uncertainty. It is more useful in situations when cost of technical difficulties is involved or uniqueness of the situation under study makes it difficult/impossible to cover enough observations to quantify the models with real data. Consequently, experts provide opinions in terms of basic probability assignment for focal elements. Usually, it is seen that experts provide basic probability assignment for interval (or crisp) focal elements. However, due to presence of uncertainty focal elements can sometimes be treated as normal/generalized triangular fuzzy number (TFN in short) instead of intervals or crisp sets. TFN encodes only most likely value (mode) and the spread. This paper presents an attempt to combine Dempster–Shafer structures (DSS in short) with generalized/normal fuzzy focal elements using possibilistic sampling technique. To this end, human health risk assessment is carried out under such setting.

Quantification of margins and mixed uncertainties using evidence theory and stochastic expansions

The objective of this paper is to implement Dempster–Shafer Theory of Evidence (DSTE) in the presence of mixed (aleatory and multiple sources of epistemic) uncertainty to the reliability and performance assessment of complex engineering systems through the use of quantification of margins and uncertainties (QMU) methodology. This study focuses on quantifying the simulation uncertainties, both in the design condition and the performance boundaries along with the determination of margins. To address the possibility of multiple sources and intervals for epistemic uncertainty characterization, DSTE is used for uncertainty quantification. An approach to incorporate aleatory uncertainty in Dempster–Shafer structures is presented by discretizing the aleatory variable distributions into sets of intervals. In view of excessive computational costs for large scale applications and repetitive simulations needed for DSTE analysis, a stochastic response surface based on point-collocation non-intrusive polynomial chaos (NIPC) has been implemented as the surrogate for the model response. The technique is demonstrated on a model problem with non-linear analytical functions representing the outputs and performance boundaries of two coupled systems. Finally, the QMU approach is demonstrated on a multi-disciplinary analysis of a high speed civil transport (HSCT).

A MIXED UNCERTAINTY QUANTIFICATION APPROACH USING EVIDENCE THEORY AND STOCHASTIC EXPANSIONS

Uncertainty quantification (UQ) is the process of quantitative characterization and propagation of input uncertainties to the response measure of interest in experimental and computational models. The input uncertainties in computational models can be either aleatory i.e. irreducible inherent variations or epistemic i.e. reducible variability which arises from lack of knowledge. Previously, it has been shown that Dempster-Shafer Theory of Evidence (DSTE) can be applied to model epistemic uncertainty in case of uncertainty information coming from multiple sources. The objective of this paper is to model and propagate mixed uncertainty (aleatory and epistemic) using DSTE. In specific, the aleatory variables are modeled as Dempster-Shafer structures by discretizing them into sets of intervals according to their respective probability distributions. In order to avoid excessive computational cost associated with large scale applications, a stochastic response surface based on point-collocation non-intrusive polynomial chaos has been implemented as the surrogate model for the response. A convergence study for accurate representation of aleatory uncertainty in terms of minimum number of subintervals required is presented. The mixed UQ approach is demonstrated on a numerical example and high fidelity computational fluid dynamics study of transonic flow over RAE 2822 airfoil.

Analysis and computation with hybrid random set stochastic models

Engineering structures are usually modelled as input–output maps: the response is a function of input parameters and driving forces, both being nondeterministic and uncertain. The paper addresses and advertises recent methods combining stochastic processes and imprecise probability, namely random fields for the description of loads and random sets for the description of uncertain input parameters. This approach covers interval analysis, fuzzy sets, Dempster–Shafer structures and random variables and their hybrid combination with stochastic processes. The methods are demonstrated using an example from geotechnics, including an application to sensitivity analysis.