Evidential reasoning using extended fuzzy Dempster–Shafer theory for handling various facets of information deficiency

This work investigates the problem of combining deficient evidence for the purpose of quality assessment. The main focus of the work is modeling vagueness, ambiguity, and local nonspecificity in information within a unified approach. We introduce an extended fuzzy Dempster–Shafer scheme based on the simultaneous use of fuzzy interval‐grade and interval‐valued belief degree (IGIB). The latter facilitates modeling of uncertainties in terms of local ignorance associated with expert knowledge, whereas the former allows for handling the lack of information on belief degree assignments. Also, generalized fuzzy sets can be readily transformed into the proposed fuzzy IGIB structure. The reasoning for quality assessment is performed by solving nonlinear optimization problems on fuzzy Dempster–Shafer paradigm for the fuzzy IGIB structure. The application of the proposed inference method is investigated by designing a reasoning scheme for water quality monitoring and validated through the experimental data available for different sampling points in a water distribution network. © 2011 Wiley Periodicals, Inc.     

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.     

An intuitionistic view of the Dempster–Shafer belief structure

We discuss the Dempster–Shafer belief theory and describe its role in representing imprecise probabilistic information. In particular, we note its use of intervals for representing imprecise probabilities. We note in fuzzy set theory that there are two related approaches used for representing imprecise membership grades: interval-valued fuzzy sets and intuitionistic fuzzy sets. We indicate the first of these, interval-valued fuzzy sets, is in the same spirit as Dempster–Shafer representation, both use intervals. Using a relationship analogous to the type of relationship that exists between interval-valued fuzzy sets and intuitionistic fuzzy sets, we obtain from the interval-valued view of the Dempster–Shafer model an intuitionistic view of the Dempster–Shafer model. Central to this view is the use of an intuitionistic statement, pair of values, (Bel(A) Dis(A)), to convey information about the value of a variable lying in the set A. We suggest methods for combining intuitionistic statements and making inferences from these type propositions.     

Induced generalized intuitionistic fuzzy operators

We study the induced generalized aggregation operators under intuitionistic fuzzy environments. Choquet integral and Dempster–Shafer theory of evidence are applied to aggregate inuitionistic fuzzy information and some new types of aggregation operators are developed, including the induced generalized intuitionistic fuzzy Choquet integral operators and induced generalized intuitionistic fuzzy Dempster–Shafer operators. Then we investigate their various properties and some of their special cases. Additionally, we apply the developed operators to financial decision making under intuitionistic fuzzy environments. Some extensions in interval-valued intuitionistic fuzzy situations are also pointed out.     

Knowledge discovery in distributed databases using evidence theory

Distributed databases allow us to integrate data from different sources which have not previously been combined. The Dempster–Shafer theory of evidence and evidential reasoning are particularly suited to the integration of distributed databases. Evidential functions are suited to represent evidence from different sources. Evidential reasoning is carried out by the well‐known orthogonal sum. Previous work has defined linguistic summaries to discover knowledge by using fuzzy set theory and using evidence theory to define summaries. In this paper we study linguistic summaries and their applications to knowledge discovery in distributed databases. © 2000 John Wiley & Sons, Inc.     

An Evidential Axiomatic Design Approach for Decision Making Using the Evaluation of Belief Structure Satisfaction to Uncertain Target Values

Axiomatic design (AD) provides a general theory for system and product development. In recent years, the principles of AD have been successfully applied to the decision‐making field, and derived a fuzzy AD approach for fuzzy decision‐making environment. In this work, the interest is paid on the theoretical developments and applications of AD in the uncertain environment expressed by Dempster–Shafer evidence theory. Based on the concept of belief structure satisfaction to uncertain target values, an evidential AD approach is proposed for decision making by combining the independence axiom and information axiom of AD with the framework of Dempster–Shafer theory. An illustrative example has demonstrated the effectiveness of the proposed approach. This work, on the one hand, has successfully generalized the principles of AD to the Dempster–Shafer uncertain environment; on the other hand, it has presented a successful application of the concept of belief structure satisfaction.     

A class of fuzzy measures generated from a Dempster–Shafer belief structure

Here the Dempster–Shafer belief structure is viewed as providing partial information about the underlying fuzzy measure associated with a uncertain variable. In this perspective there exists many possible fuzzy measures that can be associated with a Dempster–Shafer belief structure. Typically only two of these measures have been made explicit, those being the measure of belief and plausibility. Here we introduce a whole class of fuzzy measures that can be associated with a Dempster–Shafer belief structure. As an aid to choosing between these myriad of possibilities we discuss the entropy of a fuzzy measure. ©1999 John Wiley & Sons, Inc.     

Knitted fabric defect classification for uncertain labels based on Dempster–Shafer theory of evidence

A new approach for classification of circular knitted fabric defect is proposed which is based on accepting uncertainty in labels of the learning data. In the basic classification methodologies it is assumed that correct labels are assigned to samples and these approaches concentrate on the strength of categorization. However, there are some classification problems in which a considerable amount of uncertainty exists in the labels of samples. The core of innovation in this research has been usage of the uncertain information of labeling and their combination with the Dempster–Shafer theory of evidence. The experimental results show the robustness of the proposed method in comparison with usual classification techniques of supervised learning where the certain labels are assigned to training data.     

A Dempster Shafer-based fuzzy multisensor fusion system using airborne lidar and hyperspectral imagery

ABSTRACT

Nowadays, accurate spectral reflectance information is provided by hyperspectral (HS) data while light detection and ranging (lidar) data provides precise information about the height and geometrical properties of the surfaces. In the most research papers, data fusion of disparate sensors significantly improves object classification performance compared to that of just an individual sensor. Previous researches on fusion of these two sensors had problems such as crisp classifiers or simple fuzzy decision-making systems. This article tries to overcome these weaknesses by accurate support vector machine (SVM) and Fuzzy SVM as classifiers in crisp and fuzzy decision fusion system and fusion of two sensors by two different methods based on precise theories of Bayesian and Shafer. Also, the proposed method tries to compare the results of fusion of both data using decision fusion system with stacked features strategy. This study focuses on HS and lidar fusion through three main phases. The first phase is based on the using of Noise Weighted Harsanyi-Farrand-Chang method and principal component analysis to overcome the high dimensionality problem of HS data. The second phase is based on the feature extraction and selection strategy on lidar data. Finally, fuzzy SVM and Dempster Shafer methods are applied as fuzzy classification and fuzzy decision fusion strategies on the feature spaces. A co-registered HS and lidar data set from Houston of U.S.A. by 15 classes was available to examine the effectiveness of the proposed method. The results of this study highlight that the combination of HS and lidar data enable reliable mapping of land cover.     

A new linguistic MCDM method based on multiple-criterion data fusion

Multiple-criteria decision-making (MCDM) is concerned with the ranking of decision alternatives based on preference judgements made on decision alternatives over a number of criteria. First, taking advantage of data fusion technology to comprehensively consider each criterion data is a reasonable idea to solve the MCDM problem. Second, in order to efficiently handle uncertain information in the process of decision making, some well developed mathematical tools, such as fuzzy sets theory and Dempster Shafer theory of evidence, are used to deal with MCDM. Based on the two main reasons above, a new fuzzy evidential MCDM method under uncertain environments is proposed. The rating of the criteria and the importance weight of the criteria are given by experts’ judgments, represented by triangular fuzzy numbers. Then, the weights are transformed into discounting coefficients and the ratings are transformed into basic probability assignments. The final results can be obtained through the Dempster rule of combination in a simple and straight way. A numerical example to select plant location is used to illustrate the efficiency of the proposed method.     

An interpretation of intuitionistic fuzzy sets in terms of evidence theory: Decision making aspect

This paper presents a new interpretation of intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory of evidence (DST). This interpretation makes it possible to represent all mathematical operations on intuitionistic fuzzy values as the operations on belief intervals. Such approach allows us to use directly the Dempster’s rule of combination to aggregate local criteria presented by intuitionistic fuzzy values in the decision making problem. The usefulness of the developed method is illustrated with the known example of multiple criteria decision making problem. The proposed approach and a new method for interval comparison based on DST, allow us to solve multiple criteria decision making problem without intermediate defuzzification when not only criteria, but their weights are intuitionistic fuzzy values.     

Decision-based fuzzy image restoration for noise reduction based on evidence theory

A novel decision-based fuzzy averaging (DFA) filter consisting of a D–S (Dempster–Shafer) noise detector and a two-pass noise filtering mechanism is presented in this paper. The proposed filter can effectively deal with impulsive noise, and a mix of Gaussian and impulsive noise. Bodies of evidence are extracted, and the basic belief assignment is developed using the simple support function, which avoids the counter-intuitive problem of Dempster’s combination rule. The combination belief value is the decision rule for the D–S noise detector. A fuzzy averaging method, where the weights are constructed using a predefined fuzzy set, is developed to achieve noise cancellation. A simple second-pass filter is employed to improve the final filtering performance. Experimental results confirm the effectiveness of the new DFA filter both in suppressing impulsive noise as well as a mix Gaussian and impulsive noise and in improving perceived image quality.     

Extension of belief functions to infinite-valued events

We generalise belief functions to many-valued events which are represented by elements of Lindenbaum algebra of infinite-valued ?ukasiewicz propositional logic. Our approach is based on mass assignments used in the Dempster–Shafer theory of evidence. A generalised belief function is totally monotone and it has Choquet integral representation with respect to a unique belief measure on Boolean events.     

Representation and combination of uncertainty with belief functions and possibility measures

The theory of evidence proposed by G. Shafer is gaining more and more acceptance in the field of artificial intelligence, for the purpose of managing uncertainty in knowledge bases. One of the crucial problems is combining uncertain pieces of evidence stemming from several sources, whether rules or physical sensors. This paper examines the framework of belief functions in terms of expressive power for knowledge representation. It is recalled that probability theory and Zadeh's theory of possibility are mathematically encompassed by the theory of evidence, as far as the evaluation of belief is concerned. Empirical and axiomatic foundations of belief functions and possibility measures are investigated. Then the general problem of combining uncertain evidence is addressed, with focus on Dempster rule of combination. It is pointed out that this rule is not very well adapted to the pooling of conflicting information. Alternative rules are proposed to cope with this problem and deal with specific cases such as nonreliable sources, nonexhaustive sources, inconsistent sources, and dependent sources. It is also indicated that combination rules issued from fuzzy set and possibility theory look more flexible than Dempster rule because many variants exist, and their numerical stability seems to be better.     

Managing uncertainty in location services using rough set and evidence theory

Uncertainty in service management stems from the incompleteness and vagueness of the conditioning attributes that characterize a service. In particular, location based services often have complex interaction mechanisms in terms of their neighborhood relationships. Classical location service models require rigorous parameters and conditioning attributes and offers limited flexibility to incorporate imprecise or ambiguous evidences. In this paper we have developed a formal model of uncertainty in service management. We have developed a rough set and Dempster–Shafer’s evidence theory based formalism to objectively represent uncertainty inherent in the process of service discovery, characterization, and classification. Rough set theory is ideally suited for dealing with limited resolution, vague and incomplete information, while Dempster–Shafer’s evidence theory provides a consistent approach to model an expert’s belief and ignorance in the classification decision process. Integrating these two formal approaches in spatial domain provides a way to model an expert’s belief and ignorance in service classification. In an application scenario of the model we have used a cognitive map of retail site assessment, which reflects the partially subjective assessment process. The uncertainty hidden in the cognitive map can be consistently formalized using the proposed model. Thus we provide a naturalistic means of incorporating both qualitative aspects of intuitive knowledge as well as hard empirical information for service management within a formal uncertainty framework.     

Land‐cover classification in the Andes of southern Ecuador using Landsat ETM+ data as a basis for SVAT modelling

A land‐cover classification is needed to deduce surface boundary conditions for a soil–vegetation–atmosphere transfer (SVAT) scheme that is operated by a geoecological research unit working in the Andes of southern Ecuador. Landsat Enhanced Thematic Mapper Plus (ETM+) data are used to classify distinct vegetation types in the tropical mountain forest. Besides a hard classification, a soft classification technique is applied. Dempster–Shafer evidence theory is used to analyse the quality of the spectral training sites and a modified linear spectral unmixing technique is selected to produce abundancies of the spectral endmembers. The hard classification provides very good results, with a Kappa value of 0.86. The Dempster–Shafer ambiguity underlines the good quality of the training sites and the probability guided spectral unmixing is chosen for the determination of plant functional types for the land model. A similar model run with a spatial distribution of land cover from both the hard and the soft classification processes clearly points to more realistic model results by using the land surface based on the probability guided spectral unmixing technique.     

A decision support framework for security resource allocation under ambiguity

There has been increasing interest in using Stackelberg game (known as a security game) to allocate limited security resources against different attacker types with a specific probability distribution. However, real problems of this kind often face ambiguous information, such as imprecise, unreliable and absent payoffs, and ambiguous assignments of these payoffs. To this end, based on decision theory and the Dempster–Shafer theory of evidence, this paper proposes a novel framework that can handle these common types of ambiguity. More specifically, this paper deploys the underlying principles of existing rules from decision theory, as a way to characterise different attitudes to ambiguity, during the transformation of ambiguous payoffs into point‐valued payoffs. Hence, our framework holds some good properties: (i) it subsumes traditional security games without ambiguous payoffs, (ii) a uniform margin of error will not affect the results and (iii) the influence of complete ignorance can be minimised. Also, our framework is evaluated by using nine different transformation rules, under various conditions and constraints, against 73,000 randomly generated games (a first comprehensive empirical evaluation to date). The evaluation reveals the benefits of each transformation rule and confirms that different rules can model individuals' different attitudes to ambiguity.     

Dempster–Shafer evidential theory for the automated selection of parameters for Talbot’s method contours and application to matrix exponentiation

In this paper, the Dempster–Shafer theory of evidential reasoning is applied to the problem of optimal contour parameters selection in Talbot’s method for the numerical inversion of the Laplace transform. The fundamental concept is the discrimination between rules for the parameters that define the shape of the contour based on the features of the function to invert. To demonstrate the approach, it is applied to the computation of the matrix exponential via numerical inversion of the corresponding resolvent matrix. Training for the Dempster–Shafer approach is performed on random matrices. The algorithms presented have been implemented in MATLAB. The approximated exponentials from the algorithm are compared with those from the rational approximation for the matrix exponential returned by the MATLAB expm function.     

An integrated information fusion approach based on the theory of evidence and group decision-making

Dempster–Shafer theory of evidence has been employed as a major method for reasoning with multiple evidence. The Dempster’s rule of combination is however incapable of managing highly conflicting evidence coming from different information sources at the normalization step. Extending current rules, we incorporate the ideas of group decision-making into the theory of evidence and propose an integrated approach to automatically identify and discount unreliable evidence. An adaptive robust combination rule that incorporates the information contained in the consistent focal elements is then constructed to combine such evidence. This rule adjusts the weights of the conjunctive and disjunctive rules according to a function of the consistency of focal elements. The theoretical arguments are supported by numerical experiments. Compared to existing combination rules, the proposed approach can obtain a reasonable and reliable decision, as well as the level of uncertainty about it.     

Establishing agent trust for contradictory evidence by means of fuzzy voting model: An ontology mapping case study

This paper introduces a novel trust assessment formalism for contradicting evidence in the context of multi-agent ontology mapping. Evidence combination using the Dempster rule tend to ignore contradictory evidence and the contemporary approaches for managing these conflicts introduce additional computation complexity i.e. increased response time of the system. On the Semantic Web, ontology mapping systems that need to interact with end users in real time cannot afford prolonged computation. In this work, we have made a step towards the formalisation of eliminating contradicting evidence, to utilise the original Dempster’s combination rule without introducing additional complexity. Our proposed solution incorporates the fuzzy voting model to the Dempster–Shafer theory. Finally, we present a case study where we show how our approach improves the ontology mapping problem.