Shafer Belief Structure Research Articles

Associated Probabilities in Insufficient Expert Data Analysis

Problems of modeling uncertainty and imprecision for the analysis of insufficient expert data (IED) are considered in the environment of interactive multi-group decision-making (MGDM). Based on the Choquet finite integral, a moments’ method for the IED is developed for the evaluation of the associated probabilities class (APC) of Choquet’s second-order capacity based on the informational entropy maximum principle. Based on the IED new approach of the lower and upper Choquet’s second-order capacities, identification is developed. The second pole of insufficient expert data, the data imprecision indicator, is presented in the form of a fuzzy subset and image on the alternatives set. In the environment of the Dempster–Shafer belief structure, connections between an associated possibilities class (APosC), with the APC, and an associated focal probabilities class (AFPC) are constructed. In the approach of A. Kaufman’s theory of expertons, based on the APosC and the AFPC unique fuzzy subset, the IED image on the alternatives set is constructed. Based on Sugeno’s finite integral most typical value (MTV), as a prediction on possible alternatives set, the IED is constructed. In the example, a sensitive and comparative analysis is provided for the evaluation of the new approach’s stability and reliability.

On the negation of a Dempster–Shafer belief structure based on maximum uncertainty allocation

Probability theory and Dempster–Shafer theory are two germane theories to represent and handle uncertain information. Recent study suggested a transformation to obtain the negation of a probability distribution based on the maximum entropy. Correspondingly, determining the negation of a belief structure, however, is still an open issue in Dempster–Shafer theory, which is very important in theoretical research and practical applications. In this paper, a negation transformation for belief structures is proposed based on maximum uncertainty allocation, and several important properties satisfied by the transformation have been studied. The proposed negation transformation is more general and could be totally compatible with existing transformation for probability distributions.

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.

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.

Brain MRI segmentation by combining different MRI modalities using Dempster–Shafer theory

Magnetic resonance imagings (MRIs) have different modalities, including T1- and T2-weighted, PD (proton density), and Flair images. Brain MRI segmentation is a challenge when coping with artefacts such as intensity non-uniformity, partial volume effects, and noise. As artefacts change the intensity of different part of MRI modalities, describing the intensity of these modalities is highly uncertain. Here, it is proposed that the Dempster–Shafer theory and fuzzy clustering can be combined for brain MRI segmentation because of their robustness. The purpose of this research is to offer a technique based on data fusion of different modalities to segment brain MRIs. T1, T2, PD, and Flair were employed in this study. Fuzzy clustering and specific mapping were used to form the Dempster–Shafer belief structure. In order to evaluate the efficiency of the proposed algorithm, several simulations were performed and the Dice and Jaccard coefficients were used to compare the results with those of other methods. The qualitative and quantitative results of the proposed algorithm verify the success of the proposed algorithm. This method improved 3–4% over that of the previous methods which had showed the best results.

Entailment for measure based belief structures

We discuss the Dempster–Shafer belief structure on finite universes and note its use for modeling variables that have both probabilistic uncertainty as well as imprecision. We note for these structures the probability that the variable lies in a subset cannot be precisely known but only be known to an interval value. We discuss methods for deducing this uncertainty interval. We next discuss the issue of entailment of belief structures, inferring the validity of additional belief model of a variable from an already established belief model of the variable. We next discuss a more general belief structure were the underling uncertainty rather tha0n being based on a probability distribution is based on a general measure type of uncertainty. We then extend the concept of entailment to the case where the belief structures are these more general measure based belief structures. In order to accomplish this we must extend the idea of containment from classic Dempster–Shafer belief structures to measure based belief structures.

Fuzzy rule bases with generalized belief structure inputs

We first describe the basics of fuzzy systems modeling. Fundamental to this is a collection of rules, a rule base, in which the rule antecedents are fuzzy subsets. We first look at issue of the determination of the firing level of a rule for fuzzy set inputs and the subsequent rule base output. We next consider the situation where the system input is uncertain and modeled by a Dempster–Shafer belief structure. Here our input is a collection of fuzzy subsets and the true input fuzzy set is selected based on a probability distribution over these potential input fuzzy sets. We next consider the situation where our input is modeled via a generalized belief structure where the determination of applicable input fuzzy set is modeled via a measure over these potential input fuzzy sets.

Satisfying uncertain targets using measure generalized Dempster-Shafer belief structures

We provide the basic formalisms associated with the classic Dempster–Shafer belief structure. We note it provides a model of uncertainty that has both a probabilistic uncertainty and imprecise component. We introduce the idea of generalized D–S belief structure, which, while still having an imprecise component has more a general measure based uncertainty. We discuss the idea of a target as a subset of some space X and an uncertain target is a collection of subsets each having a probability of being the true target. We consider the problem that an uncertain variable, whose value is known to be a generalized D–S structure satisfies the target, its value lies in the target set.

Maxitive Belief Structures and Imprecise Possibility Distributions

We introduce the idea of maxitive belief structures, MBS, as a framework for modeling imprecise possibility distributions in a manner analogous to the way Dempster–Shafer belief structures allows the modeling of imprecise probability distributions. We study various properties of these MBS. We consider the problem of obtaining possibility distributions compatible with these belief structures. We consider the problem of fusing multiple MBS and the performance of various arithmetic operations with these objects.

Zero-Sum Matrix Game with Payoffs of Dempster-Shafer Belief Structures and Its Applications on Sensors.

The zero-sum matrix game is one of the most classic game models, and it is widely used in many scientific and engineering fields. In the real world, due to the complexity of the decision-making environment, sometimes the payoffs received by players may be inexact or uncertain, which requires that the model of matrix games has the ability to represent and deal with imprecise payoffs. To meet such a requirement, this paper develops a zero-sum matrix game model with Dempster–Shafer belief structure payoffs, which effectively represents the ambiguity involved in payoffs of a game. Then, a decomposition method is proposed to calculate the value of such a game, which is also expressed with belief structures. Moreover, for the possible computation-intensive issue in the proposed decomposition method, as an alternative solution, a Monte Carlo simulation approach is presented, as well. Finally, the proposed zero-sum matrix games with payoffs of Dempster–Shafer belief structures is illustratively applied to the sensor selection and intrusion detection of sensor networks, which shows its effectiveness and application process.

Fuzzy relations between Dempster–Shafer belief structures

Relationships between attribute values can involve concepts such as similar or greater then. We first discuss the mathematical structure of relationships. We consider both crisp and fuzzy relationships. We introduce some important examples of mathematical relationships and describe their formal properties. We then turn to the problem of calculating the degree of relationship between attributes whose values are uncertain. We first consider the case when the uncertainties in the attribute values are expressed in terms of possibility distributions. We next look at the case when the uncertainties are in the form of a standard Dempster–Shafer belief structure. We finally consider the situation when the uncertain information associated with the attribute values are expressed in terms of Dempster-Shafer belief structures whose focal elements are fuzzy sets.

Belief structures, weight generating functions and decision-making

We describe the Dempster---Shafer belief structure and provide some of its basic properties. We introduce the plausibility and belief measures associated with a belief structure. We note that these are not the only measures that can be associated with a belief structure. We describe a general approach for generating a class of measures that can be associated with a belief structure using a monotonic function on the unit interval, called a weight generating function. We study a number of these functions and the measures that result. We show how to use weight-generating functions to obtain dual measures from a belief structure. We show the role of belief structures in representing imprecise probability distributions. We describe the use of dual measures, other then plausibility and belief, to provide alternative bounding intervals for the imprecise probabilities associated with a belief structure. We investigate the problem of decision making under belief structure type uncertain. We discuss two approaches to this decision problem. One of which is based on an expected value of the OWA aggregation of the payoffs associated with the focal elements. The second approach is based on using the Choquet integral of a measure generated from the belief structure. We show the equivalence of these approaches.

Combining various types of belief structures

We first discuss the basic ideas of the Dempster–Shafer belief structure. We particularly emphasize the associated measures of plausibility and belief. We discuss the cumulative distribution function associated with a belief structure. We next turn to the issue of combining multiple belief structures. Two different types of combination of belief structures are investigated. The first, which we refer to as the fusion of belief structures, occurs when the belief structures being combined are providing information about the same variable. Here we make use of the Dempster rule. The second type, which we refer to as the joining of belief structures, occurs when the belief structures being combined are providing information about different variables. Here we make use of a belief structures cumulative distribution function and the Sklar theorem. The classic belief structures typically have a finite number of focal elements; here we began to look at belief structures which have a continuum of focal elements.

Dempster–Shafer belief structures for decision making under uncertainty

We discuss the need for tools for representing various types of uncertain information in decision-making. We introduce the Dempster–Shafer belief structure and discuss how it provides a formal mathematical framework for representing various types of uncertain information. We provide some fundamental ideas and mechanisms related to these structures. We then investigate their role in the important task of decision-making under uncertainty.

Jeffrey’s rule of conditioning with various forms for uncertainty

We first introduce Jeffrey’s rule of conditioning and explain how it allows us to determine the probability of an event related to one variable from information about a collection of conditional probabilities of that event conditioned on the state another variable. We note that in the original Jeffrey paradigm we have the uncertainty about the state of the conditioning variable expressed as a probability distribution. Here we extend this by allowing alternative formulations for the uncertainty about the conditioning variable. We first consider the case where our uncertainty is expressed in terms of a measure. This allows us to consider the case where our uncertainty is a possibility distribution. We next consider the case where our uncertainty about the conditioning variable is expressed in terms of a Dempster–Shafer belief structure. Finally we consider the case where we are ignorant about the underlying distribution and must use the decision maker’s subjective attitude about the nature of uncertainty to provide the necessary information to use in the Jeffrey rule.

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.

On the interrelation between Dempster–Shafer Belief Structures and their Belief Cumulative Distribution Functions

We consider Dempster–Shafer belief structures (DSBSs) m having finitely many non-empty compact intervals as focal elements and prove several results describing the interrelation between DSBSs and their so-called Belief Cumulative Distribution Functions (BCDFs) x↦F(x)=(F̲(x),F¯(x)) induced by the corresponding belief and plausibility measures. In particular, we answer a question by Ronald R. Yager on the injectivity of the assignment m↦F.

Some hybrid aggregating operators in linguistic decision making with Dempster–Shafer belief structure

In this paper, we have studied the Dempster–Shafer theory of evidence in situations of decision making with linguistic information and we develop a new aggregation operator: belief structure generalized linguistic hybrid averaging (BS-GLHA) operator and a wide range of particular cases. we have developed the new decision making model with Dempster–Shafer belief structure that uses linguistic information in order to manage uncertain situations that cannot be managed in a probabilistic way. We have seen that all these approaches are very useful for representing the new approaches in a more complete way selecting for each situation the particular case that it is closest to our interests in the specific problem analyzed. Finally, a numerical example is used to illustrate the applicability and effectiveness of the proposed method. We have pointed out that the results and decisions are dependent on the linguistic aggregation operator used in the decision making process.

Joint cumulative distribution functions for Dempster–Shafer belief structures using copulas

We first introduce the Dempster---Shafer belief structure and highlight its role in the representation of information about a random variable for which our knowledge of the probabilities is interval-valued. We investigate the formation of the cumulative distribution function (CDF) for these types of variables. It is noted that this is also interval-valued and is expressible in terms of plausibility and belief measures. The class of aggregation operators known as copulas are introduced and a number of their properties are provided. We discuss Sklar's theorem, which provides for the use of copulas in the formulation of joint CDFs from the marginal CDFs of classic random variables. We then look to extend these ideas to the case of joining the marginal CDFs associated with Dempster---Shafer belief structures. Finally we look at the formulation CDFs obtained from functions of multiple D---S belief structures.

A novel fuzzy Dempster–Shafer inference system for brain MRI segmentation

Brain Magnetic Resonance Imaging (MRI) segmentation is a challenging task due to the complex anatomical structure of brain tissues as well as intensity non-uniformity, partial volume effects and noise. Segmentation methods based on fuzzy approaches have been developed to overcome the uncertainty caused by these effects. In this study, a novel combination of fuzzy inference system and Dempster–Shafer Theory is applied to brain MRI for the purpose of segmentation where the pixel intensity and the spatial information are used as features. In the proposed modeling, the consequent part of rules is a Dempster–Shafer belief structure. The novelty aspect of this work is that the rules are paraphrased as evidences. The results show that the proposed algorithm, called FDSIS has satisfactory outputs on both simulated and real brain MRI datasets.