Nonparametric regression analysis of uncertain and imprecise data using belief functions

AbstractThis paper introduces a new approach to regression analysis based on a fuzzy extension of belief function theory. For a given input vector x, the method provides a prediction regarding the value of the output variable y, in the form of a fuzzy belief assignment (FBA), defined as a collection of fuzzy sets of values with associated masses of belief. The output FBA is computed using a nonparametric, instance-based approach: training samples in the neighborhood of x are considered as sources of partial information on the response variable; the pieces of evidence are discounted as a function of their distance to x, and pooled using Dempster’s rule of combination. The method can cope with heterogeneous training data, including numbers, intervals, fuzzy numbers, and, more generally, fuzzy belief assignments, a convenient formalism for modeling unreliable and imprecise information provided by experts or multi-sensor systems. The performances of the method are compared to those of standard regression techniques using several simulated data sets

Conjunctive and Disjunctive Combination of Belief Functions Induced by Non Distinct Bodies of Evidence

Dempster’s rule plays a central role in the theory of belief functions. However, it assumes the combined bodies of evidence to be distinct, an assumption which is not always verified in practice. In this paper, a new operator, the cautious rule of combination, is introduced. This operator is commutative, associative and idempotent. This latter property makes it suitable to combine belief functions induced by reliable, but possibly overlapping bodies of evidence. A dual operator, the bold disjunctive rule, is also introduced. This operator is also commutative, associative and idempotent, and can be used to combine belief functions issues from possibly overlapping and unreliable sources. Finally, the cautious and bold rules are shown to be particular members of infinite families of conjunctive and disjunctive combination rules based on triangular norms and conorms

Constructing belief functions from sample data using multinomial confidence regions

The Transferable Belief Model is a subjectivist model of uncertainty in which an agent’s beliefs at a given time are modeled using the formalism of belief functions. Belief functions that enter the model are usually either elicited from experts, or must be constructed from observation data. There are, however, few simple and opera-tional methods available for building belief functions from data. Such a method is proposed in this paper. More precisely, we tackle the problem of quantifying beliefs held by an agent about the realization of a discrete random variable X with unknown probability distribution PX, having observed a realization of an independent, identi-cally distributed random sample with the same distribution. The solution is obtained using simultaneous confidence intervals for multinomial proportions, several of which have been proposed in the statistical literature. The proposed solution verifies two “reasonable ” properties with respect to PX: it is less committed than PX with some user-defined probability, and it converges towards PX in probability as the size of the sample tends to infinity. A general formulation is given, and a useful approximation with a simple analytical expression is presented, in the important special case where the domain of X is ordered

Modeling Vague Beliefs Using Fuzzy-Valued Belief Structures

This paper presents a rational approach to the representation and manipulation of imprecise degrees of belief in the framework of evidence theory. We adopt as a starting point the non probabilistic interpretation of belief functions provided by Smets' Transferable Belief Model, as well as previous generalizations of evidence theory allowing to deal with fuzzy propositions. We then introduce the concepts of interval-valued and fuzzy-valued belief structures, defined, respectively, as crisp and fuzzy sets of belief structures verifying hard or elastic constraints. We then proceed with a generalization of various concepts of Dempster-Shafer theory including those of belief and plausibility functions, combination rules and normalization procedures. Most calculations implied by the manipulation of these concepts are based on simple forms of linear programming problems for which analytical solutions exist, making the whole scheme computationally tractable. We discuss the application of this ..