Fuzzy set connectives as combinations of belief structures

doi:10.1016/0020-0255(92)90096-Q

Information Sciences

Volume 66, Issue 3, 15 December 1992, Pages 245-276

https://doi.org/10.1016/0020-0255(92)90096-Q Get rights and content

Abstract

Consonant belief structures provide a representation for fuzzy sets owing to the fact that their plausibility measures are essentially possibility measures. We note that two belief structures are equivalent if their plausibility and belief functions are equal. This observation leads us to provide a multiple number of equivalent representations for any belief structure. Commensurate representations can be induced for two different belief structures by forcing the same number of focal elements with the same weights. We show that if we represent two consonant belief structures in a commensurate manner then their aggregations are closed with respect to consonance, provided that the additional requirement that the underlying probability distributions satisfy a condition of correlation is imposed. The results of this work allow us to use belief structure representations for the manipulation of fuzzy subsets under various logical combinations. All basic fuzzy set connectives can thus be interpreted in the framework of the theory of evidence.

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In the paper, we formalize the notion of contradiction between belief functions: we argue that belief functions are not contradictory if they provide non-contradictory models for decision-making. To elaborate on this idea, we take the decision rule from imprecise probabilities and show that sources of information described by belief functions are not contradictory iff the intersection of corresponding credal sets is not empty. We demonstrate that evidential conjunctive and disjunctive rules fit with this idea and they are justified in a probabilistic setting. In the case of contradictory sources of information, we analyze possible conjunctions and show how the result can be described by generalized credal sets. Based on generalized credal sets, we propose a measure of contradiction between information sources and find its axiomatics. We show how the contradiction correction can be produced based on generalized credal sets and how it can be done on sets of surely desirable gambles.
Idempotent conjunctive and disjunctive combination of belief functions by distance minimization
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However, if idempotence is a consequence of the LCP, satisfying idempotence does not imply satisfying the LCP. As shown by Dubois and Yager [6], there is virtually an infinity of ways to derive idempotent combination rules, not all of them necessarily following a least-commitment principle. For instance, Cattaneo [7] provides an idempotent rule following a conflict-minimization approach, which may lead to non-least committed results [8].
Idempotence is a desirable property when cautiousness is wanted in an information fusion process, since in this case combining identical information should not lead to the reinforcement of some hypothesis. Idempotent operators also guarantee that identical information items are not counted twice in the fusion process, a very important property in decentralized applications where the information origin cannot always be tracked (ad-hoc wireless networks are typical examples). In the theory of belief functions, a sound way to combine conjunctively multiple information items is to design a combination rule that selects the least informative element among a subset of belief functions more informative than each of the combined ones. In contrast, disjunctive rules can be retrieved by selecting the most informative element among a subset of belief functions less informative than each of the combined ones. One interest of such approaches is that they provide idempotent rules by construction.
The notions of less and more informative are often formalized through partial orderings extending usual set-inclusion, yet the only two informative partial orders that provide a straightforward idempotent rule leading to a unique result are those based on the conjunctive and disjunctive weight functions. In this article, we show that other partial orders can achieve a similar goal when the problem is slightly relaxed into a distance optimization one. Building upon previous work, this paper investigates the use of distances compatible with informative partial orders to determine a unique solution to the combination problem. The obtained operators are conjunctive/disjunctive, idempotent and commutative, but lack associativity. They are, however, quasi-associative allowing sequential combinations at no extra complexity. Some experiments demonstrate interesting discrepancies as compared to existing approaches, notably with the aforementioned rules relying on weight functions.
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An evidential fusion approach for gender profiling
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For combining difference pieces of information, DS theory distinguishes two cases, i.e., whether pieces of information are from distinct, or non-distinct, sources. Many combination rules are proposed for information from distinct sources, among which are the well-known Dempster’s rule [22], Smets’ rule [24], Yager’s rule [29], and Dubois & Prade’s hybrid rule [6], etc. In [5], two combination rules, i.e., the cautious rule and the bold disjunctive rule, for information from non-distinct sources are proposed.
CCTV (Closed-Circuit TeleVision) systems are broadly deployed in the present world. To ensure in-time reaction for intelligent surveillance, it is a fundamental task for real-world applications to determine the gender of people of interest. However, normal video algorithms for gender profiling (usually face profiling) have three drawbacks. First, the profiling result is always uncertain. Second, the profiling result is not stable. The degree of certainty usually varies over time, sometimes even to the extent that a male is classified as a female, and vice versa. Third, for a robust profiling result in cases that a person’s face is not visible, other features, such as body shape, are required. These algorithms may provide different recognition results - at the very least, they will provide different degrees of certainties. To overcome these problems, in this paper, we introduce an Dempster-Shafer (DS) evidential approach that makes use of profiling results from multiple algorithms over a period of time, in particular, Denoeux’s cautious rule is applied for fusing mass functions through time lines. Experiments show that this approach does provide better results than single profiling results and classic fusion results. Furthermore, it is found that if severe mis-classification has occurred at the beginning of the time line, the combination can yield undesirable results. To remedy this weakness, we further propose three extensions to the evidential approach proposed above incorporating notions of time-window, time-attenuation, and time-discounting, respectively. These extensions also applies Denoeux’s rule along with time lines and take the DS approach as a special case. Experiments show that these three extensions do provide better results than their predecessor when mis-classifications occur.
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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.
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2015, International Journal of Approximate Reasoning
Citation Excerpt :
There are also works, see for example [1], where you can find argumentation that the classical D–S rule is not justified in probability theory. In this paper we investigate the generalized Dempster–Shafer (GD–S) rules that were firstly introduced by Dubois and Yager [15], where each GD–S rule is defined as follows. In the D–S theory each source of information can be described by a random set.

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Fuzzy set connectives as combinations of belief structures

Abstract

Fuzzy Sets Syst.

J. Math. Anal. Appl.

Europ. J. Operations Res.

A Mathematical Theory of Evidence

Decision, Order and Time in Human Affairs

On several representations of an uncertain body of evidence

Possibility Theory: an Approach to Computerized Processing of Uncertainty

Identification of fuzzy sets with a class of cannonically induced random sets

Interpretation of membership functions of fuzzy sets in terms of plausibility and belief