Interpreting evidential distances by connecting them to partial orders: Application to belief function approximation

Klein, John; Destercke, Sébastien; Colot, Olivier

doi:10.1016/j.ijar.2016.01.001

Cited by 12 publications

(17 citation statements)

References 28 publications

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“…Considering results in [20], the operator f,k can be applied for f ∈ {pl, q}. To our knowledge, there is no evidential distance reported to be w , d or s -compatible in the literature, hence that f,k can be easily applied for f ∈ {w, d, s} is not guaranteed.…”

Section: New Conjunctive Operators From Soft Lcpmentioning

confidence: 99%

“…An evidential distance is a function d : M × M → [0, ∞] that satisfies the symmetry, definiteness and triangle inequality properties. In [20], we have formalized the idea of compatibility between a distance and a partial order in the following way: Definition 1. Given a partial order f defined on M, an evidential distance d is said to be f -compatible (in the strict sense) if for any mass functions m 1 , m 2 and m 3 such that m 1 f m 2 f m 3 , we have:…”

Section: Evidential Distances and Their Compatibility With Partial Ormentioning

confidence: 99%

“…According to corollary 3 in [20], we know that this problem is a convex optimization problem with a unique solution if the chosen distance d f,k is fcompatible and if 2 ≤ k < ∞. Let f,k denote this operator which is formally defined as follows: Definition 3. for any set of functions {m 1 , .., m }, we have…”

Section: Disjunctive Combination Using Partial Ordersmentioning

confidence: 99%

“…In this paper, we take inspiration from some of our previous work [20] studying the consistency of distances with partial orders comparing informative contents to propose a new way to derive cautious combination rules. Our approach departs from previous ones, as it is formulated as an optimization problem (similarly to what is done by Cattaneo [3] for conflict minimization) that naturally satisfies the LCP principle.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Idempotent conjunctive and disjunctive combination of belief functions by distance minimization

Klein

Destercke

Colot

2018

International Journal of Approximate Reasoning

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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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Section: New Conjunctive Operators From Soft Lcpmentioning

confidence: 99%

Section: Evidential Distances and Their Compatibility With Partial Ormentioning

confidence: 99%

Section: Disjunctive Combination Using Partial Ordersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Idempotent conjunctive and disjunctive combination of belief functions by distance minimization

Klein

Destercke

Colot

2018

International Journal of Approximate Reasoning

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“…These distances are not consistent with informational partial orders that generalize set inclusion. See[20] for a definition of the consistency of mass function distances with partial orders.…”

mentioning

confidence: 99%

Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces

Klein

2019

International Journal of Approximate Reasoning

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We prove that intersections and unions of independent random sets in finite spaces achieve a form of Lipschitz continuity. More precisely, given the distribution of a random set Ξ, the function mapping any random set distribution to the distribution of its intersection (under independence assumption) with Ξ is Lipschitz continuous with unit Lipschitz constant if the space of random set distributions is endowed with a metric defined as the L k norm distance between inclusion functionals also known as commonalities. Moreover, the function mapping any random set distribution to the distribution of its union (under independence assumption) with Ξ is Lipschitz continuous with unit Lipschitz constant if the space of random set distributions is endowed with a metric defined as the L k norm distance between hitting functionals also known as plausibilities.Using the epistemic random set interpretation of belief functions, we also discuss the ability of these distances to yield conflict measures. All the proofs in this paper are derived in the framework of Dempster-Shafer belief functions. Let alone the discussion on conflict measures, it is straightforward to transcribe the proofs into the general (non necessarily epistemic) random set terminology.

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An association coefficient of a belief function and its application in a target recognition system

Pan

Deng

2019

Int J Intell Syst

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The conflict problem in D-S evidence theory has attracted the attention of many scholars. Conflict coefficients are proposed to describe conflicts between bodies of evidence. The association coefficient as the opposite of the conflict coefficient is also used to measure the conflict. The larger the association coefficient, the smaller the conflict degree, and the higher the similarity between the evidence bodies, and vice versa.In this paper, the degree of association is defined by Deng Entropy, and a new association coefficient is proposed based on the basic inequality. The nature of the new association coefficient and conflict coefficients is explored using examples. Finally, the association coefficient combined with the D-S combination rule is applied to the target recognition system, and accurate results are obtained.association coefficient, conflict coefficients, conflict problem, Deng entropy, D-S evidence theory, target recognition system | INTRODUCTIONThe D-S evidence theory was first proposed by Dempster 1 in 1967 and later developed by his student Shafer 2 in 1976 as an imprecise reasoning theory, which is widely promoted. [3][4][5][6] The theory was first applied to expert systems, 7-9 and it was gradually applied in the fields of information fusion, 10-12 intelligence analysis, 9,13-16 evidential reasoning, [17][18][19] and multiattribute decision analysis. [20][21][22][23][24] As an indeterminate reasoning method which satisfies the conditions weaker than Bayesian probability theory; it has the ability to directly express "uncertainty" 25 and "don't know." 26 Therefore, the D-S evidence theory is flexible and convenient for describing uncertainties. However, D-S evidence theory has produced counterintuitive results in the fusion of a highly conflicting example proposed by Zadeh. 27 Therefore, the issue of conflict between bodies of evidence has attracted more and more researchers so far. [28][29][30][31][32] Domestic and foreign scholars have proposed a large number of improved methods, which can be divided into two major categories: (a) modification of the combination rules of the classical D-S evidence theory to achieve the redistribution of conflicts in the case of evidence conflicts. [33][34][35][36] Such measures alleviate the conflict between the bodies of evidence to a certain extent, but at the same time, it is difficult for the new rules to maintain the original properties of the D-S combination rules. (b) Keep the classic combination rules unchanged, and preprocess conflict data before combination. [37][38][39][40][41][42] Haenni 43 believes that the modification of the data model is in engineering, mathematics and philosophically more reasonable. However, regardless of the type of method used to deal with conflicts between evidence, the conflict coefficient is an extremely important parameter. Therefore, the most critical step before choosing the right combination of evidence is the measure of evidence conflict.To measure the conflict between evidence bodies, many researchers have proposed...

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Interpreting evidential distances by connecting them to partial orders: Application to belief function approximation

Cited by 12 publications

References 28 publications

Idempotent conjunctive and disjunctive combination of belief functions by distance minimization

Idempotent conjunctive and disjunctive combination of belief functions by distance minimization

Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces

An association coefficient of a belief function and its application in a target recognition system

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