New distances between bodies of evidence based on Dempsterian specialization matrices and their consistency with the conjunctive combination rule

Loudahi, Mehena; Klein, John; Vannobel, Jean-Marc; Colot, Olivier

doi:10.1016/j.ijar.2014.02.007

Cited by 17 publications

(19 citation statements)

References 26 publications

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“…See D for proof. As compared to previous Lipschitz continuity results [25,26], specialization distances have a greater time complexity as compared to commonality ones. Indeed, although the construction of specialization distances can be sped up [24], the time complexity for the specialization distance is quadratic in N. More precisely, the time complexity to build a specialization matrix is…”

Section: Main Results On Lipschitz Continuity For the Conjunctive Rulementioning

confidence: 54%

“…Proving that a fusion operator achieves Lipschitz continuity is not trivial because M is not finite but instead a compact subset of an uncountable space. In [25], Loudahi et al established the consistency of the L 1 and L ∞ based specialization distances w.r.t. the conjunctive and disjunctive rules.…”

Section: Union Intersection and Random Set Distribution Distancesmentioning

confidence: 99%

“…The authors proved that the consistency under study holds between some belief function distances and combination operators that yield the distribution of the intersection or union of independent random sets. The results that we introduce in this paper involves distances that are computationally more tractable than those introduced in [25,26] but also rely on independence assumptions. In the random set literature, many results regarding unions of i.i.d.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Main Results On Lipschitz Continuity For the Conjunctive Rulementioning

confidence: 54%

Section: Union Intersection and Random Set Distribution Distancesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…An evidence discount thought based on the evidence source reliability proposed by Shafer has corrected the evidence source belief function but has not provided the specific algorithm for the discount rate. Many scholars have studied the specific algorithm for the discount rate from different perspectives, such as evidence processing based on the evidence weights (Van‐Nam, Nakamori, Tu‐Bao, & Murai, ; Xu et al, ; Yang et al, ), determination of correction factors based on the evidence distance (Jousselme & Maupin, ; Loudahi, Klein, Vannobel, & Colot, ), and determination of factors based on the evidence‐related (Wen, Zhengyou, & Yaode, ). However, determination of discount factors is provided with some limitations: (a) Discount factors are determined for a certain aspect related to the evidence or evidence conflict and cannot reflect the relationship among evidence sources comprehensively; (b) Correction factors are calculated mainly based on simple calculation of basic probability assignment (BPA) values of all evidence, and determination of discount factors is rather one‐sided.…”

Section: Introductionmentioning

confidence: 99%

Determination of evidence correction factors based on the neural network

Zhu

et al. 2017

Expert Systems

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A modified method to combine evidence based on artificial neural network and Dempster's rule of combination is proposed. The comprehensive discounting factor of this method adds learning and data mining capability to evidence combination, making it more suitable for the interrelated, conflicted evidence, or evidence with different importance and more suitable for group decision making. This method can be used for repeatable and verifiable systems. A study about securities market experts group prediction is conducted to verify the effectiveness of the proposed method.

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“…Jousselme and Maupin [17] analyzed and surveyed the evidential distance literature and clarified the advantages and limitations of evidential distances. Based on their comments, we formalized in [21] a number of desirable mathematical properties for evidential distances and proposed specific distances possessing these properties. These distances are obtained using the norm of the difference between matrices encoding bodies of evidence.…”

Section: Introductionmentioning

confidence: 99%

Evidential Matrix Metrics as Distances Between Meta-Data Dependent Bodies of Evidence

Loudahi

Klein

Vannobel

et al. 2016

IEEE Trans. Cybern.

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As part of the theory of belief functions, we address the problem of appraising the similarity between bodies of evidence in a relevant way using metrics. Such metrics are called evidential distances and must be computed from mathematical objects depicting the information inside bodies of evidence. Specialization matrices are such objects and, therefore, an evidential distance can be obtained by computing the norm of the difference of these matrices. Any matrix norm can be thus used to define a full metric. In this paper, we show that other matrices can be used to obtain new evidential distances. These are the α -specialization and α -generalization matrices and are closely related to the α -junctive combination rules. We prove that any L(1) norm-based distance thus defined is consistent with its corresponding α -junction. If α > 0 , these distances have in addition relevant variations induced by the poset structure of the belief function domain. Furthermore, α -junctions are meta-data dependent combination rules. The meta-data involved in α -junctions deals with the truthfulness of information sources. Consequently, the behavior of such evidential distances is analyzed in situations involving uncertain or partial meta-knowledge about information source truthfulness.

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New distances between bodies of evidence based on Dempsterian specialization matrices and their consistency with the conjunctive combination rule

Cited by 17 publications

References 26 publications

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

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

Determination of evidence correction factors based on the neural network

Evidential Matrix Metrics as Distances Between Meta-Data Dependent Bodies of Evidence

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