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تسجيل مستخدم جديدEfficient Mobius Transformations and their applications to Dempster-Shafer Theory: Clarification and implementation
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Dempster-Shafer Theory (DST) generalizes Bayesian probability theory, offering useful additional information, but suffers from a high computational burden. A lot of work has been done to reduce the complexity of computations used in information fusion with Dempsters rule. The main approaches exploit either the structure of Boolean lattices or the information contained in belief sources. Each has its merits depending on the situation. In this paper, we propose sequences of graphs for the computation of the zeta and Mobius transformations that optimally exploit both the structure of distributive semilattices and the information contained in belief sources. We call them the Efficient Mobius Transformations (EMT). We show that the complexity of the EMT is always inferior to the complexity of algorithms that consider the whole lattice, such as the Fast Mobius Transform (FMT) for all DST transformations. We then explain how to use them to fuse two belief sources. More generally, our EMTs apply to any function in any finite distributive lattice, focusing on a meet-closed or join-closed subset. This article extends our work published at the international conference on Scalable Uncertainty Management (SUM). It clarifies it, brings some minor corrections and provides implementation details such as data structures and algorithms applied to DST.
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Dempster-Shafer Theory (DST) generalizes Bayesian probability theory, offering useful additional information, but suffers from a much higher computational burden. A lot of work has been done to reduce the time complexity of information fusion with De
mpsters rule, which is a pointwise multiplication of two zeta transforms, and optimal general algorithms have been found to get the complete definition of these transforms. Yet, it is shown in this paper that the zeta transform and its inverse, the Mobius transform, can be exactly simplified, fitting the quantity of information contained in belief functions. Beyond that, this simplification actually works for any function on any partially ordered set. It relies on a new notion that we call focal point and that constitutes the smallest domain on which both the zeta and Mobius transforms can be defined. We demonstrate the interest of these general results for DST, not only for the reduction in complexity of most transformations between belief representations and their fusion, but also for theoretical purposes. Indeed, we provide a new generalization of the conjunctive decomposition of evidence and formulas uncovering how each decomposition weight is tied to the corresponding mass function.
The article provides a review of the publications on the current trends and developments in Dempster-Shafer theory and its different applications in science, engineering, and technologies. The review took account of the following provisions with a fo
cus on some specific aspects of the theory. Firstly, the article considers the research directions whose results are known not only in scientific and academic community but understood by a wide circle of potential designers and developers of advanced engineering solutions and technologies. Secondly, the article shows the theory applications in some important areas of human activity such as manufacturing systems, diagnostics of technological processes, materials and products, building and construction, product quality control, economic and social systems. The particular attention is paid to the current state of research in the domains under consideration and, thus, the papers published, as a rule, in recent years and presenting the achievements of modern research on Dempster-Shafer theory and its application are selected and analyzed.
In analogy with the regularity lemma of Szemeredi, regularity lemmas for polynomials shown by Green and Tao (Contrib. Discrete Math. 2009) and by Kaufman and Lovett (FOCS 2008) modify a given collection of polynomials calF = {P_1,...,P_m} to a new co
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