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Register a new userFocal points and their implications for Mobius Transforms and Dempster-Shafer Theory
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Maxime Chaveroche
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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 Dempsters 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.
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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.
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 focus 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.
The orthocomplemented modular lattice of subspaces L[H(d)], of a quantum system with d- dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities, is violated by quantum probabilities in the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)]). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H1,H2), which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H1),P(H2), to the subspaces H1,H2. As an application, it is shown that the proof of CHSH inequalities for a system of two spin 1/2 particles, is valid for Kolmogorov probabilities, but it is not valid for Dempster- Shafer probabilities. The violation of these inequalities in experiments, supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.
Tightly focused light beams can exhibit electric fields spinning around any axis including the one transverse to the beams propagation direction. At certain focal positions, the corresponding local polarization ellipse can degenerate into a perfect circle, representing a point of circular polarization, or C-point. We consider the most fundamental case of a linearly polarized Gaussian beam, where - upon tight focusing - those C-points created by transversely spinning fields can form the center of 3D optical polarization topologies when choosing the plane of observation appropriately. Due to the high symmetry of the focal field, these polarization topologies exhibit non trivial structures similar to Mobius strips. We use a direct physical measure to find C-points with an arbitrarily oriented spinning axis of the electric field and experimentally investigate the fully three-dimensional polarization topologies surrounding these C-points by exploiting an amplitude and phase reconstruction technique.
The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A.M. Whitneys density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Polya frequency functions, and Polya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.
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