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    Wyświetlanie 1-3 z 3
    Tytuł:
    Information geometry of divergence functions
    Autorzy:
    Amari, S.
    Cichocki, A.
    Powiązania:
    https://bibliotekanauki.pl/articles/199901.pdf
    Data publikacji:
    2010
    Wydawca:
    Polska Akademia Nauk. Czytelnia Czasopism PAN
    Tematy:
    information geometry
    divergence functions
    Opis:
    Measures of divergence between two points play a key role in many engineering problems. One such measure is a distance function, but there are many important measures which do not satisfy the properties of the distance. The Bregman divergence, Kullback-Leibler divergence and f-divergence are such measures. In the present article, we study the differential-geometrical structure of a manifold induced by a divergence function. It consists of a Riemannian metric, and a pair of dually coupled affine connections, which are studied in information geometry. The class of Bregman divergences are characterized by a dually flat structure, which is originated from the Legendre duality. A dually flat space admits a generalized Pythagorean theorem. The class of f-divergences, defined on a manifold of probability distributions, is characterized by information monotonicity, and the Kullback-Leibler divergence belongs to the intersection of both classes. The f-divergence always gives the -geometry, which consists of the Fisher information metric and a dual pair of š-connections. The -divergence is a special class of f-divergences. This is unique, sitting at the intersection of the f-divergence and Bregman divergence classes in a manifold of positive measures. The geometry derived from the Tsallis q-entropy and related divergences are also addressed.
    Źródło:
    Bulletin of the Polish Academy of Sciences. Technical Sciences; 2010, 58, 1; 183-195
    0239-7528
    Pojawia się w:
    Bulletin of the Polish Academy of Sciences. Technical Sciences
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Information Geometry of Frechet Distributions
    Autorzy:
    Arwini, Khadiga Ali
    Powiązania:
    https://bibliotekanauki.pl/articles/1030590.pdf
    Data publikacji:
    2020
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    Frechet distribution
    extreme value distributions
    information geometry
    statistical manifold
    Opis:
    Using the Fisher information matrix (FIM) as a Riemannian metric, the family of Frechet distributions determines a two dimensional Riemannian manifold. In this paper we illustrates the information geometry of the Frechet space, and derive the α-geometry as; α-connections, α-curvature tensor, α-Ricci curvature with its eigenvalues and eigenvectors, α-sectional curvature, α-mean curvature, and α-scalar curvature, where we show that Frechet space has a constant α-scalar curvature. The special case where α = 0 corresponds to the geometry induced by the Levi-Civita connection. In addition, we consider three special cases of Frechet distributions as submanifolds with dimension one, and discuss their geometrical structures, then we prove that one of these submanifolds is an isometric isomorph of the exponential manifold, which is important in stochastic process since exponential distributions represent intervals between events for Poisson processes. After that, we introduce log-Frechet distributions, and show that this family of distributions determines a Riemannian 2-manifold which is isometric with the origin manifold. Finally, an explicit expressions for some distances in Frechet space are obtained as, Kullback-Leibler distance, and J-divergence.
    Źródło:
    World Scientific News; 2020, 144; 296-312
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Geometrical Structures of Bivariate Gamma Exponential Distributions
    Autorzy:
    Arwini, Khadiga Ali
    Powiązania:
    https://bibliotekanauki.pl/articles/1030111.pdf
    Data publikacji:
    2020
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    Pareto distribution
    bivariate distributions
    bivariate gamma exponential distribution
    gamma distribution
    information geometry
    statistical manifold
    Opis:
    This paper is devoted to the information geometry of the family of bivariate gamma exponential distributions, which have gamma and Pareto marginals, and discuss some of its applications. We begin by considering the parameter bivariate gamma exponential manifold as a Riemannian 3-manifold; by following Rao’s idea to use the Fisher information matrix (FIM), and derive the α-geometry as: α-connections, α-curvature tensor, α-Ricci curvature with its eigenvalues and eigenvectors, and α-scalar curvature. Where here the 0-geometry corresponds to the geometry induced by the Levi-Civita connection, and we show that this space has a non-constant negative scalar curvature. In addition, we consider four submanifolds as special cases, and discuss their geometrical structures, and we prove that one of these submanifolds is an isometric isomorph of the univariate gamma manifold. Then we introduce log-bivariate gamma exponential distributions, which have log-gamma and log-Pareto marginals, and we show that this family of distributions determines a Riemannian 3-manifold which is isometric with the origin manifold. We give an analytical solution for the geodesic equations, and obtain the explicit expressions for Kullback-Leibler distance, J-divergence and Bhattacharyya distance. Finally, we prove that the bivariate gamma exponential manifold can be realized in R4, using information theoretic immersions, and we give explicit information geometric tubular neighbourhoods for some special cases.
    Źródło:
    World Scientific News; 2020, 143; 181-202
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-3 z 3

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