Research On Applications Of Geometric Theory On Manifold Of Elliptical Distributions

Information geometry,as a synthesis of information theory,probability theory and differential geometry,is a new subject developed from the research on intrinsic geometric properties of statistical manifolds.It provides a powerful theoretical tool for system control,signal processing,neural network and etc.Besides,there are many applications involving non-Gauss population in theory and practice,so the class of multivariate elliptical distributions(MEDs)has received extensive attention.In this thesis,we study the geometric theory on the manifold of MEDs and its applications to signal processing.The main achievements are summarized as follows:1.Due to no explicit expression for the Rao distance on MED manifold,we propose a class of Manhattan distances with single parameter in which each element is an upper bound for the Rao distance.As a natural intrinsic measure,the Rao distance on statistical manifold has been successfully applied in many fields,but there is still no explicit expression for it on common statistical manifolds such as the Gaussian manifold.For general MED manifold,computationally efficient or analytical tight upper bound is rarely found in existing literatures.Therefore we derive the closed form of Rao distance on a specified MED submanifold having the coordinate system with mean vector and positive real variable,and then a class of Manhattan distances(MHDs)with single parameter is constructed in which each element is an upper bound of Rao distance on MED manifold.The minimal MHD(MMHD),taking the minimum in such class,can be efficiently solved owing to its convexity with respect to the scalar parameter.In contrast,the quasi-minimal MHD(QMMHD)is an approximated but analytical version of MMHD and then can be more efficiently applied to real-time dynamic systems.The performance of the proposed MHDs for approximating the Rao distance is illustrated by examples.2.Some information-geometric methods for distributed estimation fusion are proposed by taking Manhattan distance on MED manifold as loss function.Traditional fusion methods only consider the first two moments of local estimator.In contrast,information geometry takes the intrinsic Rao distance on the statistical manifold of local posterior probability densities as the loss function in the fusion model.It is more beneficial to improve the performance of estimation fusion.In this thesis,we replace the Rao distance in the geometric fusion model with the constructed MHDs to reduce the optimization complexity,and then two efficient fusion algorithms are proposed.Simulation has illustrates that the proposed QMMHD and MMHD fusion algorithms outperform other fusion methods.In contrast to the MMHD fusion,the QMMHD fusion only takes less computational time but attains the very close performance.However,in the QMMHD fusion,a suggested covariance matrix estimate is subjectively given owing to the fused mean estimate having the same form as traditional covariance intersection fusion method.Consequently,a decoupling fusion method is proposed through taking MHD projections from all local posterior densities onto the MED submanifold with a constant mean.Numerical experiments show that the improved fusion method has better fusion performance than the QMMHD.3.We derive the analytical expressions of geodesic projection and geodesic projection distance on the MED submanifold with a fixed mean vector,and study their applications to signal processing.Getting around the difficulty with no analytical expression for Rao distance between two points,the explicit geodesic projection on the Gaussian manifold has been successfully applied.However,commonly used MED manifold has more complex geodesic equations than the Gaussian manifold.In this thesis,we derive the explicit geodesic projection from a given point in MED manifold onto its submanifold with a constant mean.Here,a novel technique is developed for solving the geodesic projection problem by constructing a symmetric matrix with the geodesic equations.Moreover,this explicit geodesic projection as the optimal approximation on the MED submanifold is provided for studying three specified applications.Firstly,the constructed test statistic extends Hotelling T-squared test for multivariate Gaussian distributions to MEDs via utilizing the geodesic projection distance.Secondly,a shrinkage geodesic projection estimator is proposed to address the joint mean-covariance estimation problem with a shortage of samples by replacing prior target with its geodesic projection.Thirdly,two distributed estimation fusion methods based on the geodesic projection and geodesic projection distance are derived respectively.Numerical experiments show that the geodesic projection and geodesic projection distance have high research values in the application of signal processing.