On Differential Geometry In Statistics
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Differential Geometry And Statistics
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Author : M.K. Murray
language : en
Publisher: Routledge
Release Date : 2017-10-19
Differential Geometry And Statistics written by M.K. Murray and has been published by Routledge this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-10-19 with Mathematics categories.
Several years ago our statistical friends and relations introduced us to the work of Amari and Barndorff-Nielsen on applications of differential geometry to statistics. This book has arisen because we believe that there is a deep relationship between statistics and differential geometry and moreoever that this relationship uses parts of differential geometry, particularly its 'higher-order' aspects not readily accessible to a statistical audience from the existing literature. It is, in part, a long reply to the frequent requests we have had for references on differential geometry! While we have not gone beyond the path-breaking work of Amari and Barndorff- Nielsen in the realm of applications, our book gives some new explanations of their ideas from a first principles point of view as far as geometry is concerned. In particular it seeks to explain why geometry should enter into parametric statistics, and how the theory of asymptotic expansions involves a form of higher-order differential geometry. The first chapter of the book explores exponential families as flat geometries. Indeed the whole notion of using log-likelihoods amounts to exploiting a particular form of flat space known as an affine geometry, in which straight lines and planes make sense, but lengths and angles are absent. We use these geometric ideas to introduce the notion of the second fundamental form of a family whose vanishing characterises precisely the exponential families.
Differential Geometry In Statistical Inference
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Author : Shun'ichi Amari
language : en
Publisher: IMS
Release Date : 1987
Differential Geometry In Statistical Inference written by Shun'ichi Amari and has been published by IMS this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987 with Geometry, Differential categories.
Differential Geometry And Statistics
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Author : M.K. Murray
language : en
Publisher: CRC Press
Release Date : 1993-04-01
Differential Geometry And Statistics written by M.K. Murray and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-04-01 with Mathematics categories.
Ever since the introduction by Rao in 1945 of the Fisher information metric on a family of probability distributions, there has been interest among statisticians in the application of differential geometry to statistics. This interest has increased rapidly in the last couple of decades with the work of a large number of researchers. Until now an impediment to the spread of these ideas into the wider community of statisticians has been the lack of a suitable text introducing the modern coordinate free approach to differential geometry in a manner accessible to statisticians. Differential Geometry and Statistics aims to fill this gap. The authors bring to this book extensive research experience in differential geometry and its application to statistics. The book commences with the study of the simplest differentiable manifolds - affine spaces and their relevance to exponential families, and goes on to the general theory, the Fisher information metric, the Amari connections and asymptotics. It culminates in the theory of vector bundles, principal bundles and jets and their applications to the theory of strings - a topic presently at the cutting edge of research in statistics and differential geometry.
On Differential Geometry In Statistics
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Author : M. K. Murray
language : en
Publisher:
Release Date : 1986
On Differential Geometry In Statistics written by M. K. Murray and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1986 with Geometry, Differential categories.
Differential Geometry And Statistics
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Author : M. K. Murray
language : en
Publisher: Springer
Release Date : 2013-10-29
Differential Geometry And Statistics written by M. K. Murray and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-10-29 with Mathematics categories.
Several years ago our statistical friends and relations introduced us to the work of Amari and Barndorff-Nielsen on applications of differential geometry to statistics. This book has arisen because we believe that there is a deep relationship between statistics and differential geometry and moreoever that this relationship uses parts of differential geometry, particularly its 'higher-order' aspects not readily accessible to a statistical audience from the existing literature. It is, in part, a long reply to the frequent requests we have had for references on differential geometry! While we have not gone beyond the path-breaking work of Amari and Barndorff Nielsen in the realm of applications, our book gives some new explanations of their ideas from a first principles point of view as far as geometry is concerned. In particular it seeks to explain why geometry should enter into parametric statistics, and how the theory of asymptotic expansions involves a form of higher-order differential geometry. The first chapter of the book explores exponential families as flat geometries. Indeed the whole notion of using log-likelihoods amounts to exploiting a particular form of flat space known as an affine geometry, in which straight lines and planes make sense, but lengths and angles are absent. We use these geometric ideas to introduce the notion of the second fundamental form of a family whose vanishing characterises precisely the exponential families.
Differential Geometry In Statistical Inference
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Author : Shunʼichi Amari
language : en
Publisher:
Release Date : 2008*
Differential Geometry In Statistical Inference written by Shunʼichi Amari and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008* with Geometry, Differential categories.
This e-book is the product of Project Euclid and its mission to advance scholarly communication in the field of theoretical and applied mathematics and statistics. Project Euclid was developed and deployed by the Cornell University Library and is jointly managed by Cornell and the Duke University Press.
Differential Geometry And Statistics
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Author : Michael K. Murray
language : en
Publisher:
Release Date : 200?
Differential Geometry And Statistics written by Michael K. Murray and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 200? with MATHEMATICS categories.
It culminates in the theory of vector bundles, principal bundles and jets and their applications to the theory of strings - a topic presently at the cutting edge of research in statistics and differential geometry.
Differential Geometrical Methods In Statistics
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Author : Shun-ichi Amari
language : en
Publisher: Springer
Release Date : 1990-02-14
Differential Geometrical Methods In Statistics written by Shun-ichi Amari and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990-02-14 with Mathematics categories.
From the reviews: "In this Lecture Note volume the author describes his differential-geometric approach to parametrical statistical problems summarizing the results he had published in a series of papers in the last five years. The author provides a geometric framework for a special class of test and estimation procedures for curved exponential families. ... ... The material and ideas presented in this volume are important and it is recommended to everybody interested in the connection between statistics and geometry ..." #Metrika#1 "More than hundred references are given showing the growing interest in differential geometry with respect to statistics. The book can only strongly be recommended to a geodesist since it offers many new insights into statistics on a familiar ground." #Manuscripta Geodaetica#2
Global Affine Differential Geometry Of Hypersurfaces
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Author : An-Min Li
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2015-08-17
Global Affine Differential Geometry Of Hypersurfaces written by An-Min Li and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-08-17 with Mathematics categories.
This book draws a colorful and widespread picture of global affine hypersurface theory up to the most recent state. Moreover, the recent development revealed that affine differential geometry – as differential geometry in general – has an exciting intersection area with other fields of interest, like partial differential equations, global analysis, convex geometry and Riemann surfaces. The second edition of this monograph leads the reader from introductory concepts to recent research. Since the publication of the first edition in 1993 there appeared important new contributions, like the solutions of two different affine Bernstein conjectures, due to Chern and Calabi, respectively. Moreover, a large subclass of hyperbolic affine spheres were classified in recent years, namely the locally strongly convex Blaschke hypersurfaces that have parallel cubic form with respect to the Levi-Civita connection of the Blaschke metric. The authors of this book present such results and new methods of proof.
Riemannian Geometric Statistics In Medical Image Analysis
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Author : Xavier Pennec
language : en
Publisher: Academic Press
Release Date : 2019-09-02
Riemannian Geometric Statistics In Medical Image Analysis written by Xavier Pennec and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-09-02 with Computers categories.
Over the past 15 years, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data. Riemannian Geometric Statistics in Medical Image Analysis is a complete reference on statistics on Riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. It provides an introduction to the core methodology followed by a presentation of state-of-the-art methods. Beyond medical image computing, the methods described in this book may also apply to other domains such as signal processing, computer vision, geometric deep learning, and other domains where statistics on geometric features appear. As such, the presented core methodology takes its place in the field of geometric statistics, the statistical analysis of data being elements of nonlinear geometric spaces. The foundational material and the advanced techniques presented in the later parts of the book can be useful in domains outside medical imaging and present important applications of geometric statistics methodology Content includes: - The foundations of Riemannian geometric methods for statistics on manifolds with emphasis on concepts rather than on proofs - Applications of statistics on manifolds and shape spaces in medical image computing - Diffeomorphic deformations and their applications As the methods described apply to domains such as signal processing (radar signal processing and brain computer interaction), computer vision (object and face recognition), and other domains where statistics of geometric features appear, this book is suitable for researchers and graduate students in medical imaging, engineering and computer science. - A complete reference covering both the foundations and state-of-the-art methods - Edited and authored by leading researchers in the field - Contains theory, examples, applications, and algorithms - Gives an overview of current research challenges and future applications