Differential Geometry And Statistics
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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: 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 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.
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 Geometrical Methods In Statistics
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Author : Shun-ichi Amari
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Differential Geometrical Methods In Statistics written by Shun-ichi Amari and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 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
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.
Aspects Of Differential Geometry Iii
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Author : Esteban Calviño-Louzao
language : en
Publisher: Springer Nature
Release Date : 2022-05-31
Aspects Of Differential Geometry Iii written by Esteban Calviño-Louzao and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-05-31 with Mathematics categories.
Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book III is aimed at the first-year graduate level but is certainly accessible to advanced undergraduates. It deals with invariance theory and discusses invariants both of Weyl and not of Weyl type; the Chern‒Gauss‒Bonnet formula is treated from this point of view. Homothety homogeneity, local homogeneity, stability theorems, and Walker geometry are discussed. Ricci solitons are presented in the contexts of Riemannian, Lorentzian, and affine geometry.
Aspects Of Differential Geometry Iv
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Author : Esteban Calviño-Louzao
language : en
Publisher: Morgan & Claypool Publishers
Release Date : 2019-04-18
Aspects Of Differential Geometry Iv written by Esteban Calviño-Louzao and has been published by Morgan & Claypool Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-04-18 with Mathematics categories.
Book IV continues the discussion begun in the first three volumes. Although it is aimed at first-year graduate students, it is also intended to serve as a basic reference for people working in affine differential geometry. It also should be accessible to undergraduates interested in affine differential geometry. We are primarily concerned with the study of affine surfaces which are locally homogeneous. We discuss affine gradient Ricci solitons, affine Killing vector fields, and geodesic completeness. Opozda has classified the affine surface geometries which are locally homogeneous; we follow her classification. Up to isomorphism, there are two simply connected Lie groups of dimension 2. The translation group R2 is Abelian and the ???? + ?? group is non-Abelian. The first chapter presents foundational material. The second chapter deals with Type ?? surfaces. These are the left-invariant affine geometries on R2. Associating to each Type ?? surface the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue ?? = -1 turns out to be a very powerful technique and plays a central role in our study as it links an analytic invariant with the underlying geometry of the surface. The third chapter deals with Type ?? surfaces; these are the left-invariant affine geometries on the ???? + ?? group. These geometries form a very rich family which is only partially understood. The only remaining homogeneous geometry is that of the sphere ??2. The fourth chapter presents relations between the geometry of an affine surface and the geometry of the cotangent bundle equipped with the neutral signature metric of the modified Riemannian extension.
Geometric Modeling In Probability And Statistics
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Author : Ovidiu Calin
language : en
Publisher: Springer
Release Date : 2014-07-17
Geometric Modeling In Probability And Statistics written by Ovidiu Calin and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-17 with Mathematics categories.
This book covers topics of Informational Geometry, a field which deals with the differential geometric study of the manifold probability density functions. This is a field that is increasingly attracting the interest of researchers from many different areas of science, including mathematics, statistics, geometry, computer science, signal processing, physics and neuroscience. It is the authors’ hope that the present book will be a valuable reference for researchers and graduate students in one of the aforementioned fields. This textbook is a unified presentation of differential geometry and probability theory, and constitutes a text for a course directed at graduate or advanced undergraduate students interested in applications of differential geometry in probability and statistics. The book contains over 100 proposed exercises meant to help students deepen their understanding, and it is accompanied by software that is able to provide numerical computations of several information geometric objects. The reader will understand a flourishing field of mathematics in which very few books have been written so far.
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