Differential Geometry and Statistics

Author: M.K. Murray
Publisher: Routledge
ISBN: 9781351455114
Release Date: 2017-10-19
Genre: Mathematics

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

Author: M.K. Murray
Publisher: Routledge
ISBN: 9781351455114
Release Date: 2017-10-19
Genre: Mathematics

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.

Geometrical Foundations of Asymptotic Inference

Author: Robert E. Kass
Publisher: John Wiley & Sons
ISBN: 9781118165973
Release Date: 2011-09-09
Genre: Mathematics

Differential geometry provides an aesthetically appealing and often revealing view of statistical inference. Beginning with an elementary treatment of one-parameter statistical models and ending with an overview of recent developments, this is the first book to provide an introduction to the subject that is largely accessible to readers not already familiar with differential geometry. It also gives a streamlined entry into the field to readers with richer mathematical backgrounds. Much space is devoted to curved exponential families, which are of interest not only because they may be studied geometrically but also because they are analytically convenient, so that results may be derived rigorously. In addition, several appendices provide useful mathematical material on basic concepts in differential geometry. Topics covered include the following: * Basic properties of curved exponential families * Elements of second-order, asymptotic theory * The Fisher-Efron-Amari theory of information loss and recovery * Jeffreys-Rao information-metric Riemannian geometry * Curvature measures of nonlinearity * Geometrically motivated diagnostics for exponential family regression * Geometrical theory of divergence functions * A classification of and introduction to additional work in the field

Methods of Information Geometry

Author: Shun-ichi Amari
Publisher: American Mathematical Soc.
ISBN: 0821843028
Release Date: 2007
Genre: Mathematics

Information geometry provides the mathematical sciences with a new framework of analysis. It has emerged from the investigation of the natural differential geometric structure on manifolds of probability distributions, which consists of a Riemannian metric defined by the Fisher information and a one-parameter family of affine connections called the $\alpha$-connections. The duality between the $\alpha$-connection and the $(-\alpha)$-connection together with the metric play an essential role in this geometry. This kind of duality, having emerged from manifolds of probability distributions, is ubiquitous, appearing in a variety of problems which might have no explicit relation to probability theory. Through the duality, it is possible to analyze various fundamental problems in a unified perspective. The first half of this book is devoted to a comprehensive introduction to the mathematical foundation of information geometry, including preliminaries from differential geometry, the geometry of manifolds or probability distributions, and the general theory of dual affine connections. The second half of the text provides an overview of many areas of applications, such as statistics, linear systems, information theory, quantum mechanics, convex analysis, neural networks, and affine differential geometry. The book can serve as a suitable text for a topics course for advanced undergraduates and graduate students.

Differential Geometrical Methods in Statistics

Author: Shun-ichi Amari
Publisher: Springer Science & Business Media
ISBN: 9781461250562
Release Date: 2012-12-06
Genre: Mathematics

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

Algebraic Geometry and Statistical Learning Theory

Author: Sumio Watanabe
Publisher: Cambridge University Press
ISBN: 9780521864671
Release Date: 2009-08-13
Genre: Computers

Sure to be influential, Watanabe's book lays the foundations for the use of algebraic geometry in statistical learning theory. Many models/machines are singular: mixture models, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are major examples. The theory achieved here underpins accurate estimation techniques in the presence of singularities.

Applications of Differential Geometry to Econometrics

Author: Paul Marriott
Publisher: Cambridge University Press
ISBN: 0521651166
Release Date: 2000-08-31
Genre: Business & Economics

Although geometry has always aided intuition in econometrics, more recently differential geometry has become a standard tool in the analysis of statistical models, offering a deeper appreciation of existing methodologies and highlighting the essential issues which can be hidden in an algebraic development of a problem. Originally published in 2000, this volume was an early example of the application of these techniques to econometrics. An introductory chapter provides a brief tutorial for those unfamiliar with the tools of Differential Geometry. The topics covered in the following chapters demonstrate the power of the geometric method to provide practical solutions and insight into problems of econometric inference.

An Introduction to the Bootstrap

Author: Bradley Efron
Publisher: CRC Press
ISBN: 0412042312
Release Date: 1994-05-15
Genre: Mathematics

Statistics is a subject of many uses and surprisingly few effective practitioners. The traditional road to statistical knowledge is blocked, for most, by a formidable wall of mathematics. The approach in An Introduction to the Bootstrap avoids that wall. It arms scientists and engineers, as well as statisticians, with the computational techniques they need to analyze and understand complicated data sets.

Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis

Author: Victor Patrangenaru
Publisher: CRC Press
ISBN: 9781439820513
Release Date: 2015-09-18
Genre: Mathematics

A New Way of Analyzing Object Data from a Nonparametric Viewpoint Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis provides one of the first thorough treatments of the theory and methodology for analyzing data on manifolds. It also presents in-depth applications to practical problems arising in a variety of fields, including statistics, medical imaging, computer vision, pattern recognition, and bioinformatics. The book begins with a survey of illustrative examples of object data before moving to a review of concepts from mathematical statistics, differential geometry, and topology. The authors next describe theory and methods for working on various manifolds, giving a historical perspective of concepts from mathematics and statistics. They then present problems from a wide variety of areas, including diffusion tensor imaging, similarity shape analysis, directional data analysis, and projective shape analysis for machine vision. The book concludes with a discussion of current related research and graduate-level teaching topics as well as considerations related to computational statistics. Researchers in diverse fields must combine statistical methodology with concepts from projective geometry, differential geometry, and topology to analyze data objects arising from non-Euclidean object spaces. An expert-driven guide to this approach, this book covers the general nonparametric theory for analyzing data on manifolds, methods for working with specific spaces, and extensive applications to practical research problems. These problems show how object data analysis opens a formidable door to the realm of big data analysis.

Analysis for Diffusion Processes on Riemannian Manifolds

Author: Feng-Yu Wang
Publisher: World Scientific
ISBN: 9789814452663
Release Date: 2013-09-23
Genre: Mathematics

Stochastic analysis on Riemannian manifolds without boundary has been well established. However, the analysis for reflecting diffusion processes and sub-elliptic diffusion processes is far from complete. This book contains recent advances in this direction along with new ideas and efficient arguments, which are crucial for further developments. Many results contained here (for example, the formula of the curvature using derivatives of the semigroup) are new among existing monographs even in the case without boundary. Contents:PreliminariesDiffusion Processes on Riemannian Manifolds without BoundaryReflecting Diffusion Processes on Manifolds with BoundaryStochastic Analysis on Path Space over Manifolds with BoundarySubelliptic Diffusion Processes Readership: Graduate students, researchers and professionals in probability theory, differential geometry and partial differential equations. Keywords:Diffusion Pocess;Reflecting Diffusion Process;Neumann Semigroup;Curvature;Second Fundamental Form;ManifoldKey Features:First book where the key theory and machinery of the reflecting diffusion processes on Riemannian manifolds with boundary are systematically introducedFirst book to clarify intrinsic links between the semigroup properties on one hand and geometric quantities (curvature and second fundamental form) on the other, and these links are introduced in an easy to understand manner: by formulating geometric quantities using short time behaviors of derivatives of the semigroup, whereby a reader can easily comprehend the equivalence of semigroup properties associated with lower bounds of these geometric quantitiesFirst book where stochastic analysis on Riemannian manifolds with boundary are introduced

Nonparametric Inference on Manifolds

Author: Abhishek Bhattacharya
Publisher: Cambridge University Press
ISBN: 9781107019584
Release Date: 2012-04-05
Genre: Mathematics

A systematic introduction to a general nonparametric theory of statistics on manifolds, with emphasis on manifolds of shapes.

Information Geometry

Author: Khadiga Arwini
Publisher: Springer
ISBN: 9783540693932
Release Date: 2008-07-19
Genre: Mathematics

This volume uses information geometry to give a common differential geometric framework for a wide range of illustrative applications including amino acid sequence spacings, cryptology studies, clustering of communications and galaxies, and cosmological voids.

Introduction to High Dimensional Statistics

Author: Christophe Giraud
Publisher: CRC Press
ISBN: 9781482237955
Release Date: 2014-12-17
Genre: Business & Economics

Ever-greater computing technologies have given rise to an exponentially growing volume of data. Today massive data sets (with potentially thousands of variables) play an important role in almost every branch of modern human activity, including networks, finance, and genetics. However, analyzing such data has presented a challenge for statisticians and data analysts and has required the development of new statistical methods capable of separating the signal from the noise. Introduction to High-Dimensional Statistics is a concise guide to state-of-the-art models, techniques, and approaches for handling high-dimensional data. The book is intended to expose the reader to the key concepts and ideas in the most simple settings possible while avoiding unnecessary technicalities. Offering a succinct presentation of the mathematical foundations of high-dimensional statistics, this highly accessible text: Describes the challenges related to the analysis of high-dimensional data Covers cutting-edge statistical methods including model selection, sparsity and the lasso, aggregation, and learning theory Provides detailed exercises at the end of every chapter with collaborative solutions on a wikisite Illustrates concepts with simple but clear practical examples Introduction to High-Dimensional Statistics is suitable for graduate students and researchers interested in discovering modern statistics for massive data. It can be used as a graduate text or for self-study.

Random Fields and Geometry

Author: R. J. Adler
Publisher: Springer Science & Business Media
ISBN: 0387481168
Release Date: 2009-01-29
Genre: Mathematics

This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined. "Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory.

Geometric Modeling in Probability and Statistics

Author: Ovidiu Calin
Publisher: Springer
ISBN: 9783319077796
Release Date: 2014-07-17
Genre: Mathematics

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.