Geometric Modeling In Probability And Statistics

Author : Ovidiu Calin,Constantin Udrişte
Publisher : Springer
Page : 375 pages
File Size : 43,8 Mb
Release : 2014-07-17
Category : Mathematics
ISBN : 9783319077796

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Geometric Modeling in Probability and Statistics by Ovidiu Calin,Constantin Udrişte Pdf

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.

Author : Ovidiu Calin,Constantin Udriste
Publisher : Springer
Page : 375 pages
File Size : 49,9 Mb
Release : 2014-08-01
Category : Mathematics
ISBN : 3319077783

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Geometric Modeling in Probability and Statistics by Ovidiu Calin,Constantin Udriste Pdf

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.

Author : Bruce G. Lindsay
Publisher : IMS
Page : 184 pages
File Size : 47,9 Mb
Release : 1995
Category : Mathematics
ISBN : 0940600323

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Mixture Models by Bruce G. Lindsay Pdf

Author : Hamid Krim,Anthony Yezzi
Publisher : Springer Science & Business Media
Page : 396 pages
File Size : 50,9 Mb
Release : 2007-12-31
Category : Mathematics
ISBN : 9780817644819

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Statistics and Analysis of Shapes by Hamid Krim,Anthony Yezzi Pdf

The subject of pattern analysis and recognition pervades many aspects of our daily lives, including user authentication in banking, object retrieval from databases in the consumer sector, and the omnipresent surveillance and security measures around sensitive areas. Shape analysis, a fundamental building block in many approaches to these applications, is also used in statistics, biomedical applications (Magnetic Resonance Imaging), and many other related disciplines. With contributions from some of the leading experts and pioneers in the field, this self-contained, unified volume is the first comprehensive treatment of theory, methods, and algorithms available in a single resource. Developments are discussed from a rapidly increasing number of research papers in diverse fields, including the mathematical and physical sciences, engineering, and medicine.

Author : Philipp Grohs,Martin Holler,Andreas Weinmann
Publisher : Springer Nature
Page : 701 pages
File Size : 55,5 Mb
Release : 2020-04-03
Category : Mathematics
ISBN : 9783030313517

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Handbook of Variational Methods for Nonlinear Geometric Data by Philipp Grohs,Martin Holler,Andreas Weinmann Pdf

This book covers different, current research directions in the context of variational methods for non-linear geometric data. Each chapter is authored by leading experts in the respective discipline and provides an introduction, an overview and a description of the current state of the art. Non-linear geometric data arises in various applications in science and engineering. Examples of nonlinear data spaces are diverse and include, for instance, nonlinear spaces of matrices, spaces of curves, shapes as well as manifolds of probability measures. Applications can be found in biology, medicine, product engineering, geography and computer vision for instance. Variational methods on the other hand have evolved to being amongst the most powerful tools for applied mathematics. They involve techniques from various branches of mathematics such as statistics, modeling, optimization, numerical mathematics and analysis. The vast majority of research on variational methods, however, is focused on data in linear spaces. Variational methods for non-linear data is currently an emerging research topic. As a result, and since such methods involve various branches of mathematics, there is a plethora of different, recent approaches dealing with different aspects of variational methods for nonlinear geometric data. Research results are rather scattered and appear in journals of different mathematical communities. The main purpose of the book is to account for that by providing, for the first time, a comprehensive collection of different research directions and existing approaches in this context. It is organized in a way that leading researchers from the different fields provide an introductory overview of recent research directions in their respective discipline. As such, the book is a unique reference work for both newcomers in the field of variational methods for non-linear geometric data, as well as for established experts that aim at to exploit new research directions or collaborations. Chapter 9 of this book is available open access under a CC BY 4.0 license at link.springer.com.

Author : Nihat Ay,Jürgen Jost,Hông Vân Lê,Lorenz Schwachhöfer
Publisher : Springer
Page : 407 pages
File Size : 54,5 Mb
Release : 2017-08-25
Category : Mathematics
ISBN : 9783319564784

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Information Geometry by Nihat Ay,Jürgen Jost,Hông Vân Lê,Lorenz Schwachhöfer Pdf

The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory. Parametrised measure models are defined as fundamental geometric objects, which can be both finite or infinite dimensional. Based on these models, canonical tensor fields are introduced and further studied, including the Fisher metric and the Amari-Chentsov tensor, and embeddings of statistical manifolds are investigated. This novel foundation then leads to application highlights, such as generalizations and extensions of the classical uniqueness result of Chentsov or the Cramér-Rao inequality. Additionally, several new application fields of information geometry are highlighted, for instance hierarchical and graphical models, complexity theory, population genetics, or Markov Chain Monte Carlo. The book will be of interest to mathematicians who are interested in geometry, information theory, or the foundations of statistics, to statisticians as well as to scientists interested in the mathematical foundations of complex systems.

Author : Wilfrid S. Kendall
Publisher : Routledge
Page : 419 pages
File Size : 50,7 Mb
Release : 2019-06-10
Category : Mathematics
ISBN : 9781351413725

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Stochastic Geometry by Wilfrid S. Kendall Pdf

Stochastic geometry involves the study of random geometric structures, and blends geometric, probabilistic, and statistical methods to provide powerful techniques for modeling and analysis. Recent developments in computational statistical analysis, particularly Markov chain Monte Carlo, have enormously extended the range of feasible applications. Stochastic Geometry: Likelihood and Computation provides a coordinated collection of chapters on important aspects of the rapidly developing field of stochastic geometry, including: o a "crash-course" introduction to key stochastic geometry themes o considerations of geometric sampling bias issues o tesselations o shape o random sets o image analysis o spectacular advances in likelihood-based inference now available to stochastic geometry through the techniques of Markov chain Monte Carlo

Author : Frank Nielsen
Publisher : Springer
Page : 392 pages
File Size : 42,9 Mb
Release : 2018-11-19
Category : Technology & Engineering
ISBN : 9783030025205

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Geometric Structures of Information by Frank Nielsen Pdf

This book focuses on information geometry manifolds of structured data/information and their advanced applications featuring new and fruitful interactions between several branches of science: information science, mathematics and physics. It addresses interrelations between different mathematical domains like shape spaces, probability/optimization & algorithms on manifolds, relational and discrete metric spaces, computational and Hessian information geometry, algebraic/infinite dimensional/Banach information manifolds, divergence geometry, tensor-valued morphology, optimal transport theory, manifold & topology learning, and applications like geometries of audio-processing, inverse problems and signal processing. The book collects the most important contributions to the conference GSI’2017 – Geometric Science of Information.

Author : Ian L. Dryden,John T. Kent
Publisher : John Wiley & Sons
Page : 432 pages
File Size : 47,9 Mb
Release : 2015-09-03
Category : Mathematics
ISBN : 9781118866603

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Geometry Driven Statistics by Ian L. Dryden,John T. Kent Pdf

A timely collection of advanced, original material in the area of statistical methodology motivated by geometric problems, dedicated to the influential work of Kanti V. Mardia This volume celebrates Kanti V. Mardia's long and influential career in statistics. A common theme unifying much of Mardia’s work is the importance of geometry in statistics, and to highlight the areas emphasized in his research this book brings together 16 contributions from high-profile researchers in the field. Geometry Driven Statistics covers a wide range of application areas including directional data, shape analysis, spatial data, climate science, fingerprints, image analysis, computer vision and bioinformatics. The book will appeal to statisticians and others with an interest in data motivated by geometric considerations. Summarizing the state of the art, examining some new developments and presenting a vision for the future, Geometry Driven Statistics will enable the reader to broaden knowledge of important research areas in statistics and gain a new appreciation of the work and influence of Kanti V. Mardia.

Author : David J. Saville,Graham R. Wood
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 46,6 Mb
Release : 1996-06-20
Category : Mathematics
ISBN : 0387947051

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Statistical Methods by David J. Saville,Graham R. Wood Pdf

"[Suitable for] general interest readers as well as university students in their first or second year ... linear or vector geometry students who desire the illumination provided by a concrete application of the theory"--Page [1].

Author : Robert Simon Fong,Peter Tino
Publisher : Springer Nature
Page : 171 pages
File Size : 44,9 Mb
Release : 2022-05-17
Category : Technology & Engineering
ISBN : 9783031042935

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Population-Based Optimization on Riemannian Manifolds by Robert Simon Fong,Peter Tino Pdf

Manifold optimization is an emerging field of contemporary optimization that constructs efficient and robust algorithms by exploiting the specific geometrical structure of the search space. In our case the search space takes the form of a manifold. Manifold optimization methods mainly focus on adapting existing optimization methods from the usual “easy-to-deal-with” Euclidean search spaces to manifolds whose local geometry can be defined e.g. by a Riemannian structure. In this way the form of the adapted algorithms can stay unchanged. However, to accommodate the adaptation process, assumptions on the search space manifold often have to be made. In addition, the computations and estimations are confined by the local geometry. This book presents a framework for population-based optimization on Riemannian manifolds that overcomes both the constraints of locality and additional assumptions. Multi-modal, black-box manifold optimization problems on Riemannian manifolds can be tackled using zero-order stochastic optimization methods from a geometrical perspective, utilizing both the statistical geometry of the decision space and Riemannian geometry of the search space. This monograph presents in a self-contained manner both theoretical and empirical aspects of stochastic population-based optimization on abstract Riemannian manifolds.

Author : Shun-ichi Amari
Publisher : Springer
Page : 378 pages
File Size : 40,7 Mb
Release : 2016-02-02
Category : Mathematics
ISBN : 9784431559788

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Information Geometry and Its Applications by Shun-ichi Amari Pdf

This is the first comprehensive book on information geometry, written by the founder of the field. It begins with an elementary introduction to dualistic geometry and proceeds to a wide range of applications, covering information science, engineering, and neuroscience. It consists of four parts, which on the whole can be read independently. A manifold with a divergence function is first introduced, leading directly to dualistic structure, the heart of information geometry. This part (Part I) can be apprehended without any knowledge of differential geometry. An intuitive explanation of modern differential geometry then follows in Part II, although the book is for the most part understandable without modern differential geometry. Information geometry of statistical inference, including time series analysis and semiparametric estimation (the Neyman–Scott problem), is demonstrated concisely in Part III. Applications addressed in Part IV include hot current topics in machine learning, signal processing, optimization, and neural networks. The book is interdisciplinary, connecting mathematics, information sciences, physics, and neurosciences, inviting readers to a new world of information and geometry. This book is highly recommended to graduate students and researchers who seek new mathematical methods and tools useful in their own fields.

Author : Dirk P. Kroese,Joshua C.C. Chan
Publisher : Springer Science & Business Media
Page : 400 pages
File Size : 42,6 Mb
Release : 2013-11-18
Category : Computers
ISBN : 9781461487753

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Statistical Modeling and Computation by Dirk P. Kroese,Joshua C.C. Chan Pdf

This textbook on statistical modeling and statistical inference will assist advanced undergraduate and graduate students. Statistical Modeling and Computation provides a unique introduction to modern Statistics from both classical and Bayesian perspectives. It also offers an integrated treatment of Mathematical Statistics and modern statistical computation, emphasizing statistical modeling, computational techniques, and applications. Each of the three parts will cover topics essential to university courses. Part I covers the fundamentals of probability theory. In Part II, the authors introduce a wide variety of classical models that include, among others, linear regression and ANOVA models. In Part III, the authors address the statistical analysis and computation of various advanced models, such as generalized linear, state-space and Gaussian models. Particular attention is paid to fast Monte Carlo techniques for Bayesian inference on these models. Throughout the book the authors include a large number of illustrative examples and solved problems. The book also features a section with solutions, an appendix that serves as a MATLAB primer, and a mathematical supplement.​

Author : Herbert Solomon
Publisher : SIAM
Page : 180 pages
File Size : 47,7 Mb
Release : 1978-06-01
Category : Mathematics
ISBN : 9780898710250

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Geometric Probability by Herbert Solomon Pdf

Topics include: ways modern statistical procedures can yield estimates of pi more precisely than the original Buffon procedure traditionally used; the question of density and measure for random geometric elements that leave probability and expectation statements invariant under translation and rotation; and much more.

Author : Dietrich Stoyan,Wilfrid S. Kendall,Joseph Mecke
Publisher : Wiley-Blackwell
Page : 480 pages
File Size : 53,7 Mb
Release : 1995
Category : Mathematics
ISBN : UOM:39015037816017

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Stochastic Geometry and Its Applications by Dietrich Stoyan,Wilfrid S. Kendall,Joseph Mecke Pdf

The exposition is mathematically precise and takes into account the latest results. However, in many cases proofs are omitted. Applied scientists who may not wish to follow the mathematical arguments in detail will still be able to interpret and use the formulae.