Gaussian Processes

Author : Carl Edward Rasmussen,Christopher K. I. Williams
Publisher : MIT Press
Page : 266 pages
File Size : 46,6 Mb
Release : 2005-11-23
Category : Computers
ISBN : 9780262182539

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Gaussian Processes for Machine Learning by Carl Edward Rasmussen,Christopher K. I. Williams Pdf

A comprehensive and self-contained introduction to Gaussian processes, which provide a principled, practical, probabilistic approach to learning in kernel machines. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics. The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.

Author : Mikhail Lifshits
Publisher : Springer Science & Business Media
Page : 129 pages
File Size : 51,8 Mb
Release : 2012-01-11
Category : Mathematics
ISBN : 9783642249396

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Lectures on Gaussian Processes by Mikhail Lifshits Pdf

Gaussian processes can be viewed as a far-reaching infinite-dimensional extension of classical normal random variables. Their theory presents a powerful range of tools for probabilistic modelling in various academic and technical domains such as Statistics, Forecasting, Finance, Information Transmission, Machine Learning - to mention just a few. The objective of these Briefs is to present a quick and condensed treatment of the core theory that a reader must understand in order to make his own independent contributions. The primary intended readership are PhD/Masters students and researchers working in pure or applied mathematics. The first chapters introduce essentials of the classical theory of Gaussian processes and measures with the core notions of reproducing kernel, integral representation, isoperimetric property, large deviation principle. The brevity being a priority for teaching and learning purposes, certain technical details and proofs are omitted. The later chapters touch important recent issues not sufficiently reflected in the literature, such as small deviations, expansions, and quantization of processes. In university teaching, one can build a one-semester advanced course upon these Briefs.​

Author : I.A. Ibragimov,Y.A. Rozanov
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 49,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461262756

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Gaussian Random Processes by I.A. Ibragimov,Y.A. Rozanov Pdf

The book deals mainly with three problems involving Gaussian stationary processes. The first problem consists of clarifying the conditions for mutual absolute continuity (equivalence) of probability distributions of a "random process segment" and of finding effective formulas for densities of the equiva lent distributions. Our second problem is to describe the classes of spectral measures corresponding in some sense to regular stationary processes (in par ticular, satisfying the well-known "strong mixing condition") as well as to describe the subclasses associated with "mixing rate". The third problem involves estimation of an unknown mean value of a random process, this random process being stationary except for its mean, i. e. , it is the problem of "distinguishing a signal from stationary noise". Furthermore, we give here auxiliary information (on distributions in Hilbert spaces, properties of sam ple functions, theorems on functions of a complex variable, etc. ). Since 1958 many mathematicians have studied the problem of equivalence of various infinite-dimensional Gaussian distributions (detailed and sys tematic presentation of the basic results can be found, for instance, in [23]). In this book we have considered Gaussian stationary processes and arrived, we believe, at rather definite solutions. The second problem mentioned above is closely related with problems involving ergodic theory of Gaussian dynamic systems as well as prediction theory of stationary processes.

Author : Juš Kocijan
Publisher : Springer
Page : 267 pages
File Size : 43,8 Mb
Release : 2015-11-21
Category : Technology & Engineering
ISBN : 9783319210216

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Modelling and Control of Dynamic Systems Using Gaussian Process Models by Juš Kocijan Pdf

This monograph opens up new horizons for engineers and researchers in academia and in industry dealing with or interested in new developments in the field of system identification and control. It emphasizes guidelines for working solutions and practical advice for their implementation rather than the theoretical background of Gaussian process (GP) models. The book demonstrates the potential of this recent development in probabilistic machine-learning methods and gives the reader an intuitive understanding of the topic. The current state of the art is treated along with possible future directions for research. Systems control design relies on mathematical models and these may be developed from measurement data. This process of system identification, when based on GP models, can play an integral part of control design in data-based control and its description as such is an essential aspect of the text. The background of GP regression is introduced first with system identification and incorporation of prior knowledge then leading into full-blown control. The book is illustrated by extensive use of examples, line drawings, and graphical presentation of computer-simulation results and plant measurements. The research results presented are applied in real-life case studies drawn from successful applications including: a gas–liquid separator control; urban-traffic signal modelling and reconstruction; and prediction of atmospheric ozone concentration. A MATLAB® toolbox, for identification and simulation of dynamic GP models is provided for download.

Author : Hemachandran K,Shubham Tayal,Preetha Mary George,Parveen Singla,Utku Kose
Publisher : CRC Press
Page : 165 pages
File Size : 49,7 Mb
Release : 2022-04-14
Category : Business & Economics
ISBN : 9781000569599

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Bayesian Reasoning and Gaussian Processes for Machine Learning Applications by Hemachandran K,Shubham Tayal,Preetha Mary George,Parveen Singla,Utku Kose Pdf

This book introduces Bayesian reasoning and Gaussian processes into machine learning applications. Bayesian methods are applied in many areas, such as game development, decision making, and drug discovery. It is very effective for machine learning algorithms in handling missing data and extracting information from small datasets. Bayesian Reasoning and Gaussian Processes for Machine Learning Applications uses a statistical background to understand continuous distributions and how learning can be viewed from a probabilistic framework. The chapters progress into such machine learning topics as belief network and Bayesian reinforcement learning, which is followed by Gaussian process introduction, classification, regression, covariance, and performance analysis of Gaussian processes with other models. FEATURES Contains recent advancements in machine learning Highlights applications of machine learning algorithms Offers both quantitative and qualitative research Includes numerous case studies This book is aimed at graduates, researchers, and professionals in the field of data science and machine learning.

Author : Michael B. Marcus,Jay Rosen
Publisher : Cambridge University Press
Page : 4 pages
File Size : 51,6 Mb
Release : 2006-07-24
Category : Mathematics
ISBN : 9781139458832

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Markov Processes, Gaussian Processes, and Local Times by Michael B. Marcus,Jay Rosen Pdf

This book was first published in 2006. Written by two of the foremost researchers in the field, this book studies the local times of Markov processes by employing isomorphism theorems that relate them to certain associated Gaussian processes. It builds to this material through self-contained but harmonized 'mini-courses' on the relevant ingredients, which assume only knowledge of measure-theoretic probability. The streamlined selection of topics creates an easy entrance for students and experts in related fields. The book starts by developing the fundamentals of Markov process theory and then of Gaussian process theory, including sample path properties. It then proceeds to more advanced results, bringing the reader to the heart of contemporary research. It presents the remarkable isomorphism theorems of Dynkin and Eisenbaum and then shows how they can be applied to obtain new properties of Markov processes by using well-established techniques in Gaussian process theory. This original, readable book will appeal to both researchers and advanced graduate students.

Author : Takeyuki Hida,Masuyuki Hitsuda
Publisher : American Mathematical Soc.
Page : 208 pages
File Size : 41,6 Mb
Release : 2024-07-03
Category : Mathematics
ISBN : 0821887637

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Gaussian Processes by Takeyuki Hida,Masuyuki Hitsuda Pdf

Aimed at students and researchers in mathematics, communications engineering, and economics, this book describes the probabilistic structure of a Gaussian process in terms of its canonical representation (or its innovation process). Multiple Markov properties of a Gaussian process and equivalence problems of Gaussian processes are clearly presented. The authors' approach is unique, involving causality in time evolution and information-theoretic aspects. Because the book is self-contained and only requires background in the fundamentals of probability theory and measure theory, it would be suitable as a textbook at the senior undergraduate or graduate level.

Author : Harry Dym,Henry P. McKean
Publisher : Courier Corporation
Page : 354 pages
File Size : 44,5 Mb
Release : 2008-01-01
Category : Mathematics
ISBN : 9780486462790

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Gaussian Processes, Function Theory, and the Inverse Spectral Problem by Harry Dym,Henry P. McKean Pdf

This text offers background in function theory, Hardy functions, and probability as preparation for surveys of Gaussian processes, strings and spectral functions, and strings and spaces of integral functions. It addresses the relationship between the past and the future of a real, one-dimensional, stationary Gaussian process. 1976 edition.

Author : Jian Qing Shi,Taeryon Choi
Publisher : CRC Press
Page : 214 pages
File Size : 40,6 Mb
Release : 2011-07-01
Category : Mathematics
ISBN : 9781439837740

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Gaussian Process Regression Analysis for Functional Data by Jian Qing Shi,Taeryon Choi Pdf

Gaussian Process Regression Analysis for Functional Data presents nonparametric statistical methods for functional regression analysis, specifically the methods based on a Gaussian process prior in a functional space. The authors focus on problems involving functional response variables and mixed covariates of functional and scalar variables.Coveri

Author : Robert B. Gramacy
Publisher : CRC Press
Page : 560 pages
File Size : 41,8 Mb
Release : 2020-03-10
Category : Mathematics
ISBN : 9781000766202

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Surrogates by Robert B. Gramacy Pdf

Computer simulation experiments are essential to modern scientific discovery, whether that be in physics, chemistry, biology, epidemiology, ecology, engineering, etc. Surrogates are meta-models of computer simulations, used to solve mathematical models that are too intricate to be worked by hand. Gaussian process (GP) regression is a supremely flexible tool for the analysis of computer simulation experiments. This book presents an applied introduction to GP regression for modelling and optimization of computer simulation experiments. Features: • Emphasis on methods, applications, and reproducibility. • R code is integrated throughout for application of the methods. • Includes more than 200 full colour figures. • Includes many exercises to supplement understanding, with separate solutions available from the author. • Supported by a website with full code available to reproduce all methods and examples. The book is primarily designed as a textbook for postgraduate students studying GP regression from mathematics, statistics, computer science, and engineering. Given the breadth of examples, it could also be used by researchers from these fields, as well as from economics, life science, social science, etc.

Author : Mikhail Lifshits
Publisher : Springer Science & Business Media
Page : 129 pages
File Size : 52,5 Mb
Release : 2012-01-13
Category : Mathematics
ISBN : 9783642249389

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Lectures on Gaussian Processes by Mikhail Lifshits Pdf

Gaussian processes can be viewed as a far-reaching infinite-dimensional extension of classical normal random variables. Their theory presents a powerful range of tools for probabilistic modelling in various academic and technical domains such as Statistics, Forecasting, Finance, Information Transmission, Machine Learning - to mention just a few. The objective of these Briefs is to present a quick and condensed treatment of the core theory that a reader must understand in order to make his own independent contributions. The primary intended readership are PhD/Masters students and researchers working in pure or applied mathematics. The first chapters introduce essentials of the classical theory of Gaussian processes and measures with the core notions of reproducing kernel, integral representation, isoperimetric property, large deviation principle. The brevity being a priority for teaching and learning purposes, certain technical details and proofs are omitted. The later chapters touch important recent issues not sufficiently reflected in the literature, such as small deviations, expansions, and quantization of processes. In university teaching, one can build a one-semester advanced course upon these Briefs.​

Author : Yuliya Mishura,Mounir Zili
Publisher : Elsevier
Page : 210 pages
File Size : 48,5 Mb
Release : 2018-05-26
Category : Mathematics
ISBN : 9780081023631

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Stochastic Analysis of Mixed Fractional Gaussian Processes by Yuliya Mishura,Mounir Zili Pdf

Stochastic Analysis of Mixed Fractional Gaussian Processes presents the main tools necessary to characterize Gaussian processes. The book focuses on the particular case of the linear combination of independent fractional and sub-fractional Brownian motions with different Hurst indices. Stochastic integration with respect to these processes is considered, as is the study of the existence and uniqueness of solutions of related SDE's. Applications in finance and statistics are also explored, with each chapter supplying a number of exercises to illustrate key concepts. Presents both mixed fractional and sub-fractional Brownian motions Provides an accessible description for mixed fractional gaussian processes that is ideal for Master's and PhD students Includes different Hurst indices

Author : Robert J. Adler
Publisher : IMS
Page : 198 pages
File Size : 52,7 Mb
Release : 1990
Category : Mathematics
ISBN : 094060017X

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An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes by Robert J. Adler Pdf

Author : Vladimir I. Piterbarg
Publisher : American Mathematical Soc.
Page : 222 pages
File Size : 54,9 Mb
Release : 2012-03-28
Category : Mathematics
ISBN : 9780821883310

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Asymptotic Methods in the Theory of Gaussian Processes and Fields by Vladimir I. Piterbarg Pdf

This book is devoted to a systematic analysis of asymptotic behavior of distributions of various typical functionals of Gaussian random variables and fields. The text begins with an extended introduction, which explains fundamental ideas and sketches the basic methods fully presented later in the book. Good approximate formulas and sharp estimates of the remainders are obtained for a large class of Gaussian and similar processes. The author devotes special attention to the development of asymptotic analysis methods, emphasizing the method of comparison, the double-sum method and the method of moments. The author has added an extended introduction and has significantly revised the text for this translation, particularly the material on the double-sum method.

Author : Yuxin Zhao
Publisher : Linköping University Electronic Press
Page : 51 pages
File Size : 40,9 Mb
Release : 2019-02-27
Category : Electronic
ISBN : 9789176851623

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Gaussian Processes for Positioning Using Radio Signal Strength Measurements by Yuxin Zhao Pdf

Estimation of unknown parameters is considered as one of the major research areas in statistical signal processing. In the most recent decades, approaches in estimation theory have become more and more attractive in practical applications. Examples of such applications may include, but are not limited to, positioning using various measurable radio signals in indoor environments, self-navigation for autonomous cars, image processing, radar tracking and so on. One issue that is usually encountered when solving an estimation problem is to identify a good system model, which may have great impacts on the estimation performance. In this thesis, we are interested in studying estimation problems particularly in inferring the unknown positions from noisy radio signal measurements. In addition, the modeling of the system is studied by investigating the relationship between positions and radio signal strength measurements. One of the main contributions of this thesis is to propose a novel indoor positioning framework based on proximity measurements, which are obtained by quantizing the received signal strength measurements. Sequential Monte Carlo methods, to be more specific particle filter and smoother, are utilized for estimating unknown positions from proximity measurements. The Cramér-Rao bounds for proximity-based positioning are further derived as a benchmark for the positioning accuracy in this framework. Secondly, to improve the estimation performance, Bayesian non-parametric modeling, namely Gaussian processes, have been adopted to provide more accurate and flexible models for both dynamic motions and radio signal strength measurements. Then, the Cramér-Rao bounds for Gaussian process based system models are derived and evaluated in an indoor positioning scenario. In addition, we estimate the positions of stationary devices by comparing the individual signal strength measurements with a pre-constructed fingerprinting database. The positioning accuracy is further compared to the case where a moving device is positioned using a time series of radio signal strength measurements. Moreover, Gaussian processes have been applied to sports analytics, where trajectory modeling for athletes is studied. The proposed framework can be further utilized to carry out, for instance, performance prediction and analysis, health condition monitoring, etc. Finally, a grey-box modeling is proposed to analyze the forces, particularly in cross-country skiing races, by combining a deterministic kinetic model with Gaussian process.