Bayesian Hierarchical Gaussian Process Models for Functional Data Analysis

Bayesian Hierarchical Gaussian Process Models for Functional Data Analysis
Author: Cecilia Ann Earls
Publisher:
Total Pages: 334
Release: 2014
Genre:
ISBN:

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This dissertation encompasses a breadth of topics in the area of functional data analysis where each function is modeled as a Gaussian process within the framework of a Bayesian hierarchical model. As Gaussian processes cannot be worked with directly in this context, a foundational aspect of this work illustrates that using a finite approximation to each process is sufficient to provide good estimates throughout the entire process. More importantly, it is established that using a finite approximation of a bivariate random process within the estimation procedure also results in providing good estimates throughout the entire bivariate process. With this result, the mean and covariance functions associated with a Gaussian process can be considered as random effects within a Bayesian hierarchical model. Inference for both parameters is based upon their posterior distributions which provide not only estimates of these parameters, but also quantifies variation in these parameters. Here we also propose Bayesian hierarchical models for smoothing, functional linear regression, and functional registration. The registration model introduced here is shown to favorably compare with the best registration methods currently available as measured by the Sobolev Least Squares criterion. Within this registration framework, an Adapted Variational Bayes algorithm is introduced to address the computational costs associated with inference in high-dimensional Bayesian models. With multiple examples, both simulated and using real data, it is shown that this algorithm results in registered function estimates that closely agree with corresponding estimates obtained from an MCMC sampling scheme. With this algorithm, functional prediction is considered for the first time in a registration context. The final area of inference for functional data that is proposed for the first time here is a combined registration and factor analysis model. This model is shown to outperform currently available registration methods for data in which the registered functions vary in more than one functional direction. The models presented here are applied to several simulated data sets as well as data from the Berkeley Growth Study, functional sea-surface temperature data, and a juggling data set.

Hierarchical Modeling and Analysis for Spatial Data

Hierarchical Modeling and Analysis for Spatial Data
Author: Sudipto Banerjee
Publisher: CRC Press
Total Pages: 583
Release: 2014-09-12
Genre: Mathematics
ISBN: 1439819181

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Keep Up to Date with the Evolving Landscape of Space and Space-Time Data Analysis and ModelingSince the publication of the first edition, the statistical landscape has substantially changed for analyzing space and space-time data. More than twice the size of its predecessor, Hierarchical Modeling and Analysis for Spatial Data, Second Edition reflec

Bayesian Analysis with Python

Bayesian Analysis with Python
Author: Osvaldo Martin
Publisher: Packt Publishing Ltd
Total Pages: 282
Release: 2016-11-25
Genre: Computers
ISBN: 1785889850

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Unleash the power and flexibility of the Bayesian framework About This Book Simplify the Bayes process for solving complex statistical problems using Python; Tutorial guide that will take the you through the journey of Bayesian analysis with the help of sample problems and practice exercises; Learn how and when to use Bayesian analysis in your applications with this guide. Who This Book Is For Students, researchers and data scientists who wish to learn Bayesian data analysis with Python and implement probabilistic models in their day to day projects. Programming experience with Python is essential. No previous statistical knowledge is assumed. What You Will Learn Understand the essentials Bayesian concepts from a practical point of view Learn how to build probabilistic models using the Python library PyMC3 Acquire the skills to sanity-check your models and modify them if necessary Add structure to your models and get the advantages of hierarchical models Find out how different models can be used to answer different data analysis questions When in doubt, learn to choose between alternative models. Predict continuous target outcomes using regression analysis or assign classes using logistic and softmax regression. Learn how to think probabilistically and unleash the power and flexibility of the Bayesian framework In Detail The purpose of this book is to teach the main concepts of Bayesian data analysis. We will learn how to effectively use PyMC3, a Python library for probabilistic programming, to perform Bayesian parameter estimation, to check models and validate them. This book begins presenting the key concepts of the Bayesian framework and the main advantages of this approach from a practical point of view. Moving on, we will explore the power and flexibility of generalized linear models and how to adapt them to a wide array of problems, including regression and classification. We will also look into mixture models and clustering data, and we will finish with advanced topics like non-parametrics models and Gaussian processes. With the help of Python and PyMC3 you will learn to implement, check and expand Bayesian models to solve data analysis problems. Style and approach Bayes algorithms are widely used in statistics, machine learning, artificial intelligence, and data mining. This will be a practical guide allowing the readers to use Bayesian methods for statistical modelling and analysis using Python.

Bayesian Data Analysis, Third Edition

Bayesian Data Analysis, Third Edition
Author: Andrew Gelman
Publisher: CRC Press
Total Pages: 677
Release: 2013-11-01
Genre: Mathematics
ISBN: 1439840954

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Now in its third edition, this classic book is widely considered the leading text on Bayesian methods, lauded for its accessible, practical approach to analyzing data and solving research problems. Bayesian Data Analysis, Third Edition continues to take an applied approach to analysis using up-to-date Bayesian methods. The authors—all leaders in the statistics community—introduce basic concepts from a data-analytic perspective before presenting advanced methods. Throughout the text, numerous worked examples drawn from real applications and research emphasize the use of Bayesian inference in practice. New to the Third Edition Four new chapters on nonparametric modeling Coverage of weakly informative priors and boundary-avoiding priors Updated discussion of cross-validation and predictive information criteria Improved convergence monitoring and effective sample size calculations for iterative simulation Presentations of Hamiltonian Monte Carlo, variational Bayes, and expectation propagation New and revised software code The book can be used in three different ways. For undergraduate students, it introduces Bayesian inference starting from first principles. For graduate students, the text presents effective current approaches to Bayesian modeling and computation in statistics and related fields. For researchers, it provides an assortment of Bayesian methods in applied statistics. Additional materials, including data sets used in the examples, solutions to selected exercises, and software instructions, are available on the book’s web page.

Functional Data Analysis with R and MATLAB

Functional Data Analysis with R and MATLAB
Author: James Ramsay
Publisher: Springer Science & Business Media
Total Pages: 213
Release: 2009-06-29
Genre: Computers
ISBN: 0387981853

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The book provides an application-oriented overview of functional analysis, with extended and accessible presentations of key concepts such as spline basis functions, data smoothing, curve registration, functional linear models and dynamic systems Functional data analysis is put to work in a wide a range of applications, so that new problems are likely to find close analogues in this book The code in R and Matlab in the book has been designed to permit easy modification to adapt to new data structures and research problems

Gaussian Process Regression Analysis for Functional Data

Gaussian Process Regression Analysis for Functional Data
Author: Jian Qing Shi
Publisher: CRC Press
Total Pages: 218
Release: 2011-07-01
Genre: Mathematics
ISBN: 1439837732

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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. Covering the basics of Gaussian process regression, the first several chapters discuss functional data analysis, theoretical aspects based on the asymptotic properties of Gaussian process regression models, and new methodological developments for high dimensional data and variable selection. The remainder of the text explores advanced topics of functional regression analysis, including novel nonparametric statistical methods for curve prediction, curve clustering, functional ANOVA, and functional regression analysis of batch data, repeated curves, and non-Gaussian data. Many flexible models based on Gaussian processes provide efficient ways of model learning, interpreting model structure, and carrying out inference, particularly when dealing with large dimensional functional data. This book shows how to use these Gaussian process regression models in the analysis of functional data. Some MATLAB® and C codes are available on the first author’s website.

Statistical Modeling Using Bayesian Latent Gaussian Models

Statistical Modeling Using Bayesian Latent Gaussian Models
Author: Birgir Hrafnkelsson
Publisher: Springer Nature
Total Pages: 256
Release: 2023-12-10
Genre: Mathematics
ISBN: 3031397916

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This book focuses on the statistical modeling of geophysical and environmental data using Bayesian latent Gaussian models. The structure of these models is described in a thorough introductory chapter, which explains how to construct prior densities for the model parameters, how to infer the parameters using Bayesian computation, and how to use the models to make predictions. The remaining six chapters focus on the application of Bayesian latent Gaussian models to real examples in glaciology, hydrology, engineering seismology, seismology, meteorology and climatology. These examples include: spatial predictions of surface mass balance; the estimation of Antarctica’s contribution to sea-level rise; the estimation of rating curves for the projection of water level to discharge; ground motion models for strong motion; spatial modeling of earthquake magnitudes; weather forecasting based on numerical model forecasts; and extreme value analysis of precipitation on a high-dimensional grid. The book is aimed at graduate students and experts in statistics, geophysics, environmental sciences, engineering, and related fields.

Bayesian Analysis with Python

Bayesian Analysis with Python
Author: Osvaldo Martin
Publisher: Packt Publishing Ltd
Total Pages: 395
Release: 2024-01-31
Genre: Computers
ISBN: 1805125419

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Learn the fundamentals of Bayesian modeling using state-of-the-art Python libraries, such as PyMC, ArviZ, Bambi, and more, guided by an experienced Bayesian modeler who contributes to these libraries Key Features Conduct Bayesian data analysis with step-by-step guidance Gain insight into a modern, practical, and computational approach to Bayesian statistical modeling Enhance your learning with best practices through sample problems and practice exercises Purchase of the print or Kindle book includes a free PDF eBook. Book DescriptionThe third edition of Bayesian Analysis with Python serves as an introduction to the main concepts of applied Bayesian modeling using PyMC, a state-of-the-art probabilistic programming library, and other libraries that support and facilitate modeling like ArviZ, for exploratory analysis of Bayesian models; Bambi, for flexible and easy hierarchical linear modeling; PreliZ, for prior elicitation; PyMC-BART, for flexible non-parametric regression; and Kulprit, for variable selection. In this updated edition, a brief and conceptual introduction to probability theory enhances your learning journey by introducing new topics like Bayesian additive regression trees (BART), featuring updated examples. Refined explanations, informed by feedback and experience from previous editions, underscore the book's emphasis on Bayesian statistics. You will explore various models, including hierarchical models, generalized linear models for regression and classification, mixture models, Gaussian processes, and BART, using synthetic and real datasets. By the end of this book, you will possess a functional understanding of probabilistic modeling, enabling you to design and implement Bayesian models for your data science challenges. You'll be well-prepared to delve into more advanced material or specialized statistical modeling if the need arises.What you will learn Build probabilistic models using PyMC and Bambi Analyze and interpret probabilistic models with ArviZ Acquire the skills to sanity-check models and modify them if necessary Build better models with prior and posterior predictive checks Learn the advantages and caveats of hierarchical models Compare models and choose between alternative ones Interpret results and apply your knowledge to real-world problems Explore common models from a unified probabilistic perspective Apply the Bayesian framework's flexibility for probabilistic thinking Who this book is for If you are a student, data scientist, researcher, or developer looking to get started with Bayesian data analysis and probabilistic programming, this book is for you. The book is introductory, so no previous statistical knowledge is required, although some experience in using Python and scientific libraries like NumPy is expected.

Bayesian Reasoning and Gaussian Processes for Machine Learning Applications

Bayesian Reasoning and Gaussian Processes for Machine Learning Applications
Author: Hemachandran K
Publisher: CRC Press
Total Pages: 147
Release: 2022-04-14
Genre: Business & Economics
ISBN: 1000569586

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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.

Probability and Bayesian Modeling

Probability and Bayesian Modeling
Author: Jim Albert
Publisher: CRC Press
Total Pages: 553
Release: 2019-12-06
Genre: Mathematics
ISBN: 1351030132

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Probability and Bayesian Modeling is an introduction to probability and Bayesian thinking for undergraduate students with a calculus background. The first part of the book provides a broad view of probability including foundations, conditional probability, discrete and continuous distributions, and joint distributions. Statistical inference is presented completely from a Bayesian perspective. The text introduces inference and prediction for a single proportion and a single mean from Normal sampling. After fundamentals of Markov Chain Monte Carlo algorithms are introduced, Bayesian inference is described for hierarchical and regression models including logistic regression. The book presents several case studies motivated by some historical Bayesian studies and the authors’ research. This text reflects modern Bayesian statistical practice. Simulation is introduced in all the probability chapters and extensively used in the Bayesian material to simulate from the posterior and predictive distributions. One chapter describes the basic tenets of Metropolis and Gibbs sampling algorithms; however several chapters introduce the fundamentals of Bayesian inference for conjugate priors to deepen understanding. Strategies for constructing prior distributions are described in situations when one has substantial prior information and for cases where one has weak prior knowledge. One chapter introduces hierarchical Bayesian modeling as a practical way of combining data from different groups. There is an extensive discussion of Bayesian regression models including the construction of informative priors, inference about functions of the parameters of interest, prediction, and model selection. The text uses JAGS (Just Another Gibbs Sampler) as a general-purpose computational method for simulating from posterior distributions for a variety of Bayesian models. An R package ProbBayes is available containing all of the book datasets and special functions for illustrating concepts from the book. A complete solutions manual is available for instructors who adopt the book in the Additional Resources section.