Dimension Reduction And Sufficient Graphical Models
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Dimension Reduction And Sufficient Graphical Models
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Author : Kyongwon Kim
language : en
Publisher:
Release Date : 2020
Dimension Reduction And Sufficient Graphical Models written by Kyongwon Kim and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with categories.
The methods I develop in my thesis are based on linear or nonlinear sufficient dimension reduction. The basic principle of linear sufficient dimension reduction is to extract a small number of linear combinations of predictor variables, which can represent original predictor variables without loss of information on the conditional distribution of response variable given predictor variables. Nonlinear sufficient dimension reduction is a more generalized version of linear sufficient dimension reduction to the nonlinear context. I am focusing on applying sufficient dimension reduction methods into two areas, regression modeling and graphical models. The first project is about statistical inference in regression context after sufficient dimension reduction. Second, I apply nonlinear sufficient dimension reduction method to the well known statistical graphical models in machine learning. These projects have consistency in a context that discovering areas that sufficient dimension reduction can be applied and establishing statistical theory behind their applications. My first project is about post sufficient dimension reduction statistical inference. The methodologies of sufficient dimension reduction have undergone extensive developments in the past three decades. However, there has been a lack of systematic and rigorous development of post dimension reduction inference, which has seriously hindered its applications. The current common practice is to treat the estimated sufficient predictors as the true predictors and use them as the starting point of the downstream statistical inference. However, this naive inference approach would grossly overestimate the confidence level of an interval, or the power of a test, leading to the distorted results. In this project, we develop a general and comprehensive framework of post dimension reduction inference, which can accommodate any dimension reduction method and model building method, as long as their corresponding influence functions are available. Within this general framework, we derive the influence functions and present the explicit post reduction formulas for the combinations of numerous dimension reduction and model building methods. We then develop post reduction inference methods for both confidence interval and hypothesis testing. We investigate the finite-sample performance of our procedures by simulations and a real data analysis. My second project is about applying nonlinear dimension reduction technique to graphical models. We introduce the Sufficient Graphical Model by applying the recently developed nonlinear sufficient dimension reduction techniques to the evaluation of conditional independence. Graphical model is nonparametric in nature, as it does not make distributional assumptions such as the Gaussian or copula Gaussian assumptions. However, unlike fully nonparametric graphical model, which relies on the high-dimensional kernel to characterize a conditional independence, our graphical model is based on a conditional independence given a set of sufficient predictors with a substantially reduced dimension. In this way, we avoid the curse of dimensionality that comes with a high-dimensional kernel. We develop the population-level properties, convergence rate, and consistency of our estimate. By simulation comparisons and an analysis of the DREAM 4 Challenge data set, we demonstrate that our method outperforms the existing methods when the Gaussian or copula Gaussian assumptions are violated, and its performance remains excellent in the high-dimensional setting.
Sufficient Dimension Reduction
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Author : Bing Li
language : en
Publisher: CRC Press
Release Date : 2018-04-27
Sufficient Dimension Reduction written by Bing Li and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-04-27 with Mathematics categories.
Sufficient dimension reduction is a rapidly developing research field that has wide applications in regression diagnostics, data visualization, machine learning, genomics, image processing, pattern recognition, and medicine, because they are fields that produce large datasets with a large number of variables. Sufficient Dimension Reduction: Methods and Applications with R introduces the basic theories and the main methodologies, provides practical and easy-to-use algorithms and computer codes to implement these methodologies, and surveys the recent advances at the frontiers of this field. Features Provides comprehensive coverage of this emerging research field. Synthesizes a wide variety of dimension reduction methods under a few unifying principles such as projection in Hilbert spaces, kernel mapping, and von Mises expansion. Reflects most recent advances such as nonlinear sufficient dimension reduction, dimension folding for tensorial data, as well as sufficient dimension reduction for functional data. Includes a set of computer codes written in R that are easily implemented by the readers. Uses real data sets available online to illustrate the usage and power of the described methods. Sufficient dimension reduction has undergone momentous development in recent years, partly due to the increased demands for techniques to process high-dimensional data, a hallmark of our age of Big Data. This book will serve as the perfect entry into the field for the beginning researchers or a handy reference for the advanced ones. The author Bing Li obtained his Ph.D. from the University of Chicago. He is currently a Professor of Statistics at the Pennsylvania State University. His research interests cover sufficient dimension reduction, statistical graphical models, functional data analysis, machine learning, estimating equations and quasilikelihood, and robust statistics. He is a fellow of the Institute of Mathematical Statistics and the American Statistical Association. He is an Associate Editor for The Annals of Statistics and the Journal of the American Statistical Association.
Sparse Graphical Modeling For High Dimensional Data
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Author : Faming Liang
language : en
Publisher: CRC Press
Release Date : 2023-08-02
Sparse Graphical Modeling For High Dimensional Data written by Faming Liang and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-08-02 with Mathematics categories.
This book provides a general framework for learning sparse graphical models with conditional independence tests. It includes complete treatments for Gaussian, Poisson, multinomial, and mixed data; unified treatments for covariate adjustments, data integration, and network comparison; unified treatments for missing data and heterogeneous data; efficient methods for joint estimation of multiple graphical models; effective methods of high-dimensional variable selection; and effective methods of high-dimensional inference. The methods possess an embarrassingly parallel structure in performing conditional independence tests, and the computation can be significantly accelerated by running in parallel on a multi-core computer or a parallel architecture. This book is intended to serve researchers and scientists interested in high-dimensional statistics, and graduate students in broad data science disciplines. Key Features: A general framework for learning sparse graphical models with conditional independence tests Complete treatments for different types of data, Gaussian, Poisson, multinomial, and mixed data Unified treatments for data integration, network comparison, and covariate adjustment Unified treatments for missing data and heterogeneous data Efficient methods for joint estimation of multiple graphical models Effective methods of high-dimensional variable selection Effective methods of high-dimensional inference
Elements Of Dimensionality Reduction And Manifold Learning
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Author : Benyamin Ghojogh
language : en
Publisher: Springer Nature
Release Date : 2023-02-02
Elements Of Dimensionality Reduction And Manifold Learning written by Benyamin Ghojogh and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-02-02 with Computers categories.
Dimensionality reduction, also known as manifold learning, is an area of machine learning used for extracting informative features from data for better representation of data or separation between classes. This book presents a cohesive review of linear and nonlinear dimensionality reduction and manifold learning. Three main aspects of dimensionality reduction are covered: spectral dimensionality reduction, probabilistic dimensionality reduction, and neural network-based dimensionality reduction, which have geometric, probabilistic, and information-theoretic points of view to dimensionality reduction, respectively. The necessary background and preliminaries on linear algebra, optimization, and kernels are also explained to ensure a comprehensive understanding of the algorithms. The tools introduced in this book can be applied to various applications involving feature extraction, image processing, computer vision, and signal processing. This book is applicable to a wide audience who would like to acquire a deep understanding of the various ways to extract, transform, and understand the structure of data. The intended audiences are academics, students, and industry professionals. Academic researchers and students can use this book as a textbook for machine learning and dimensionality reduction. Data scientists, machine learning scientists, computer vision scientists, and computer scientists can use this book as a reference. It can also be helpful to statisticians in the field of statistical learning and applied mathematicians in the fields of manifolds and subspace analysis. Industry professionals, including applied engineers, data engineers, and engineers in various fields of science dealing with machine learning, can use this as a guidebook for feature extraction from their data, as the raw data in industry often require preprocessing. The book is grounded in theory but provides thorough explanations and diverse examples to improve the reader’s comprehension of the advanced topics. Advanced methods are explained in a step-by-step manner so that readers of all levels can follow the reasoning and come to a deep understanding of the concepts. This book does not assume advanced theoretical background in machine learning and provides necessary background, although an undergraduate-level background in linear algebra and calculus is recommended.
Handbook Of Graphical Models
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Author : Marloes Maathuis
language : en
Publisher: CRC Press
Release Date : 2018-11-12
Handbook Of Graphical Models written by Marloes Maathuis and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-12 with Mathematics categories.
A graphical model is a statistical model that is represented by a graph. The factorization properties underlying graphical models facilitate tractable computation with multivariate distributions, making the models a valuable tool with a plethora of applications. Furthermore, directed graphical models allow intuitive causal interpretations and have become a cornerstone for causal inference. While there exist a number of excellent books on graphical models, the field has grown so much that individual authors can hardly cover its entire scope. Moreover, the field is interdisciplinary by nature. Through chapters by leading researchers from different areas, this handbook provides a broad and accessible overview of the state of the art. Key features: * Contributions by leading researchers from a range of disciplines * Structured in five parts, covering foundations, computational aspects, statistical inference, causal inference, and applications * Balanced coverage of concepts, theory, methods, examples, and applications * Chapters can be read mostly independently, while cross-references highlight connections The handbook is targeted at a wide audience, including graduate students, applied researchers, and experts in graphical models.
Regression Graphics
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Author : R. Dennis Cook
language : en
Publisher: John Wiley & Sons
Release Date : 1998-09-30
Regression Graphics written by R. Dennis Cook and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-09-30 with Mathematics categories.
An exploration of regression graphics through computer graphics. Recent developments in computer technology have stimulated new and exciting uses for graphics in statistical analyses. Regression Graphics, one of the first graduate-level textbooks on the subject, demonstrates how statisticians, both theoretical and applied, can use these exciting innovations. After developing a relatively new regression context that requires few scope-limiting conditions, Regression Graphics guides readers through the process of analyzing regressions graphically and assessing and selecting models. This innovative reference makes use of a wide range of graphical tools, including 2D and 3D scatterplots, 3D binary response plots, and scatterplot matrices. Supplemented by a companion ftp site, it features numerous data sets and applied examples that are used to elucidate the theory. Other important features of this book include: * Extensive coverage of a relatively new regression context based on dimension-reduction subspaces and sufficient summary plots * Graphical regression, an iterative visualization process for constructing sufficient regression views * Graphics for regressions with a binary response * Graphics for model assessment, including residual plots * Net-effects plots for assessing predictor contributions * Graphics for predictor and response transformations * Inverse regression methods * Access to a Web site of supplemental plots, data sets, and 3D color displays. An ideal text for students in graduate-level courses on statistical analysis, Regression Graphics is also an excellent reference for professional statisticians.
Festschrift In Honor Of R Dennis Cook
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Author : Efstathia Bura
language : en
Publisher: Springer Nature
Release Date : 2021-04-27
Festschrift In Honor Of R Dennis Cook written by Efstathia Bura and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-04-27 with Mathematics categories.
In honor of professor and renowned statistician R. Dennis Cook, this festschrift explores his influential contributions to an array of statistical disciplines ranging from experimental design and population genetics, to statistical diagnostics and all areas of regression-related inference and analysis. Since the early 1990s, Prof. Cook has led the development of dimension reduction methodology in three distinct but related regression contexts: envelopes, sufficient dimension reduction (SDR), and regression graphics. In particular, he has made fundamental and pioneering contributions to SDR, inventing or co-inventing many popular dimension reduction methods, such as sliced average variance estimation, the minimum discrepancy approach, model-free variable selection, and sufficient dimension reduction subspaces. A prolific researcher and mentor, Prof. Cook is known for his ability to identify research problems in statistics that are both challenging and important, as well as his deep appreciation for the applied side of statistics. This collection of Prof. Cook's collaborators, colleagues, friends, and former students reflects the broad array of his contributions to the research and instructional arenas of statistics.
Density Ratio Estimation In Machine Learning
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Author : Masashi Sugiyama
language : en
Publisher: Cambridge University Press
Release Date : 2012-02-20
Density Ratio Estimation In Machine Learning written by Masashi Sugiyama and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-02-20 with Computers categories.
This book introduces theories, methods and applications of density ratio estimation, a newly emerging paradigm in the machine learning community.
Advances In Contemporary Statistics And Econometrics
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Author : Abdelaati Daouia
language : en
Publisher: Springer Nature
Release Date : 2021-06-14
Advances In Contemporary Statistics And Econometrics written by Abdelaati Daouia and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-14 with Mathematics categories.
This book presents a unique collection of contributions on modern topics in statistics and econometrics, written by leading experts in the respective disciplines and their intersections. It addresses nonparametric statistics and econometrics, quantiles and expectiles, and advanced methods for complex data, including spatial and compositional data, as well as tools for empirical studies in economics and the social sciences. The book was written in honor of Christine Thomas-Agnan on the occasion of her 65th birthday. Given its scope, it will appeal to researchers and PhD students in statistics and econometrics alike who are interested in the latest developments in their field.