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Author : Lipu Tian
Publisher :
ISBN 13 :
Total Pages : 38 pages
Book Rating : 4.:/5 (794 download)
Download or read book A Simulation Study on Using Moment Functions for Sufficient Dimension Reduction written by Lipu Tian and published by . This book was released on 2012 with total page 38 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Jingyue Lu
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (144 download)
Download or read book Sufficient Dimension Reduction written by Jingyue Lu and published by . This book was released on 2017 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In regression analysis, it is difficult to uncover the dependence relationship between a response variable and a covariate vector when the dimension of the covariate vector is high. To reduce the dimension of the covariate vector, one approach is sufficient dimension reduction. Sufficient dimension reduction is based on the assumption that the response variable relates to only a few linear combinations of the covariate vector. Thus, by replacing the covariate vector with these linear combinations, sufficient dimension reduction achieves dimension reduction. The goal of sufficient dimension reduction is to estimate the space spanned by these linear combinations of the covariate vector. We denote this space by S. In this thesis, we give an introductory review on three important sufficient dimension reduction methods. They are Sliced Inverse Regression (SIR), Sliced Average Variance Estimate (SAVE) and Principle Hessian Directions (pHd). Li proposed SIR in 1991. SIR is a method that exploits the simplicity of the inverse regression. Given the univariate response variable and the high dimensional covariate, it is much easier to regress the covariate against the response variable than the other way around. Motivated by a theorem that connects forward regression and inverse regression, SIR estimates S using inverse regression lines. Since SIR uses first moments only, it fails when there exists symmetry dependence between the response variable and the covariate. To make up for this defect, Cook proposed SAVE in a comment on SIR in 1991. SAVE follows the general lines of SIR but uses second moments as well as first moments to estimate S. pHd is also a second moment method. Li developed pHd in 1992 based on the observation that the eigenvectors for the Hessian matrices of the regression function are closely related to the basis vectors of S. Therefore pHd provides an estimate of S by using these eigenvectors. To compare these methods, a simulation study is presented at the end. From the simulation results, SIR is the most efficient method and SAVE is the most time consuming method. Since SIR fails when symmetry dependence exists, we recommend pHd when symmetry dependence presents and SIR in other cases.
Author : Prabha Shrestha
Publisher :
ISBN 13 :
Total Pages : 133 pages
Book Rating : 4.:/5 (122 download)
Download or read book Application of Influence Function in Sufficient Dimension Reduction Models written by Prabha Shrestha and published by . This book was released on 2020 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: In regression analysis, sufficient dimension reduction (SDR) models have gained significant popularity in the past three decades. While many methods have been proposed in the literature regarding the analysis of SDR models, the vast majority are of the type called inverse regression methods, pioneered by the sliced inverse regression method (Li \cite{Li91}). Most of these inverse regression methods rely on a matrix, commonly known as the central matrix. One of the main goals of the analysis of SDR models is the estimation of the central space. An influence function (IF) is a tool that analyzes the performance of a statistical estimator. In this dissertation, we focus on the application of IF on the analysis of SDR models. There are various inverse regression methods in existence. But none of them stands out in all cases, and it is not clear which central matrix one should use out of numerous options existing in the literature. We propose an IF-based approach for selection of a best performing central matrix from a class of inverse regression methods, and we extend this approach to the situation where the data are partially contaminated. Asymptotic results are established, and an extensive simulation study is conducted to examine the performance of the proposed algorithm. Another issue in an SDR model is the estimation of the dimension of its central space. Based on the IF, we propose a measure that combines the eigenvalues of the central matrix and an IF measure to estimate the dimension of the central space. In addition, we analyze the IF of the functional of Benasseni's measure for a specific inverse regression method, the $k{\text-th}$ moment method.
Author : Wei Luo
Publisher :
ISBN 13 :
Total Pages : 40 pages
Book Rating : 4.:/5 (13 download)
Download or read book Inverse Moment Methods for Sufficient Forecasting Using High-Dimensional Predictors written by Wei Luo and published by . This book was released on 2017 with total page 40 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider forecasting a single time series using high-dimensional predictors in the presence of a possible nonlinear forecast function. The sufficient forecasting (Fan et al., 2016) used sliced inverse regression to estimate lower-dimensional sufficient indices for non-parametric forecasting using factor models. However, Fan et al. (2016) is fundamentally limited to the inverse first-moment method, by assuming the restricted fixed number of factors, linearity condition for factors, and monotone effect of factors on the response. In this work, we study the inverse second-moment method using directional regression and the inverse third-moment method to extend the methodology and applicability of the sufficient forecasting. As the number of factors diverges with the dimension of predictors, the proposed method relaxes the distributional assumption of the predictor and enhances the capability of capturing the non-monotone effect of factors on the response. We not only provide a high-dimensional analysis of inverse moment methods such as exhaustiveness and rate of convergence, but also prove their model selection consistency. The power of our proposed methods is demonstrated in both simulation studies and an empirical study of forecasting monthly macroeconomic data from Q1 1959 to Q1 2016. During our theoretical development, we prove an invariance result for inverse moment methods, which make a separate contribution to the sufficient dimension reduction.
Author : Bing Li
Publisher : CRC Press
ISBN 13 : 1351645730
Total Pages : 362 pages
Book Rating : 4.3/5 (516 download)
Download or read book Sufficient Dimension Reduction written by Bing Li and published by CRC Press. This book was released on 2018-04-27 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : Jun Song
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (1 download)
Download or read book Dimension Reduction for Functional Data written by Jun Song and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In regression problems, sufficient dimension reduction (SDR) allows us to reduce the dimension of predictor variables without losing any regression information which is considered to be a part of supervised machine learning. In particular, new theories and methodologies are in increasing demand to adapt to complex types of data with a drastically increased dimension such as functional data of infinite dimension. In this work, we established theories and methods of dimension reduction for functional data in three ways: (1) nonlinear supervised dimension reduction, (2) linear supervised dimension reduction, (3) nonlinear unsupervised dimension reduction. The fundamental idea of the theories is the construction of a feature space over a function space where real data live. We construct a feature space by using reproducing kernel Hilbert space (RKHS) in a nested way, called nested RKHS. Which allows us to treat functional data and capture nonlinear characteristics of data at the same time. In addition, nested RKHS can be used to develop weak conditional moments for developing general theories and methods for linear dimension reduction. We employ additive structure over the functional data so that the methods work for multivariate functional data. We develop two methods of nonlinear SDR for functional data, three methods of linear SDR for functional data, and a general framework of nonlinear functional PCA. Then asymptotic results, dimension determination, and its consistency have been studied for parts of methods. Simulation studies and real data application results show that the methods can reduce the dimension of functional data, and can be used for functional classification with high effectiveness.
Author : Chong Wang
Publisher :
ISBN 13 :
Total Pages : 96 pages
Book Rating : 4.:/5 (995 download)
Download or read book A Study of Sufficient Dimension Reduction Methods written by Chong Wang and published by . This book was released on 2017 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Ning Wang
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (135 download)
Download or read book Dimension Reduction and Regression for Tensor Data and Mixture Models written by Ning Wang and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In modern statistics, many data sets are of complex structure, including but not limited to high dimensionality, higher-order, and heterogeneity. Recently, there has been growing interest in developing valid and efficient statistical methods for these data sets. In my thesis, we studied three types of data complexity: (1) tensor data (a.k.a. array valued random objects); (2) heavy-tailed data; (3) data from heterogeneous subpopulations. We address these three challenges by developing novel methodologies and efficient algorithms. Specifically, we proposed likelihood-based dimension folding methods for tensor data, studied the robust tensor $\td$ regression by a proposed tensor $\td$ distribution, and developed an algorithm and theory for high-dimensional mixture linear regression. My work on these three topics is elaborated as follows. In recent years, traditional multivariate analysis tools, such as multivariate regression and discriminant analysis, are generalized from modeling random vectors and matrices to higher-order random tensors (a.k.a.~array-valued random objects). Equipped with tensor algebra and high-dimensional computation techniques, concise and interpretable statistical models and estimation procedures prevail in various applications. One challenge for tensor data analysis is caused by the large dimensions of the tensor. Many statistical methods such as linear discriminant analysis and quadratic discriminant analysis are not applicable or unstable for data sets with the dimension that is larger than the sample size. Sufficient dimension reduction methods are flexible tools for data visualization and exploratory analysis, typically in a regression of a univariate response on a multivariate predictor. For regressions with tensor predictors, a general framework of dimension folding and several moment-based estimation procedures have been proposed in the literature. In this essay, we propose two likelihood-based dimension folding methods motivated by quadratic discriminant analysis for tensor data: the maximum likelihood estimators are derived under a general covariance setting and a structured envelope covariance setting. We study the asymptotic properties of both estimators and show using simulation studies and a real-data analysis that they are more accurate than existing moment-based estimators. Another challenge to statistical tensor models is the non-Gaussian nature of many real-world data. Unfortunately, existing approaches are either restricted to normality or implicitly using least squares type objective functions that are computationally efficient but sensitive to data contamination. Motivated by this, we adopt a simple tensor $\td$-distribution that is, unlike the commonly used matrix $\td$-distributions, compatible with tensor operators and reshaping of the data. We study the tensor response regression with tensor $\td$-error, and develop penalized likelihood-based estimation and a novel one-step estimation. We study the asymptotic relative efficiency of various estimators and establish the one-step estimator's oracle properties and near-optimal asymptotic efficiency. We further propose a high-dimensional modification to the one-step estimation procedure and showed that it attains the minimax optimal rate in estimation. Numerical studies show the excellent performance of the one-step estimator. In the last chapter, we consider the high-dimensional mixture linear regression. The expectation-maximization (EM) algorithm and its variants are widely used in statistics. In high-dimensional mixture linear regression, the model is assumed to be a finite mixture of linear regression forms and the number of predictors is much larger than the sample size. The standard EM algorithm, which attempts to find the maximum likelihood estimator, becomes infeasible. We devise a penalized EM algorithm and study its statistical properties. Existing theoretical results of regularized EM algorithms often rely on dividing the sample into many independent batches and employing a fresh batch of sample in each iteration of the algorithm. Our algorithm and theoretical analysis do not require sample-splitting. The proposed method also has encouraging performances in simulation studies and a real data example.
Author : Seifedine Kadry
Publisher : Springer
ISBN 13 : 331907167X
Total Pages : 807 pages
Book Rating : 4.3/5 (19 download)
Download or read book Numerical Methods for Reliability and Safety Assessment written by Seifedine Kadry and published by Springer. This book was released on 2014-09-30 with total page 807 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers unique insight on structural safety and reliability by combining computational methods that address multiphysics problems, involving multiple equations describing different physical phenomena and multiscale problems, involving discrete sub-problems that together describe important aspects of a system at multiple scales. The book examines a range of engineering domains and problems using dynamic analysis, nonlinear methods, error estimation, finite element analysis and other computational techniques. This book also: · Introduces novel numerical methods · Illustrates new practical applications · Examines recent engineering applications · Presents up-to-date theoretical results · Offers perspective relevant to a wide audience, including teaching faculty/graduate students, researchers and practicing engineers.
Author :
Publisher :
ISBN 13 :
Total Pages : 838 pages
Book Rating : 4.3/5 (91 download)
Download or read book Statistica Sinica written by and published by . This book was released on 2008 with total page 838 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Ricardo A. Maronna
Publisher : John Wiley & Sons
ISBN 13 : 1119214688
Total Pages : 466 pages
Book Rating : 4.1/5 (192 download)
Download or read book Robust Statistics written by Ricardo A. Maronna and published by John Wiley & Sons. This book was released on 2019-01-04 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: A new edition of this popular text on robust statistics, thoroughly updated to include new and improved methods and focus on implementation of methodology using the increasingly popular open-source software R. Classical statistics fail to cope well with outliers associated with deviations from standard distributions. Robust statistical methods take into account these deviations when estimating the parameters of parametric models, thus increasing the reliability of fitted models and associated inference. This new, second edition of Robust Statistics: Theory and Methods (with R) presents a broad coverage of the theory of robust statistics that is integrated with computing methods and applications. Updated to include important new research results of the last decade and focus on the use of the popular software package R, it features in-depth coverage of the key methodology, including regression, multivariate analysis, and time series modeling. The book is illustrated throughout by a range of examples and applications that are supported by a companion website featuring data sets and R code that allow the reader to reproduce the examples given in the book. Unlike other books on the market, Robust Statistics: Theory and Methods (with R) offers the most comprehensive, definitive, and up-to-date treatment of the subject. It features chapters on estimating location and scale; measuring robustness; linear regression with fixed and with random predictors; multivariate analysis; generalized linear models; time series; numerical algorithms; and asymptotic theory of M-estimates. Explains both the use and theoretical justification of robust methods Guides readers in selecting and using the most appropriate robust methods for their problems Features computational algorithms for the core methods Robust statistics research results of the last decade included in this 2nd edition include: fast deterministic robust regression, finite-sample robustness, robust regularized regression, robust location and scatter estimation with missing data, robust estimation with independent outliers in variables, and robust mixed linear models. Robust Statistics aims to stimulate the use of robust methods as a powerful tool to increase the reliability and accuracy of statistical modelling and data analysis. It is an ideal resource for researchers, practitioners, and graduate students in statistics, engineering, computer science, and physical and social sciences.
Author : Kenneth Train
Publisher : Cambridge University Press
ISBN 13 : 0521766559
Total Pages : 399 pages
Book Rating : 4.5/5 (217 download)
Download or read book Discrete Choice Methods with Simulation written by Kenneth Train and published by Cambridge University Press. This book was released on 2009-07-06 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the new generation of discrete choice methods, focusing on the many advances that are made possible by simulation. Researchers use these statistical methods to examine the choices that consumers, households, firms, and other agents make. Each of the major models is covered: logit, generalized extreme value, or GEV (including nested and cross-nested logits), probit, and mixed logit, plus a variety of specifications that build on these basics. Simulation-assisted estimation procedures are investigated and compared, including maximum stimulated likelihood, method of simulated moments, and method of simulated scores. Procedures for drawing from densities are described, including variance reduction techniques such as anithetics and Halton draws. Recent advances in Bayesian procedures are explored, including the use of the Metropolis-Hastings algorithm and its variant Gibbs sampling. The second edition adds chapters on endogeneity and expectation-maximization (EM) algorithms. No other book incorporates all these fields, which have arisen in the past 25 years. The procedures are applicable in many fields, including energy, transportation, environmental studies, health, labor, and marketing.
Author :
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ISBN 13 :
Total Pages : 800 pages
Book Rating : 4.F/5 ( download)
Download or read book Dissertation Abstracts International written by and published by . This book was released on 2008 with total page 800 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : R. Dennis Cook
Publisher : John Wiley & Sons
ISBN 13 : 0470317779
Total Pages : 378 pages
Book Rating : 4.4/5 (73 download)
Download or read book Regression Graphics written by R. Dennis Cook and published by John Wiley & Sons. This book was released on 2009-09-25 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : Rick Wicklin
Publisher : SAS Institute
ISBN 13 : 1612903320
Total Pages : 363 pages
Book Rating : 4.6/5 (129 download)
Download or read book Simulating Data with SAS written by Rick Wicklin and published by SAS Institute. This book was released on 2013 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: Data simulation is a fundamental technique in statistical programming and research. Rick Wicklin's Simulating Data with SAS brings together the most useful algorithms and the best programming techniques for efficient data simulation in an accessible how-to book for practicing statisticians and statistical programmers. This book discusses in detail how to simulate data from common univariate and multivariate distributions, and how to use simulation to evaluate statistical techniques. It also covers simulating correlated data, data for regression models, spatial data, and data with given moments. It provides tips and techniques for beginning programmers, and offers libraries of functions for advanced practitioners. As the first book devoted to simulating data across a range of statistical applications, Simulating Data with SAS is an essential tool for programmers, analysts, researchers, and students who use SAS software. This book is part of the SAS Press program.
Author : Roman Vershynin
Publisher : Cambridge University Press
ISBN 13 : 1108415199
Total Pages : 299 pages
Book Rating : 4.1/5 (84 download)
Download or read book High-Dimensional Probability written by Roman Vershynin and published by Cambridge University Press. This book was released on 2018-09-27 with total page 299 pages. Available in PDF, EPUB and Kindle. Book excerpt: An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
Author : Jacob Cohen
Publisher : Routledge
ISBN 13 : 1134742770
Total Pages : 625 pages
Book Rating : 4.1/5 (347 download)
Download or read book Statistical Power Analysis for the Behavioral Sciences written by Jacob Cohen and published by Routledge. This book was released on 2013-05-13 with total page 625 pages. Available in PDF, EPUB and Kindle. Book excerpt: Statistical Power Analysis is a nontechnical guide to power analysis in research planning that provides users of applied statistics with the tools they need for more effective analysis. The Second Edition includes: * a chapter covering power analysis in set correlation and multivariate methods; * a chapter considering effect size, psychometric reliability, and the efficacy of "qualifying" dependent variables and; * expanded power and sample size tables for multiple regression/correlation.
