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Sufficient Dimension Reduction With Missing Data
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Book Synopsis Sufficient Dimension Reduction with Missing Data by : QI XIA
Download or read book Sufficient Dimension Reduction with Missing Data written by QI XIA and published by . This book was released on 2017 with total page 143 pages. Available in PDF, EPUB and Kindle. Book excerpt: Existing sufficient dimension reduction (SDR) methods typically consider cases with no missing data. The dissertation aims to propose methods to facilitate the SDR methods when the response can be missing. The first part of the dissertation focuses on the seminal sliced inverse regression (SIR) approach proposed by Li (1991). We show that missing responses generally affect the validity of the inverse regressions under the mechanism of missing at random. We then propose a simple and effective adjustment with inverse probability weighting that guarantees the validity of SIR. Furthermore, a marginal coordinate test is introduced for this adjusted estimator. The proposed method share the simplicity of SIR and requires the linear conditional mean assumption. The second part of the dissertation proposes two new estimating equation procedures: the complete case estimating equation approach and the inverse probability weighted estimating equation approach. The two approaches are applied to a family of dimension reduction methods, which includes ordinary least squares, principal Hessian directions, and SIR. By solving the estimating equations, the two approaches are able to avoid the common assumptions in the SDR literature, the linear conditional mean assumption, and the constant conditional variance assumption. For all the aforementioned methods, the asymptotic properties are established, and their superb finite sample performances are demonstrated through extensive numerical studies as well as a real data analysis. In addition, existing estimators of the central mean space have uneven performances across different types of link functions. To address this limitation, a new hybrid SDR estimator is proposed that successfully recovers the central mean space for a wide range of link functions. Based on the new hybrid estimator, we further study the order determination procedure and the marginal coordinate test. The superior performance of the hybrid estimator over existing methods is demonstrated in simulation studies. Note that the proposed procedures dealing with the missing response at random can be simply adapted to this hybrid method.
Book Synopsis Transforms in Sufficient Dimension Reduction and Their Applications in High Dimensional Data by : Jiaying Weng
Download or read book Transforms in Sufficient Dimension Reduction and Their Applications in High Dimensional Data written by Jiaying Weng and published by . This book was released on 2019 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Sufficient Dimension Reduction by : Bing Li
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 307 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.
Book Synopsis Dimension Reduction and Sufficient Graphical Models by : Kyongwon Kim
Download or read book Dimension Reduction and Sufficient Graphical Models written by Kyongwon Kim and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Statistical Methods for Handling Incomplete Data by : Jae Kwang Kim
Download or read book Statistical Methods for Handling Incomplete Data written by Jae Kwang Kim and published by CRC Press. This book was released on 2021-11-19 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: Due to recent theoretical findings and advances in statistical computing, there has been a rapid development of techniques and applications in the area of missing data analysis. Statistical Methods for Handling Incomplete Data covers the most up-to-date statistical theories and computational methods for analyzing incomplete data. Features Uses the mean score equation as a building block for developing the theory for missing data analysis Provides comprehensive coverage of computational techniques for missing data analysis Presents a rigorous treatment of imputation techniques, including multiple imputation fractional imputation Explores the most recent advances of the propensity score method and estimation techniques for nonignorable missing data Describes a survey sampling application Updated with a new chapter on Data Integration Now includes a chapter on Advanced Topics, including kernel ridge regression imputation and neural network model imputation The book is primarily aimed at researchers and graduate students from statistics, and could be used as a reference by applied researchers with a good quantitative background. It includes many real data examples and simulated examples to help readers understand the methodologies.
Book Synopsis On Sufficient Dimension Reduction Via Asymmetric Least Squares by : Abdul-Nasah Soale
Download or read book On Sufficient Dimension Reduction Via Asymmetric Least Squares written by Abdul-Nasah Soale and published by . This book was released on 2021 with total page 76 pages. Available in PDF, EPUB and Kindle. Book excerpt: Accompanying the advances in computer technology is an increase collection of high dimensional data in many scientific and social studies. Sufficient dimension reduction (SDR) is a statistical method that enable us to reduce the dimension ofpredictors without loss of regression information. In this dissertation, we introduce principal asymmetric least squares (PALS) as a unified framework for linear and nonlinear sufficient dimension reduction. Classical methods such as sliced inverse regression (Li, 1991) and principal support vector machines (Li, Artemiou and Li, 2011) often do not perform well in the presence of heteroscedastic error, while our proposal addresses this limitation by synthesizing different expectile levels. Through extensive numerical studies, we demonstrate the superior performance of PALS in terms of both computation time and estimation accuracy. For the asymptotic analysis of PALS for linear sufficient dimension reduction, we develop new tools to compute the derivative of an expectation of a non-Lipschitz function. PALS is not designed to handle symmetric link function between the response and the predictors. As a remedy, we develop expectile-assisted inverse regression estimation (EA-IRE) as a unified framework for moment-based inverse regression. We propose to first estimate the expectiles through kernel expectile regression, and then carry out dimension reduction based on random projections of the regression expectiles. Several popular inverse regression methods in the literature including slice inverse regression, slice average variance estimation, and directional regression are extended under this general framework. The proposed expectile-assisted methods outperform existing moment-based dimension reduction methods in both numerical studies and an analysis of the Big Mac data.
Book Synopsis Modern Dimension Reduction by : Philip D. Waggoner
Download or read book Modern Dimension Reduction written by Philip D. Waggoner and published by Cambridge University Press. This book was released on 2021-08-05 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: Data are not only ubiquitous in society, but are increasingly complex both in size and dimensionality. Dimension reduction offers researchers and scholars the ability to make such complex, high dimensional data spaces simpler and more manageable. This Element offers readers a suite of modern unsupervised dimension reduction techniques along with hundreds of lines of R code, to efficiently represent the original high dimensional data space in a simplified, lower dimensional subspace. Launching from the earliest dimension reduction technique principal components analysis and using real social science data, I introduce and walk readers through application of the following techniques: locally linear embedding, t-distributed stochastic neighbor embedding (t-SNE), uniform manifold approximation and projection, self-organizing maps, and deep autoencoders. The result is a well-stocked toolbox of unsupervised algorithms for tackling the complexities of high dimensional data so common in modern society. All code is publicly accessible on Github.
Book Synopsis Dimension Reduction by : Christopher J. C. Burges
Download or read book Dimension Reduction written by Christopher J. C. Burges and published by Now Publishers Inc. This book was released on 2010 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: We give a tutorial overview of several foundational methods for dimension reduction. We divide the methods into projective methods and methods that model the manifold on which the data lies. For projective methods, we review projection pursuit, principal component analysis (PCA), kernel PCA, probabilistic PCA, canonical correlation analysis (CCA), kernel CCA, Fisher discriminant analysis, oriented PCA, and several techniques for sufficient dimension reduction. For the manifold methods, we review multidimensional scaling (MDS), landmark MDS, Isomap, locally linear embedding, Laplacian eigenmaps, and spectral clustering. Although the review focuses on foundations, we also provide pointers to some more modern techniques. We also describe the correlation dimension as one method for estimating the intrinsic dimension, and we point out that the notion of dimension can be a scale-dependent quantity. The Nystr m method, which links several of the manifold algorithms, is also reviewed. We use a publicly available dataset to illustrate some of the methods. The goal is to provide a self-contained overview of key concepts underlying many of these algorithms, and to give pointers for further reading.
Book Synopsis Advances in Forward Sufficient Dimension Reduction Methods for Statistical Learning by : Harris Quach
Download or read book Advances in Forward Sufficient Dimension Reduction Methods for Statistical Learning written by Harris Quach and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern information collection methods continue to generate an abundance of data from which practitioners, in fields ranging from the social sciences and humanities to the natural and biomedical sciences, attempt to draw new insights and discoveries through statistical analyses. Against this backdrop, dimension reduction methods have come to prominence as a tool for extracting important or informative features from high dimensional data sets. In particular, Sufficient Dimension Reduction (SDR) methods have become a popular and effective tool for supervised dimension reduction. Since the seminal paper by K.-C. Li (1991), Sufficient Dimension Reduction has developed at a rapid pace, with inverse regression methods having undergone substantial development and becoming widely applicable. Alternative approaches to sufficient dimension reduction have received relatively less attention, despite being less restrictive and more effective in many scenarios. The objective of this thesis is to advance the forward regression approach for sufficient dimension reduction by developing methods that are more effective for categorical and ordinal responses, and methods that apply to functional data.
Book Synopsis Sufficient Dimension Reduction Based on Normal and Wishart Inverse Models by : Liliana Forzani
Download or read book Sufficient Dimension Reduction Based on Normal and Wishart Inverse Models written by Liliana Forzani and published by . This book was released on 2007 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Sufficient Dimension Reduction and Variable Selection by : Xin Chen
Download or read book Sufficient Dimension Reduction and Variable Selection written by Xin Chen and published by . This book was released on 2010 with total page 69 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Topics on Sufficient Dimension Reduction by : Son Nguyen
Download or read book Topics on Sufficient Dimension Reduction written by Son Nguyen and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Journal of the American Statistical Association by :
Download or read book Journal of the American Statistical Association written by and published by . This book was released on 2009 with total page 896 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Study of Sufficient Dimension Reduction Methods by : Chong Wang
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:
Book Synopsis Dimension Reduction in Statistical Modeling by : Linquan Ma (Ph.D.)
Download or read book Dimension Reduction in Statistical Modeling written by Linquan Ma (Ph.D.) and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: When the data object is described by a large number of features, it is often beneficial to reduce the dimension of the data, so that the statistical analysis can have better efficiencies. Recently, a new dimension reduction method called the envelope method by Cook, Li, and Chiaromonte (2010) has been proposed in multivariate regressions. It has the potential to gain substantial efficiency over the standard least squares estimator. Chapter 2 proposes an approach to use the envelope method when the predictors and/or the responses are missing at random. When there exists missing data, the envelope method using the complete case observations may lead to biased and inefficient results. We incorporate the envelope structure in the expectation-maximization (EM) algorithm. Our method is guaranteed to be more efficient, or at least as efficient as, the standard EM algorithm. We give asymptotic properties of our method under both normal and non-normal cases. Chapter 3 extends the envelope model to the mixed effects model for longitudinal data with possibly unbalanced design and time-varying predictors. We show that our model provides more efficient estimators than the standard estimators in mixed effects models. Chapter 4 proposes a semiparametric variant of the inner envelope model (Su and Cook, 2012) that does not rely on the linear model nor the normality assumption. We show that our proposal leads to globally and locally efficient estimators of the inner envelope spaces. We also present a computationally tractable algorithm to estimate the inner envelope. The instrumental variables (IV) are frequently used in observational studies to recover the effect of exposure in the presence of unmeasured confounding. A key fact is that the strength of IV matters: an IV with a stronger association with the exposure results in a more accurate estimation of a causal effect. While it is hard to find a stronger IV, we generalize a sufficient dimension method to remove immaterial IVs. Chapter 5 investigates two different ways of incorporating the envelope method into IV regression. We show that the first stage envelope method does not yield any efficiency gain on the standard IV estimator, however, it may reduce the finite sample bias. The second stage envelope can achieve substantial efficiency gain under certain conditions.
Book Synopsis Bayesian Nonparametrics for Causal Inference and Missing Data by : Michael J. Daniels
Download or read book Bayesian Nonparametrics for Causal Inference and Missing Data written by Michael J. Daniels and published by CRC Press. This book was released on 2023-08-23 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bayesian Nonparametrics for Causal Inference and Missing Data provides an overview of flexible Bayesian nonparametric (BNP) methods for modeling joint or conditional distributions and functional relationships, and their interplay with causal inference and missing data. This book emphasizes the importance of making untestable assumptions to identify estimands of interest, such as missing at random assumption for missing data and unconfoundedness for causal inference in observational studies. Unlike parametric methods, the BNP approach can account for possible violations of assumptions and minimize concerns about model misspecification. The overall strategy is to first specify BNP models for observed data and then to specify additional uncheckable assumptions to identify estimands of interest. The book is divided into three parts. Part I develops the key concepts in causal inference and missing data and reviews relevant concepts in Bayesian inference. Part II introduces the fundamental BNP tools required to address causal inference and missing data problems. Part III shows how the BNP approach can be applied in a variety of case studies. The datasets in the case studies come from electronic health records data, survey data, cohort studies, and randomized clinical trials. Features • Thorough discussion of both BNP and its interplay with causal inference and missing data • How to use BNP and g-computation for causal inference and non-ignorable missingness • How to derive and calibrate sensitivity parameters to assess sensitivity to deviations from uncheckable causal and/or missingness assumptions • Detailed case studies illustrating the application of BNP methods to causal inference and missing data • R code and/or packages to implement BNP in causal inference and missing data problems The book is primarily aimed at researchers and graduate students from statistics and biostatistics. It will also serve as a useful practical reference for mathematically sophisticated epidemiologists and medical researchers.
Book Synopsis Multiple Imputation of Missing Data Using SAS by : Patricia Berglund
Download or read book Multiple Imputation of Missing Data Using SAS written by Patricia Berglund and published by SAS Institute. This book was released on 2014-07-01 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: Find guidance on using SAS for multiple imputation and solving common missing data issues. Multiple Imputation of Missing Data Using SAS provides both theoretical background and constructive solutions for those working with incomplete data sets in an engaging example-driven format. It offers practical instruction on the use of SAS for multiple imputation and provides numerous examples that use a variety of public release data sets with applications to survey data. Written for users with an intermediate background in SAS programming and statistics, this book is an excellent resource for anyone seeking guidance on multiple imputation. The authors cover the MI and MIANALYZE procedures in detail, along with other procedures used for analysis of complete data sets. They guide analysts through the multiple imputation process, including evaluation of missing data patterns, choice of an imputation method, execution of the process, and interpretation of results. Topics discussed include how to deal with missing data problems in a statistically appropriate manner, how to intelligently select an imputation method, how to incorporate the uncertainty introduced by the imputation process, and how to incorporate the complex sample design (if appropriate) through use of the SAS SURVEY procedures. Discover the theoretical background and see extensive applications of the multiple imputation process in action. This book is part of the SAS Press program.
