Concentration Of Maxima And Fundamental Limits In High Dimensional Testing And Inference
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| Author | : Zheng Gao |
| Publisher | : Springer Nature |
| Total Pages | : 147 |
| Release | : 2021-09-07 |
| Genre | : Mathematics |
| ISBN | : 3030809641 |
This book provides a unified exposition of some fundamental theoretical problems in high-dimensional statistics. It specifically considers the canonical problems of detection and support estimation for sparse signals observed with noise. Novel phase-transition results are obtained for the signal support estimation problem under a variety of statistical risks. Based on a surprising connection to a concentration of maxima probabilistic phenomenon, the authors obtain a complete characterization of the exact support recovery problem for thresholding estimators under dependent errors.
| Author | : Roman Vershynin |
| Publisher | : Cambridge University Press |
| Total Pages | : 299 |
| Release | : 2018-09-27 |
| Genre | : Business & Economics |
| ISBN | : 1108415199 |
An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
| Author | : Christophe Giraud |
| Publisher | : CRC Press |
| Total Pages | : 364 |
| Release | : 2021-08-25 |
| Genre | : Computers |
| ISBN | : 1000408329 |
Praise for the first edition: "[This book] succeeds singularly at providing a structured introduction to this active field of research. ... it is arguably the most accessible overview yet published of the mathematical ideas and principles that one needs to master to enter the field of high-dimensional statistics. ... recommended to anyone interested in the main results of current research in high-dimensional statistics as well as anyone interested in acquiring the core mathematical skills to enter this area of research." —Journal of the American Statistical Association Introduction to High-Dimensional Statistics, Second Edition preserves the philosophy of the first edition: to be a concise guide for students and researchers discovering the area and interested in the mathematics involved. The main concepts and ideas are presented in simple settings, avoiding thereby unessential technicalities. High-dimensional statistics is a fast-evolving field, and much progress has been made on a large variety of topics, providing new insights and methods. Offering a succinct presentation of the mathematical foundations of high-dimensional statistics, this new edition: Offers revised chapters from the previous edition, with the inclusion of many additional materials on some important topics, including compress sensing, estimation with convex constraints, the slope estimator, simultaneously low-rank and row-sparse linear regression, or aggregation of a continuous set of estimators. Introduces three new chapters on iterative algorithms, clustering, and minimax lower bounds. Provides enhanced appendices, minimax lower-bounds mainly with the addition of the Davis-Kahan perturbation bound and of two simple versions of the Hanson-Wright concentration inequality. Covers cutting-edge statistical methods including model selection, sparsity and the Lasso, iterative hard thresholding, aggregation, support vector machines, and learning theory. Provides detailed exercises at the end of every chapter with collaborative solutions on a wiki site. Illustrates concepts with simple but clear practical examples.
| Author | : Christine Sinoquet |
| Publisher | : OUP Oxford |
| Total Pages | : 415 |
| Release | : 2014-09-18 |
| Genre | : Science |
| ISBN | : 0191019208 |
Nowadays bioinformaticians and geneticists are faced with myriad high-throughput data usually presenting the characteristics of uncertainty, high dimensionality and large complexity. These data will only allow insights into this wealth of so-called 'omics' data if represented by flexible and scalable models, prior to any further analysis. At the interface between statistics and machine learning, probabilistic graphical models (PGMs) represent a powerful formalism to discover complex networks of relations. These models are also amenable to incorporating a priori biological information. Network reconstruction from gene expression data represents perhaps the most emblematic area of research where PGMs have been successfully applied. However these models have also created renewed interest in genetics in the broad sense, in particular regarding association genetics, causality discovery, prediction of outcomes, detection of copy number variations, and epigenetics. This book provides an overview of the applications of PGMs to genetics, genomics and postgenomics to meet this increased interest. A salient feature of bioinformatics, interdisciplinarity, reaches its limit when an intricate cooperation between domain specialists is requested. Currently, few people are specialists in the design of advanced methods using probabilistic graphical models for postgenomics or genetics. This book deciphers such models so that their perceived difficulty no longer hinders their use and focuses on fifteen illustrations showing the mechanisms behind the models. Probabilistic Graphical Models for Genetics, Genomics and Postgenomics covers six main themes: (1) Gene network inference (2) Causality discovery (3) Association genetics (4) Epigenetics (5) Detection of copy number variations (6) Prediction of outcomes from high-dimensional genomic data. Written by leading international experts, this is a collection of the most advanced work at the crossroads of probabilistic graphical models and genetics, genomics, and postgenomics. The self-contained chapters provide an enlightened account of the pros and cons of applying these powerful techniques.
| Author | : |
| Publisher | : |
| Total Pages | : 88 |
| Release | : 1969-02 |
| Genre | : |
| ISBN | : |
The Bulletin of the Atomic Scientists is the premier public resource on scientific and technological developments that impact global security. Founded by Manhattan Project Scientists, the Bulletin's iconic "Doomsday Clock" stimulates solutions for a safer world.
| Author | : Evarist Giné |
| Publisher | : Cambridge University Press |
| Total Pages | : 706 |
| Release | : 2021-03-25 |
| Genre | : Mathematics |
| ISBN | : 1009022784 |
In nonparametric and high-dimensional statistical models, the classical Gauss–Fisher–Le Cam theory of the optimality of maximum likelihood estimators and Bayesian posterior inference does not apply, and new foundations and ideas have been developed in the past several decades. This book gives a coherent account of the statistical theory in infinite-dimensional parameter spaces. The mathematical foundations include self-contained 'mini-courses' on the theory of Gaussian and empirical processes, approximation and wavelet theory, and the basic theory of function spaces. The theory of statistical inference in such models - hypothesis testing, estimation and confidence sets - is presented within the minimax paradigm of decision theory. This includes the basic theory of convolution kernel and projection estimation, but also Bayesian nonparametrics and nonparametric maximum likelihood estimation. In a final chapter the theory of adaptive inference in nonparametric models is developed, including Lepski's method, wavelet thresholding, and adaptive inference for self-similar functions. Winner of the 2017 PROSE Award for Mathematics.
| Author | : Jens Ledet Jensen |
| Publisher | : Oxford University Press |
| Total Pages | : 348 |
| Release | : 1995 |
| Genre | : Mathematics |
| ISBN | : 9780198522959 |
This book explains the ideas behind the saddlepoint approximations as well as giving a detailed mathematical description of the subject and many worked out examples.
| Author | : Jianqing Fan |
| Publisher | : CRC Press |
| Total Pages | : 942 |
| Release | : 2020-09-21 |
| Genre | : Mathematics |
| ISBN | : 0429527616 |
Statistical Foundations of Data Science gives a thorough introduction to commonly used statistical models, contemporary statistical machine learning techniques and algorithms, along with their mathematical insights and statistical theories. It aims to serve as a graduate-level textbook and a research monograph on high-dimensional statistics, sparsity and covariance learning, machine learning, and statistical inference. It includes ample exercises that involve both theoretical studies as well as empirical applications. The book begins with an introduction to the stylized features of big data and their impacts on statistical analysis. It then introduces multiple linear regression and expands the techniques of model building via nonparametric regression and kernel tricks. It provides a comprehensive account on sparsity explorations and model selections for multiple regression, generalized linear models, quantile regression, robust regression, hazards regression, among others. High-dimensional inference is also thoroughly addressed and so is feature screening. The book also provides a comprehensive account on high-dimensional covariance estimation, learning latent factors and hidden structures, as well as their applications to statistical estimation, inference, prediction and machine learning problems. It also introduces thoroughly statistical machine learning theory and methods for classification, clustering, and prediction. These include CART, random forests, boosting, support vector machines, clustering algorithms, sparse PCA, and deep learning.
| Author | : |
| Publisher | : |
| Total Pages | : 704 |
| Release | : 1995 |
| Genre | : Aeronautics |
| ISBN | : |
| Author | : Jianfeng Yao |
| Publisher | : Cambridge University Press |
| Total Pages | : 0 |
| Release | : 2015-03-26 |
| Genre | : Mathematics |
| ISBN | : 9781107065178 |
High-dimensional data appear in many fields, and their analysis has become increasingly important in modern statistics. However, it has long been observed that several well-known methods in multivariate analysis become inefficient, or even misleading, when the data dimension p is larger than, say, several tens. A seminal example is the well-known inefficiency of Hotelling's T2-test in such cases. This example shows that classical large sample limits may no longer hold for high-dimensional data; statisticians must seek new limiting theorems in these instances. Thus, the theory of random matrices (RMT) serves as a much-needed and welcome alternative framework. Based on the authors' own research, this book provides a first-hand introduction to new high-dimensional statistical methods derived from RMT. The book begins with a detailed introduction to useful tools from RMT, and then presents a series of high-dimensional problems with solutions provided by RMT methods.
