Tensor Methods In Statistics

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Tensor Methods in Statistics

Author : P. McCullagh
Publisher : CRC Press
ISBN 13 : 1351094017
Total Pages : 185 pages
Book Rating : 4.3/5 (51 download)

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Book Synopsis Tensor Methods in Statistics by : P. McCullagh

Download or read book Tensor Methods in Statistics written by P. McCullagh and published by CRC Press. This book was released on 2018-01-18 with total page 185 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a systematic development of tensor methods in statistics, beginning with the study of multivariate moments and cumulants. The effect on moment arrays and on cumulant arrays of making linear or affine transformations of the variables is studied. Because of their importance in statistical theory, invariant functions of the cumulants are studied in some detail. This is followed by an examination of the effect of making a polynomial transformation of the original variables. The fundamental operation of summing over complementary set partitions is introduced at this stage. This operation shapes the notation and pervades much of the remainder of the book. The necessary lattice-theory is discussed and suitable tables of complementary set partitions are provided. Subsequent chapters deal with asymptotic approximations based on Edgeworth expansion and saddlepoint expansion. The saddlepoint expansion is introduced via the Legendre transformation of the cumulant generating function, also known as the conjugate function of the cumulant generating function. A recurring them is that, with suitably chosen notation, multivariate calculations are often simpler and more transparent than the corresponding univariate calculations. The final two chapters deal with likelihood ratio statistics, maximum likelihood estimation and the effect on inferences of conditioning on ancillary or approximately ancillary statistics. The Bartlett adjustment factor is derived in the general case and simplified for certain types of generalized linear models. Finally, Barndorff-Nielsen's formula for the conditional distribution of the maximum liklelihood estimator is derived and discussed. More than 200 Exercises are provided to illustrate the uses of tensor methodology.

Tensor Methods in Statistics

Author : Peter McCullagh
Publisher : Courier Dover Publications
ISBN 13 : 0486823784
Total Pages : 308 pages
Book Rating : 4.4/5 (868 download)

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Book Synopsis Tensor Methods in Statistics by : Peter McCullagh

Download or read book Tensor Methods in Statistics written by Peter McCullagh and published by Courier Dover Publications. This book was released on 2018-07-18 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: A pioneering monograph on tensor methods applied to distributional problems arising in statistics, this work begins with the study of multivariate moments and cumulants. An invaluable reference for graduate students and professional statisticians. 1987 edition.

Tensor Methods in Statistics

Author : Peter McCullagh
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (959 download)

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Book Synopsis Tensor Methods in Statistics by : Peter McCullagh

Download or read book Tensor Methods in Statistics written by Peter McCullagh and published by . This book was released on 1987 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Tensor Methods in Statistics

Author : P. McCullagh
Publisher : CRC Press
ISBN 13 : 1351085565
Total Pages : 301 pages
Book Rating : 4.3/5 (51 download)

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Book Synopsis Tensor Methods in Statistics by : P. McCullagh

Download or read book Tensor Methods in Statistics written by P. McCullagh and published by CRC Press. This book was released on 2018-01-18 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a systematic development of tensor methods in statistics, beginning with the study of multivariate moments and cumulants. The effect on moment arrays and on cumulant arrays of making linear or affine transformations of the variables is studied. Because of their importance in statistical theory, invariant functions of the cumulants are studied in some detail. This is followed by an examination of the effect of making a polynomial transformation of the original variables. The fundamental operation of summing over complementary set partitions is introduced at this stage. This operation shapes the notation and pervades much of the remainder of the book. The necessary lattice-theory is discussed and suitable tables of complementary set partitions are provided. Subsequent chapters deal with asymptotic approximations based on Edgeworth expansion and saddlepoint expansion. The saddlepoint expansion is introduced via the Legendre transformation of the cumulant generating function, also known as the conjugate function of the cumulant generating function. A recurring them is that, with suitably chosen notation, multivariate calculations are often simpler and more transparent than the corresponding univariate calculations. The final two chapters deal with likelihood ratio statistics, maximum likelihood estimation and the effect on inferences of conditioning on ancillary or approximately ancillary statistics. The Bartlett adjustment factor is derived in the general case and simplified for certain types of generalized linear models. Finally, Barndorff-Nielsen's formula for the conditional distribution of the maximum liklelihood estimator is derived and discussed. More than 200 Exercises are provided to illustrate the uses of tensor methodology.

An Introduction to Algebraic Statistics with Tensors

Author : Cristiano Bocci
Publisher : Springer Nature
ISBN 13 : 3030246248
Total Pages : 235 pages
Book Rating : 4.0/5 (32 download)

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Book Synopsis An Introduction to Algebraic Statistics with Tensors by : Cristiano Bocci

Download or read book An Introduction to Algebraic Statistics with Tensors written by Cristiano Bocci and published by Springer Nature. This book was released on 2019-09-11 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to various aspects of Algebraic Statistics with the principal aim of supporting Master’s and PhD students who wish to explore the algebraic point of view regarding recent developments in Statistics. The focus is on the background needed to explore the connections among discrete random variables. The main objects that encode these relations are multilinear matrices, i.e., tensors. The book aims to settle the basis of the correspondence between properties of tensors and their translation in Algebraic Geometry. It is divided into three parts, on Algebraic Statistics, Multilinear Algebra, and Algebraic Geometry. The primary purpose is to describe a bridge between the three theories, so that results and problems in one theory find a natural translation to the others. This task requires, from the statistical point of view, a rather unusual, but algebraically natural, presentation of random variables and their main classical features. The third part of the book can be considered as a short, almost self-contained, introduction to the basic concepts of algebraic varieties, which are part of the fundamental background for all who work in Algebraic Statistics.

Tensor Regression

Author : Jiani Liu
Publisher :
ISBN 13 : 9781680838862
Total Pages : pages
Book Rating : 4.8/5 (388 download)

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Book Synopsis Tensor Regression by : Jiani Liu

Download or read book Tensor Regression written by Jiani Liu and published by . This book was released on 2021-09-27 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensor Regression is the first thorough overview of the fundamentals, motivations, popular algorithms, strategies for efficient implementation, related applications, available datasets, and software resources for tensor-based regression analysis.

Applied Matrix and Tensor Variate Data Analysis

Author : Toshio Sakata
Publisher : Springer
ISBN 13 : 4431553878
Total Pages : 144 pages
Book Rating : 4.4/5 (315 download)

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Book Synopsis Applied Matrix and Tensor Variate Data Analysis by : Toshio Sakata

Download or read book Applied Matrix and Tensor Variate Data Analysis written by Toshio Sakata and published by Springer. This book was released on 2016-02-02 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides comprehensive reviews of recent progress in matrix variate and tensor variate data analysis from applied points of view. Matrix and tensor approaches for data analysis are known to be extremely useful for recently emerging complex and high-dimensional data in various applied fields. The reviews contained herein cover recent applications of these methods in psychology (Chap. 1), audio signals (Chap. 2) , image analysis from tensor principal component analysis (Chap. 3), and image analysis from decomposition (Chap. 4), and genetic data (Chap. 5) . Readers will be able to understand the present status of these techniques as applicable to their own fields. In Chapter 5 especially, a theory of tensor normal distributions, which is a basic in statistical inference, is developed, and multi-way regression, classification, clustering, and principal component analysis are exemplified under tensor normal distributions. Chapter 6 treats one-sided tests under matrix variate and tensor variate normal distributions, whose theory under multivariate normal distributions has been a popular topic in statistics since the books of Barlow et al. (1972) and Robertson et al. (1988). Chapters 1, 5, and 6 distinguish this book from ordinary engineering books on these topics.

Tensor Spaces and Numerical Tensor Calculus

Author : Wolfgang Hackbusch
Publisher : Springer Nature
ISBN 13 : 3030355543
Total Pages : 605 pages
Book Rating : 4.0/5 (33 download)

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Book Synopsis Tensor Spaces and Numerical Tensor Calculus by : Wolfgang Hackbusch

Download or read book Tensor Spaces and Numerical Tensor Calculus written by Wolfgang Hackbusch and published by Springer Nature. This book was released on 2019-12-16 with total page 605 pages. Available in PDF, EPUB and Kindle. Book excerpt: Special numerical techniques are already needed to deal with n × n matrices for large n. Tensor data are of size n × n ×...× n=nd, where nd exceeds the computer memory by far. They appear for problems of high spatial dimensions. Since standard methods fail, a particular tensor calculus is needed to treat such problems. This monograph describes the methods by which tensors can be practically treated and shows how numerical operations can be performed. Applications include problems from quantum chemistry, approximation of multivariate functions, solution of partial differential equations, for example with stochastic coefficients, and more. In addition to containing corrections of the unavoidable misprints, this revised second edition includes new parts ranging from single additional statements to new subchapters. The book is mainly addressed to numerical mathematicians and researchers working with high-dimensional data. It also touches problems related to Geometric Algebra.

Algebraic and Computational Aspects of Real Tensor Ranks

Author : Toshio Sakata
Publisher : Springer
ISBN 13 : 4431554599
Total Pages : 112 pages
Book Rating : 4.4/5 (315 download)

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Book Synopsis Algebraic and Computational Aspects of Real Tensor Ranks by : Toshio Sakata

Download or read book Algebraic and Computational Aspects of Real Tensor Ranks written by Toshio Sakata and published by Springer. This book was released on 2016-03-18 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides comprehensive summaries of theoretical (algebraic) and computational aspects of tensor ranks, maximal ranks, and typical ranks, over the real number field. Although tensor ranks have been often argued in the complex number field, it should be emphasized that this book treats real tensor ranks, which have direct applications in statistics. The book provides several interesting ideas, including determinant polynomials, determinantal ideals, absolutely nonsingular tensors, absolutely full column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. In addition to reviews of methods to determine real tensor ranks in details, global theories such as the Jacobian method are also reviewed in details. The book includes as well an accessible and comprehensive introduction of mathematical backgrounds, with basics of positive polynomials and calculations by using the Groebner basis. Furthermore, this book provides insights into numerical methods of finding tensor ranks through simultaneous singular value decompositions.

Tensors: Geometry and Applications

Author : J. M. Landsberg
Publisher : American Mathematical Soc.
ISBN 13 : 0821869078
Total Pages : 464 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Tensors: Geometry and Applications by : J. M. Landsberg

Download or read book Tensors: Geometry and Applications written by J. M. Landsberg and published by American Mathematical Soc.. This book was released on 2011-12-14 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language. This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.

Tensor Calculus and Applications

Author : Bhaben Chandra Kalita
Publisher : CRC Press
ISBN 13 : 0429647565
Total Pages : 123 pages
Book Rating : 4.4/5 (296 download)

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Book Synopsis Tensor Calculus and Applications by : Bhaben Chandra Kalita

Download or read book Tensor Calculus and Applications written by Bhaben Chandra Kalita and published by CRC Press. This book was released on 2019-03-11 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to make the subject easier to understand. This book provides clear concepts, tools, and techniques to master the subject -tensor, and can be used in many fields of research. Special applications are discussed in the book, to remove any confusion, and for absolute understanding of the subject. In most books, they emphasize only the theoretical development, but not the methods of presentation, to develop concepts. Without knowing how to change the dummy indices, or the real indices, the concept cannot be understood. This book takes it down a notch and simplifies the topic for easy comprehension. Features Provides a clear indication and understanding of the subject on how to change indices Describes the original evolution of symbols necessary for tensors Offers a pictorial representation of referential systems required for different kinds of tensors for physical problems Presents the correlation between critical concepts Covers general operations and concepts

Nonnegative Matrix and Tensor Factorizations

Author : Andrzej Cichocki
Publisher : John Wiley & Sons
ISBN 13 : 9780470747285
Total Pages : 500 pages
Book Rating : 4.7/5 (472 download)

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Book Synopsis Nonnegative Matrix and Tensor Factorizations by : Andrzej Cichocki

Download or read book Nonnegative Matrix and Tensor Factorizations written by Andrzej Cichocki and published by John Wiley & Sons. This book was released on 2009-07-10 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a broad survey of models and efficient algorithms for Nonnegative Matrix Factorization (NMF). This includes NMF’s various extensions and modifications, especially Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD). NMF/NTF and their extensions are increasingly used as tools in signal and image processing, and data analysis, having garnered interest due to their capability to provide new insights and relevant information about the complex latent relationships in experimental data sets. It is suggested that NMF can provide meaningful components with physical interpretations; for example, in bioinformatics, NMF and its extensions have been successfully applied to gene expression, sequence analysis, the functional characterization of genes, clustering and text mining. As such, the authors focus on the algorithms that are most useful in practice, looking at the fastest, most robust, and suitable for large-scale models. Key features: Acts as a single source reference guide to NMF, collating information that is widely dispersed in current literature, including the authors’ own recently developed techniques in the subject area. Uses generalized cost functions such as Bregman, Alpha and Beta divergences, to present practical implementations of several types of robust algorithms, in particular Multiplicative, Alternating Least Squares, Projected Gradient and Quasi Newton algorithms. Provides a comparative analysis of the different methods in order to identify approximation error and complexity. Includes pseudo codes and optimized MATLAB source codes for almost all algorithms presented in the book. The increasing interest in nonnegative matrix and tensor factorizations, as well as decompositions and sparse representation of data, will ensure that this book is essential reading for engineers, scientists, researchers, industry practitioners and graduate students across signal and image processing; neuroscience; data mining and data analysis; computer science; bioinformatics; speech processing; biomedical engineering; and multimedia.

Distribution-Free Statistical Methods, Second Edition

Author : J.S. Maritz
Publisher : CRC Press
ISBN 13 : 1000153002
Total Pages : 279 pages
Book Rating : 4.0/5 (1 download)

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Book Synopsis Distribution-Free Statistical Methods, Second Edition by : J.S. Maritz

Download or read book Distribution-Free Statistical Methods, Second Edition written by J.S. Maritz and published by CRC Press. This book was released on 2020-11-26 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: Distribution-free statistical methods enable users to make statistical inferences with minimum assumptions about the population in question. They are widely used, especially in the areas of medical and psychological research. This new edition is aimed at senior undergraduate and graduate level. It also includes a discussion of new techniques that have arisen as a result of improvements in statistical computing. Interest in estimation techniques has particularly grown, and this section of the book has been expanded accordingly. Finally, Distribution-Free Statistical Methods includes more examples with actual data sets appearing in the text.

From Algebraic Structures to Tensors

Author : Gérard Favier
Publisher : John Wiley & Sons
ISBN 13 : 1786301547
Total Pages : 324 pages
Book Rating : 4.7/5 (863 download)

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Book Synopsis From Algebraic Structures to Tensors by : Gérard Favier

Download or read book From Algebraic Structures to Tensors written by Gérard Favier and published by John Wiley & Sons. This book was released on 2020-01-02 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nowadays, tensors play a central role for the representation, mining, analysis, and fusion of multidimensional, multimodal, and heterogeneous big data in numerous fields. This set on Matrices and Tensors in Signal Processing aims at giving a self-contained and comprehensive presentation of various concepts and methods, starting from fundamental algebraic structures to advanced tensor-based applications, including recently developed tensor models and efficient algorithms for dimensionality reduction and parameter estimation. Although its title suggests an orientation towards signal processing, the results presented in this set will also be of use to readers interested in other disciplines. This first book provides an introduction to matrices and tensors of higher-order based on the structures of vector space and tensor space. Some standard algebraic structures are first described, with a focus on the hilbertian approach for signal representation, and function approximation based on Fourier series and orthogonal polynomial series. Matrices and hypermatrices associated with linear, bilinear and multilinear maps are more particularly studied. Some basic results are presented for block matrices. The notions of decomposition, rank, eigenvalue, singular value, and unfolding of a tensor are introduced, by emphasizing similarities and differences between matrices and tensors of higher-order.

Matrix and Tensor Decompositions in Signal Processing, Volume 2

Author : Gérard Favier
Publisher : John Wiley & Sons
ISBN 13 : 1119700965
Total Pages : 386 pages
Book Rating : 4.1/5 (197 download)

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Book Synopsis Matrix and Tensor Decompositions in Signal Processing, Volume 2 by : Gérard Favier

Download or read book Matrix and Tensor Decompositions in Signal Processing, Volume 2 written by Gérard Favier and published by John Wiley & Sons. This book was released on 2021-08-17 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: The second volume will deal with a presentation of the main matrix and tensor decompositions and their properties of uniqueness, as well as very useful tensor networks for the analysis of massive data. Parametric estimation algorithms will be presented for the identification of the main tensor decompositions. After a brief historical review of the compressed sampling methods, an overview of the main methods of retrieving matrices and tensors with missing data will be performed under the low rank hypothesis. Illustrative examples will be provided.

Tensors for Data Processing

Author : Yipeng Liu
Publisher : Academic Press
ISBN 13 : 0323859658
Total Pages : 598 pages
Book Rating : 4.3/5 (238 download)

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Book Synopsis Tensors for Data Processing by : Yipeng Liu

Download or read book Tensors for Data Processing written by Yipeng Liu and published by Academic Press. This book was released on 2021-10-21 with total page 598 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensors for Data Processing: Theory, Methods and Applications presents both classical and state-of-the-art methods on tensor computation for data processing, covering computation theories, processing methods, computing and engineering applications, with an emphasis on techniques for data processing. This reference is ideal for students, researchers and industry developers who want to understand and use tensor-based data processing theories and methods. As a higher-order generalization of a matrix, tensor-based processing can avoid multi-linear data structure loss that occurs in classical matrix-based data processing methods. This move from matrix to tensors is beneficial for many diverse application areas, including signal processing, computer science, acoustics, neuroscience, communication, medical engineering, seismology, psychometric, chemometrics, biometric, quantum physics and quantum chemistry. Provides a complete reference on classical and state-of-the-art tensor-based methods for data processing Includes a wide range of applications from different disciplines Gives guidance for their application

Expansions and Asymptotics for Statistics

Author : Christopher G. Small
Publisher : CRC Press
ISBN 13 : 1420011022
Total Pages : 359 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis Expansions and Asymptotics for Statistics by : Christopher G. Small

Download or read book Expansions and Asymptotics for Statistics written by Christopher G. Small and published by CRC Press. This book was released on 2010-05-07 with total page 359 pages. Available in PDF, EPUB and Kindle. Book excerpt: Asymptotic methods provide important tools for approximating and analysing functions that arise in probability and statistics. Moreover, the conclusions of asymptotic analysis often supplement the conclusions obtained by numerical methods. Providing a broad toolkit of analytical methods, Expansions and Asymptotics for Statistics shows how asymptoti