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Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition

Author: Haruo Yanai

Publisher: Springer Science & Business Media

Published: 2011-04-06

Total Pages: 244

ISBN-13: 144199887X

Aside from distribution theory, projections and the singular value decomposition (SVD) are the two most important concepts for understanding the basic mechanism of multivariate analysis. The former underlies the least squares estimation in regression analysis, which is essentially a projection of one subspace onto another, and the latter underlies principal component analysis, which seeks to find a subspace that captures the largest variability in the original space. This book is about projections and SVD. A thorough discussion of generalized inverse (g-inverse) matrices is also given because it is closely related to the former. The book provides systematic and in-depth accounts of these concepts from a unified viewpoint of linear transformations finite dimensional vector spaces. More specially, it shows that projection matrices (projectors) and g-inverse matrices can be defined in various ways so that a vector space is decomposed into a direct-sum of (disjoint) subspaces. Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition will be useful for researchers, practitioners, and students in applied mathematics, statistics, engineering, behaviormetrics, and other fields.

Numerical And Symbolic Computations Of Generalized Inverses

Author: Yimin Wei

Publisher: World Scientific

Published: 2018-07-18

Total Pages: 472

ISBN-13: 9813238682

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We introduce new methods connecting numerics and symbolic computations, i.e., both the direct and iterative methods as well as the symbolic method for computing the generalized inverses. These will be useful for Engineers and Statisticians, in addition to applied mathematicians.Also, main applications of generalized inverses will be presented. Symbolic method covered in our book but not discussed in other book, which is important for numerical-symbolic computations.

Matrix Algebra for Linear Models

Author: Marvin H. J. Gruber

Publisher: John Wiley & Sons

Published: 2013-12-31

Total Pages: 391

ISBN-13: 1118592557

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A self-contained introduction to matrix analysis theory and applications in the field of statistics Comprehensive in scope, Matrix Algebra for Linear Models offers a succinct summary of matrix theory and its related applications to statistics, especially linear models. The book provides a unified presentation of the mathematical properties and statistical applications of matrices in order to define and manipulate data. Written for theoretical and applied statisticians, the book utilizes multiple numerical examples to illustrate key ideas, methods, and techniques crucial to understanding matrix algebra’s application in linear models. Matrix Algebra for Linear Models expertly balances concepts and methods allowing for a side-by-side presentation of matrix theory and its linear model applications. Including concise summaries on each topic, the book also features: Methods of deriving results from the properties of eigenvalues and the singular value decomposition Solutions to matrix optimization problems for obtaining more efficient biased estimators for parameters in linear regression models A section on the generalized singular value decomposition Multiple chapter exercises with selected answers to enhance understanding of the presented material Matrix Algebra for Linear Models is an ideal textbook for advanced undergraduate and graduate-level courses on statistics, matrices, and linear algebra. The book is also an excellent reference for statisticians, engineers, economists, and readers interested in the linear statistical model.

Linear Algebra for Pattern Processing

Author: Kenichi Kanatani

Publisher: Springer Nature

Published: 2022-06-01

Total Pages: 141

ISBN-13: 303102544X

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Linear algebra is one of the most basic foundations of a wide range of scientific domains, and most textbooks of linear algebra are written by mathematicians. However, this book is specifically intended to students and researchers of pattern information processing, analyzing signals such as images and exploring computer vision and computer graphics applications. The author himself is a researcher of this domain. Such pattern information processing deals with a large amount of data, which are represented by high-dimensional vectors and matrices. There, the role of linear algebra is not merely numerical computation of large-scale vectors and matrices. In fact, data processing is usually accompanied with "geometric interpretation." For example, we can think of one data set being "orthogonal" to another and define a "distance" between them or invoke geometric relationships such as "projecting" some data onto some space. Such geometric concepts not only help us mentally visualize abstract high-dimensional spaces in intuitive terms but also lead us to find what kind of processing is appropriate for what kind of goals. First, we take up the concept of "projection" of linear spaces and describe "spectral decomposition," "singular value decomposition," and "pseudoinverse" in terms of projection. As their applications, we discuss least-squares solutions of simultaneous linear equations and covariance matrices of probability distributions of vector random variables that are not necessarily positive definite. We also discuss fitting subspaces to point data and factorizing matrices in high dimensions in relation to motion image analysis. Finally, we introduce a computer vision application of reconstructing the 3D location of a point from three camera views to illustrate the role of linear algebra in dealing with data with noise. This book is expected to help students and researchers of pattern information processing deepen the geometric understanding of linear algebra.

International Encyclopedia of Education

Author:

Publisher: Elsevier

Published: 2009-04-17

Total Pages: 6964

ISBN-13: 0080448941

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The field of education has experienced extraordinary technological, societal, and institutional change in recent years, making it one of the most fascinating yet complex fields of study in social science. Unequalled in its combination of authoritative scholarship and comprehensive coverage, International Encyclopedia of Education, Third Edition succeeds two highly successful previous editions (1985, 1994) in aiming to encapsulate research in this vibrant field for the twenty-first century reader. Under development for five years, this work encompasses over 1,000 articles across 24 individual areas of coverage, and is expected to become the dominant resource in the field. Education is a multidisciplinary and international field drawing on a wide range of social sciences and humanities disciplines, and this new edition comprehensively matches this diversity. The diverse background and multidisciplinary subject coverage of the Editorial Board ensure a balanced and objective academic framework, with 1,500 contributors representing over 100 countries, capturing a complete portrait of this evolving field. A totally new work, revamped with a wholly new editorial board, structure and brand-new list of meta-sections and articles Developed by an international panel of editors and authors drawn from senior academia Web-enhanced with supplementary multimedia audio and video files, hotlinked to relevant references and sources for further study Incorporates ca. 1,350 articles, with timely coverage of such topics as technology and learning, demography and social change, globalization, and adult learning, to name a few Offers two content delivery options - print and online - the latter of which provides anytime, anywhere access for multiple users and superior search functionality via ScienceDirect, as well as multimedia content, including audio and video files

New Perspectives in Partial Least Squares and Related Methods

Author: Herve Abdi

Publisher: Springer Science & Business Media

Published: 2013-10-17

Total Pages: 351

ISBN-13: 1461482836

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New Perspectives in Partial Least Squares and Related Methods shares original, peer-reviewed research from presentations during the 2012 partial least squares methods meeting (PLS 2012). This was the 7th meeting in the series of PLS conferences and the first to take place in the USA. PLS is an abbreviation for Partial Least Squares and is also sometimes expanded as projection to latent structures. This is an approach for modeling relations between data matrices of different types of variables measured on the same set of objects. The twenty-two papers in this volume, which include three invited contributions from our keynote speakers, provide a comprehensive overview of the current state of the most advanced research related to PLS and related methods. Prominent scientists from around the world took part in PLS 2012 and their contributions covered the multiple dimensions of the partial least squares-based methods. These exciting theoretical developments ranged from partial least squares regression and correlation, component based path modeling to regularized regression and subspace visualization. In following the tradition of the six previous PLS meetings, these contributions also included a large variety of PLS approaches such as PLS metamodels, variable selection, sparse PLS regression, distance based PLS, significance vs. reliability, and non-linear PLS. Finally, these contributions applied PLS methods to data originating from the traditional econometric/economic data to genomics data, brain images, information systems, epidemiology, and chemical spectroscopy. Such a broad and comprehensive volume will also encourage new uses of PLS models in work by researchers and students in many fields.

Scalar, Vector, and Matrix Mathematics

Author: Dennis S. Bernstein

Publisher: Princeton University Press

Published: 2018-02-27

Total Pages: 1593

ISBN-13: 0691176531

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The essential reference book on matrices—now fully updated and expanded, with new material on scalar and vector mathematics Since its initial publication, this book has become the essential reference for users of matrices in all branches of engineering, science, and applied mathematics. In this revised and expanded edition, Dennis Bernstein combines extensive material on scalar and vector mathematics with the latest results in matrix theory to make this the most comprehensive, current, and easy-to-use book on the subject. Each chapter describes relevant theoretical background followed by specialized results. Hundreds of identities, inequalities, and facts are stated clearly and rigorously, with cross-references, citations to the literature, and helpful comments. Beginning with preliminaries on sets, logic, relations, and functions, this unique compendium covers all the major topics in matrix theory, such as transformations and decompositions, polynomial matrices, generalized inverses, and norms. Additional topics include graphs, groups, convex functions, polynomials, and linear systems. The book also features a wealth of new material on scalar inequalities, geometry, combinatorics, series, integrals, and more. Now more comprehensive than ever, Scalar, Vector, and Matrix Mathematics includes a detailed list of symbols, a summary of notation and conventions, an extensive bibliography and author index with page references, and an exhaustive subject index. Fully updated and expanded with new material on scalar and vector mathematics Covers the latest results in matrix theory Provides a list of symbols and a summary of conventions for easy and precise use Includes an extensive bibliography with back-referencing plus an author index

Matrix-Based Introduction to Multivariate Data Analysis

Author: Kohei Adachi

Publisher: Springer

Published: 2016-10-11

Total Pages: 298

ISBN-13: 9811023417

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This book enables readers who may not be familiar with matrices to understand a variety of multivariate analysis procedures in matrix forms. Another feature of the book is that it emphasizes what model underlies a procedure and what objective function is optimized for fitting the model to data. The author believes that the matrix-based learning of such models and objective functions is the fastest way to comprehend multivariate data analysis. The text is arranged so that readers can intuitively capture the purposes for which multivariate analysis procedures are utilized: plain explanations of the purposes with numerical examples precede mathematical descriptions in almost every chapter. This volume is appropriate for undergraduate students who already have studied introductory statistics. Graduate students and researchers who are not familiar with matrix-intensive formulations of multivariate data analysis will also find the book useful, as it is based on modern matrix formulations with a special emphasis on singular value decomposition among theorems in matrix algebra. The book begins with an explanation of fundamental matrix operations and the matrix expressions of elementary statistics, followed by the introduction of popular multivariate procedures with advancing levels of matrix algebra chapter by chapter. This organization of the book allows readers without knowledge of matrices to deepen their understanding of multivariate data analysis.

A Generalized Inverse-epsilon Algorithm for Constructing Intersection Projection Matrices, with Applications

Author: L. Duane Pyle

Publisher:

Published: 1966

Total Pages: 34

ISBN-13:

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Given k linear manifolds and corresponding perpendicular projection matrices, a closed formula is derived for the perpendicular projection matrix. The derivation uses results taken from the theory of generalized inverses together with an application of Wynn's Epsilon-Algorithm to a convergent sequence of matrices. A variant of this formula is then used in solving arbitrary complex linear systems by iteration and in computing generalized inverses; the latter application provides a solution to least squares linear regression problems. (Author).

Expository Moments for Pseudo Distributions

Author: Haruhiko Ogasawara

Publisher: Springer Nature

Published: 2023-01-01

Total Pages: 348

ISBN-13: 9811935254

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This book provides expository derivations for moments of a family of pseudo distributions, which is an extended family of distributions including the pseudo normal (PN) distributions recently proposed by the author. The PN includes the skew normal (SN) derived by A. Azzalini and the closed skew normal (CSN) obtained by A. Domínguez-Molina, G. González-Farías, and A. K. Gupta as special cases. It is known that the CSN includes the SN and other various distributions as special cases, which shows that the PN has a wider variety of distributions. The SN and CSN have symmetric and skewed asymmetric distributions. However, symmetric distributions are restricted to normal ones. On the other hand, symmetric distributions in the PN can be non-normal as well as normal. In this book, for the non-normal symmetric distributions, the term “kurtic normal (KN)” is used, where the coined word “kurtic” indicates “mesokurtic, leptokurtic, or platykurtic” used in statistics. The variety of the PN was made possible using stripe (tigerish) and sectional truncation in univariate and multivariate distributions, respectively. The proofs of the moments and associated results are not omitted and are often given in more than one method with their didactic explanations.