Matrix And Linear Transformation Pdf
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Independent Component Analysis and Blind Signal Separation
Author: Carlos G. Puntonet
language: en
Publisher: Springer Science & Business Media
Release Date: 2004-09-17
In many situations found both in Nature and in human-built systems, a set of mixed signalsisobserved(frequentlyalsowithnoise),anditisofgreatscienti?candtech- logicalrelevanceto beableto isolateor separatethemso thattheinformationin each ofthesignalscanbeutilized. Blind source separation (BSS) research is one of the more interesting emerging ?eldsnowadaysinthe?eldofsignalprocessing.Itdealswiththealgorithmsthatallow therecoveryoftheoriginalsourcesfromasetofmixturesonly.Theadjective"blind" is applied becausethe purposeis to estimate the originalsourceswithoutany a priori knowledgeabouteitherthesourcesorthemixingsystem.Mostofthemodelsemployed in BSS assume the hypothesisabout the independenceof the original sources. Under this hypothesis,a BSS problemcan be consideredas a particularcase of independent componentanalysis(ICA),alineartransformationtechniquethat,startingfromam- tivariate representation of the data, minimizes the statistical dependence between the componentsoftherepresentation.Itcan beclaimed thatmostoftheadvancesin ICA havebeenmotivatedbythesearchforsolutionstotheBSSproblemand,theotherway around,advancesinICAhavebeenimmediatelyappliedtoBSS. ICA and BSS algorithms start from a mixture model, whose parameters are - timated from the observed mixtures. Separation is achieved by applying the inverse mixturemodelto theobservedsignals(separatingorunmixingmodel).Mixturem- els usually fall into three broad categories: instantaneous linear models, convolutive modelsandnonlinearmodels,the?rstonebeingthesimplestbut,in general,notnear realisticapplications.Thedevelopmentandtestofthealgorithmscanbeaccomplished throughsyntheticdataorwithreal-worlddata.Obviously,themostimportantaim(and mostdif?cult)istheseparationofreal-worldmixtures.BSSandICAhavestrongre- tionsalso,apartfromsignalprocessing,withother?eldssuchasstatisticsandarti?cial neuralnetworks. As long as we can ?nd a system that emits signals propagated through a mean, andthosesignalsarereceivedbyasetofsensorsandthereisaninterestinrecovering the originalsources,we have a potential?eld ofapplication forBSS and ICA. Inside thatwiderangeofapplicationswecan?nd,forinstance:noisereductionapplications, biomedicalapplications,audiosystems,telecommunications,andmanyothers. This volume comes out just 20 years after the ?rst contributionsin ICA and BSS 1 appeared . Thereinafter,the numberof research groupsworking in ICA and BSS has been constantly growing, so that nowadays we can estimate that far more than 100 groupsareresearchinginthese?elds. Asproofoftherecognitionamongthescienti?ccommunityofICAandBSSdev- opmentstherehavebeennumerousspecialsessionsandspecialissuesinseveralwell- 1 J.Herault, B.Ans,"Circuits neuronaux à synapses modi?ables: décodage de messages c- posites para apprentissage non supervise", C.R. de l'Académie des Sciences, vol. 299, no. III-13,pp.525-528,1984.
Matrix Algebra
This book presents the theory of matrix algebra for statistical applications, explores various types of matrices encountered in statistics, and covers numerical linear algebra. Matrix algebra is one of the most important areas of mathematics in data science and in statistical theory, and previous editions had essential updates and comprehensive coverage on critical topics in mathematics. This 3rd edition offers a self-contained description of relevant aspects of matrix algebra for applications in statistics. It begins with fundamental concepts of vectors and vector spaces; covers basic algebraic properties of matrices and analytic properties of vectors and matrices in multivariate calculus; and concludes with a discussion on operations on matrices, in solutions of linear systems and in eigenanalysis. It also includes discussions of the R software package, with numerous examples and exercises. Matrix Algebra considers various types of matrices encountered in statistics, such as projection matrices and positive definite matrices, and describes special properties of those matrices; as well as describing various applications of matrix theory in statistics, including linear models, multivariate analysis, and stochastic processes. It begins with a discussion of the basics of numerical computations and goes on to describe accurate and efficient algorithms for factoring matrices, how to solve linear systems of equations, and the extraction of eigenvalues and eigenvectors. It covers numerical linear algebra—one of the most important subjects in the field of statistical computing. The content includes greater emphases on R, and extensive coverage of statistical linear models. Matrix Algebra is ideal for graduate and advanced undergraduate students, or as a supplementary text for courses in linear models or multivariate statistics. It’s also ideal for use in a course in statistical computing, or as a supplementary text forvarious courses that emphasize computations.
Linear Transformation
This book introduces linear transformation and its key results, which have applications in engineering, physics, and various branches of mathematics. Linear transformation is a difficult subject for students. This concise text provides an in-depth overview of linear trans-formation. It provides multiple-choice questions, covers enough examples for the reader to gain a clear understanding, and includes exact methods with specific shortcuts to reach solutions for particular problems. Research scholars and students working in the fields of engineering, physics, and different branches of mathematics need to learn the concepts of linear transformation to solve their problems. This book will serve their need instead of having to use the more complex texts that contain more concepts then needed. The chapters mainly discuss the definition of linear transformation, properties of linear transformation, linear operators, composition of two or more linear transformations, kernels and range of linear transformation, inverse transformation, one-to-one and onto transformation, isomorphism, matrix linear transformation, and similarity of two matrices.