Multivariate statistical analysis in the real and complex domains
This book explores topics in multivariate statistical analysis, relevant in the real and complex domains. It utilizes simplified and unified notations to render the complex subject matter both accessible and enjoyable, drawing from clear exposition and numerous illustrative examples. The book featur...
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| Language | English |
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Springer
2022
Springer Nature Springer International Publishing AG |
| Edition | 1 |
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| Abstract | This book explores topics in multivariate statistical analysis, relevant in the real and complex domains. It utilizes simplified and unified notations to render the complex subject matter both accessible and enjoyable, drawing from clear exposition and numerous illustrative examples. The book features an in-depth treatment of theory with a fair balance of applied coverage, and a classroom lecture style so that the learning process feels organic. It also contains original results, with the goal of driving research conversations forward. This will be particularly useful for researchers working in machine learning, biomedical signal processing, and other fields that increasingly rely on complex random variables to model complex-valued data. It can also be used in advanced courses on multivariate analysis. Numerous exercises are included throughout. |
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| AbstractList | This book explores topics in multivariate statistical analysis, relevant in the real and complex domains. It utilizes simplified and unified notations to render the complex subject matter both accessible and enjoyable, drawing from clear exposition and numerous illustrative examples. The book features an in-depth treatment of theory with a fair balance of applied coverage, and a classroom lecture style so that the learning process feels organic. It also contains original results, with the goal of driving research conversations forward. This will be particularly useful for researchers working in machine learning, biomedical signal processing, and other fields that increasingly rely on complex random variables to model complex-valued data. It can also be used in advanced courses on multivariate analysis. Numerous exercises are included throughout. |
| Author | Haubold, Hans J Mathai, Arak M Provost, Serge B |
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| SubjectTerms | Applied mathematics classifications cluster complex domain factor analysis Gaussian distributions mathematical statistics Mathematics Mathematics and Science matrix-variate Multivariate analysis multivariate statistical analysis Physics Probability and statistics profile analyses Statistical physics type-1 distributions type-2 distributions Wishart distribution |
| TableOfContents | 5.5.3 Samples from a p-variate Gaussian population and the Wishart density -- 5.5a.3 Sample from a complex Gaussian population and the Wishart density -- 5.5.4 Some properties of the Wishart distribution, real case -- 5.5.5 The generalized variance -- 5.5.6 Inverse Wishart distribution -- 5.5.7 Marginal distributions of a Wishart matrix -- 5.5.8 Connections to geometrical probability problems -- 5.6 The Distribution of the Sample Correlation Coefficient -- 5.6.1 The special case ρ=0 -- 5.6.2 The multiple and partial correlation coefficients -- 5.6.3 Different derivations of ρ1.(2…p) -- 5.6.4 Distributional aspects of the sample multiple correlation coefficient -- 5.6.5 The partial correlation coefficient -- 5.7 Distributions of Products and Ratios of Matrix-variate Random Variables -- 5.7.1 The density of a product of real matrices -- 5.7.2 M-convolution and fractional integralof the second kind -- 5.7.3 A pathway extension of fractional integrals -- 5.7.4 The density of a ratio of real matrices -- 5.7.5 A pathway extension of first kind integrals, real matrix-variate case -- 5.7a Density of a Product and Integrals of the Second Kind -- 5.7a.1 Density of a product and fractional integral of the second kind, complex case -- 5.7a.2 Density of a ratio and fractional integrals of the first kind, complex case -- 5.8 Densities Involving Several Matrix-variate Random Variables, Real Case -- 5.8.1 The type-1 Dirichlet density, real scalar case -- 5.8.2 The type-2 Dirichlet density, real scalar case -- 5.8.3 Some properties of Dirichlet densities in the real scalar case -- 5.8.4 Some generalizations of the Dirichlet models -- 5.8.5 A pseudo Dirichlet model -- 5.8.6 The type-1 Dirichlet model in real matrix-variate case -- 5.8.7 The type-2 Dirichlet model in the real matrix-variate case -- 5.8.8 A pseudo Dirichlet model 4.6 Sampling from a Real Matrix-variate Gaussian Density -- 4.6.1 The distribution of the sample sum of products matrix, real case -- 4.6.2 Linear functions of sample vectors -- 4.6.3 The general real matrix-variate case -- 4.6a The General Complex Matrix-variate Case -- 4.7 The Singular Matrix-variate Gaussian Distribution -- References -- 5 Matrix-Variate Gamma and Beta Distributions -- 5.1 Introduction -- 5.1a The Complex Matrix-variate Gamma -- 5.2 The Real Matrix-variate Gamma Density -- 5.2.1 The mgf of the real matrix-variate gammadistribution -- 5.2a The Matrix-variate Gamma Function and Density,Complex Case -- 5.2a.1 The mgf of the complex matrix-variate gamma distribution -- 5.3 Matrix-variate Type-1 Beta and Type-2 Beta Densities,Real Case -- 5.3.1 Some properties of real matrix-variate type-1 and type-2 beta densities -- 5.3a Matrix-variate Type-1 and Type-2 Beta Densities, Complex Case -- 5.3.2 Explicit evaluation of type-1 matrix-variate beta integrals, real case -- 5.3a.1 Evaluation of matrix-variate type-1 beta integrals, complex case -- 5.3.3 General partitions, real case -- 5.3.4 Methods avoiding integration over the Stiefel manifold -- 5.3.5 Arbitrary moments of the determinants, real gamma and beta matrices -- 5.3a.2 Arbitrary moments of the determinants in the complex case -- 5.4 The Densities of Some General Structures -- 5.4.1 The G-function -- 5.4.2 Some special cases of the G-function -- 5.4.3 The H-function -- 5.4.4 Some special cases of the H-function -- 5.5, 5.5a The Wishart Density -- 5.5.1 Explicit evaluations of the matrix-variate gamma integral, real case -- 5.5a.1 Evaluation of matrix-variate gamma integrals in the complex case -- 5.5.2 Triangularization of the Wishart matrixin the real case -- 5.5a.2 Triangularization of the Wishart matrix in the complex domain 3.5a.2 Linear functions of the sample vectors in the complex domain -- 3.5.3 Maximum likelihood estimators of the p-variate real Gaussian distribution -- 3.5a.3 MLE's in the complex p-variate Gaussian case -- 3.5a.4 Matrix derivatives in the complex domain -- 3.5.4 Properties of maximum likelihood estimators -- 3.5.5 Some limiting properties in the p-variate case -- 3.6 Elliptically Contoured Distribution, Real Case -- 3.6.1 Some properties of elliptically contoureddistributions -- 3.6.2 The density of u=r2 -- 3.6.3 Mean value vector and covariance matrix -- 3.6.4 Marginal and conditional distributions -- 3.6.5 The characteristic function of an elliptically contoured distribution -- References -- 4 The Matrix-Variate Gaussian Distribution -- 4.1 Introduction -- 4.2 Real Matrix-variate and Multivariate Gaussian Distributions -- 4.2a The Matrix-variate Gaussian Density, Complex Case -- 4.2.1 Some properties of a real matrix-variate Gaussian density -- 4.2a.1 Some properties of a complex matrix-variate Gaussian density -- 4.2.2 Additional properties in the real and complex cases -- 4.2.3 Some special cases -- 4.3 Moment Generating Function and Characteristic Function, Real Case -- 4.3a Moment Generating and Characteristic Functions, Complex Case -- 4.3.1 Distribution of the exponent, real case -- 4.3a.1 Distribution of the exponent, complex case -- 4.3.2 Linear functions in the real case -- 4.3a.2 Linear functions in the complex case -- 4.3.3 Partitioning of the parameter matrix -- 4.3.4 Distributions of quadratic and bilinear forms -- 4.4 Marginal Densities in the Real Matrix-variate Gaussian Case -- 4.4a Marginal Densities in the Complex Matrix-variate Gaussian Case -- 4.5 Conditional Densities in the Real Matrix-variate Gaussian Case -- 4.5a Conditional Densities in the Matrix-variate Complex Gaussian Case -- 4.5.1 Re-examination of the case q=1 5.8a Dirichlet Models in the Complex Domain Intro -- Preface/Special features -- Contents -- List of Symbols -- 1 Mathematical Preliminaries -- 1.1 Introduction -- 1.2 Determinants -- 1.2.1 Inverses by row operations or elementary operations -- 1.3 Determinants of Partitioned Matrices -- 1.4 Eigenvalues and Eigenvectors -- 1.5 Definiteness of Matrices, Quadratic and Hermitian Forms -- 1.5.1 Singular value decomposition -- 1.6 Wedge Product of Differentials and Jacobians -- 1.7 Differential Operators -- 1.7.1 Some basic applications of the vector differential operator -- References -- 2 The Univariate Gaussian and Related Distributions -- 2.1 Introduction -- 2.1a The Complex Scalar Gaussian Variable -- 2.1.1 Linear functions of Gaussian variables in the real domain -- 2.1a.1 Linear functions in the complex domain -- 2.1.2 The chisquare distribution in the real domain -- 2.1a.2 The chisquare distribution in the complex domain -- 2.1.3 The type-2 beta and F distributions in the real domain -- 2.1a.3 The type-2 beta and F distributions in the complex domain -- 2.1.4 Power transformation of type-1 and type-2 beta random variables -- 2.1.5 Exponentiation of real scalar type-1 and type-2 beta variables -- 2.1.6 The Student-t distribution in the real domain -- 2.1a.4 The Student-t distribution in the complex domain -- 2.1.7 The Cauchy distribution in the real domain -- 2.2 Quadratic Forms, Chisquaredness and Independence in the Real Domain -- 2.2a Hermitian Forms, Chisquaredness and Independence in the Complex Domain -- 2.2.1 Extensions of the results in the real domain -- 2.2a.1 Extensions of the results in the complex domain -- 2.3 Simple Random Samples from Real Populations and Sampling Distributions -- 2.3a Simple Random Samples from a Complex Gaussian Population -- 2.3.1 Noncentral chisquare having n degrees of freedom in the real domain 2.3.1.1 Mean value and variance, real central and non-central chisquare -- 2.3a.1 Noncentral chisquare having n degrees of freedom in the complex domain -- 2.4 Distributions of Products and Ratios and Connection to Fractional Calculus -- 2.5 General Structures -- 2.5.1 Product of real scalar gamma variables -- 2.5.2 Product of real scalar type-1 beta variables -- 2.5.3 Product of real scalar type-2 beta variables -- 2.5.4 General products and ratios -- 2.5.5 The H-function -- 2.6 A Collection of Random Variables -- 2.6.1 Chebyshev's inequality -- 2.7 Parameter Estimation: Point Estimation -- 2.7.1 The method of moments and the method of maximum likelihood -- 2.7.2 Bayes' estimates -- 2.7.3 Interval estimation -- References -- 3 The Multivariate Gaussian and Related Distributions -- 3.1 Introduction -- 3.1a The Multivariate Gaussian Density in the Complex Domain -- 3.2 The Multivariate Normal or Gaussian Distribution, Real Case -- 3.2.1 The moment generating function in the real case -- 3.2a The Moment Generating Function in the Complex Case -- 3.2a.1 Moments from the moment generating function -- 3.2a.2 Linear functions -- 3.3 Marginal and Conditional Densities, Real Case -- 3.3a Conditional and Marginal Densities in the Complex Case -- 3.4 Chisquaredness and Independence of Quadratic Forms in the Real Case -- 3.4.1 Independence of quadratic forms -- 3.4a Chisquaredness and Independence in the ComplexGaussian Case -- 3.4a.1 Independence of Hermitian forms -- 3.5 Samples from a p-variate Real Gaussian Population -- 3.5a Simple Random Sample from a p-variate Complex Gaussian Population -- 3.5.1 Some simplifications of the sample matrix in the real Gaussian case -- 3.5.2 Linear functions of the sample vectors -- 3.5a.1 Some simplifications of the sample matrix in the complex Gaussian case |
| Title | Multivariate statistical analysis in the real and complex domains |
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