Multivariate Analysis (eBook)

Kanti V. Mardia, John T. Kent, Charles C. Taylor (Autoren)

eBook Download: EPUB

2024 | 2. Auflage
1094 Seiten
John Wiley & Sons (Verlag)
978-1-118-89251-0 (ISBN)

Infos

Multivariate Analysis

Comprehensive Reference Work on Multivariate Analysis and its Applications

The first edition of this book, by Mardia, Kent and Bibby, has been used globally for over 40 years. This second edition brings many topics up to date, with a special emphasis on recent developments.

A wide range of material in multivariate analysis is covered, including the classical themes of multivariate normal theory, multivariate regression, inference, multidimensional scaling, factor analysis, cluster analysis and principal component analysis. The book also now covers modern developments such as graphical models, robust estimation, statistical learning, and high-dimensional methods. The book expertly blends theory and application, providing numerous worked examples and exercises at the end of each chapter. The reader is assumed to have a basic knowledge of mathematical statistics at an undergraduate level together with an elementary understanding of linear algebra. There are appendices which provide a background in matrix algebra, a summary of univariate statistics, a collection of statistical tables and a discussion of computational aspects. The work includes coverage of:

• Basic properties of random vectors, copulas, normal distribution theory, and estimation
• Hypothesis testing, multivariate regression, and analysis of variance
• Principal component analysis, factor analysis, and canonical correlation analysis
• Discriminant analysis, cluster analysis, and multidimensional scaling
• New advances and techniques, including supervised and unsupervised statistical learning, graphical models and regularization methods for high-dimensional data

Although primarily designed as a textbook for final year undergraduates and postgraduate students in mathematics and statistics, the book will also be of interest to research workers and applied scientists.

Kanti V. Mardia OBE is a Senior Research Professor in the Department of Statistics at the University of Leeds, Leverhulme Emeritus Fellow, and Visiting Professor in the Department of Statistics, University of Oxford.

John T. Kent and Charles C. Taylor are both Professors in the Department of Statistics, University of Leeds.

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Notation, Abbreviations, and Key Ideas

Matrices and Vectors

• Vectors are viewed as column vectors and are represented using bold lower case letters. Round brackets are generally used when a vector is expressed in terms of its elements. For example, in which the th element or component is denoted . The transpose of is denoted , so is a row vector.
• Matrices are written using bold upper case letters, e.g. and . The matrix may be written as in which is the element of the matrix in row and column . If has rows and columns, then the th row of , written as a column vector, is

  and the th column is written as

  Hence, can be expressed in various forms,

  We generally use square brackets when a matrix is expanded in terms of its elements.

  Operations on a matrix include

  1. – transpose:
  2. – determinant:
  3. – inverse:
  4. – generalized inverse:

  where for the final three operations, is assumed to be square, and for the inverse operation, is additionally assumed to be nonsingular. Different types of matrices are given in Tables A.1 and A.3. Table A.2 lists some further matrix operations.

Random Variables and Data

• In general, a random vector and a nonrandom vector are both indicated using a bold lower case letter, e.g. . Thus, the distinction between the two must be determined from the context. This convention is in contrast to the standard convention in statistics where upper case letters are used to denote random quantities, and lower case letters their observed values.
• The reason for our convention is that bold upper case letters are generally used for a data matrix , both random and fixed.
• In spite of the above convention, we very occasionally (e.g. parts of Chapters 2 and 10) use bold upper case letters for a random vector when it is important to distinguish between the random vector and a possible value .
• The phrase “high‐dimensional data” often implies , whereas the phrase “big data” often just indicates that or is large.

Parameters and Statistics

Elements of an data matrix are generally written , where indices are used to label the observations, and indices are used to label the variables.

If the rows of a data matrix are normally distributed with mean and covariance matrix , and , the following notation is used to distinguish various population and sample quantities:

Parameter

Sample

Mean vector

Covariance matrix

Unbiased covariance matrix

Concentration matrix

Correlation matrix

Distributions

The following notation is used for univariate and multivariate distributions. Appendix B summarizes the univariate distributions used in the Book.

cumulative distribution function/distribution function (d.f.)

probability density function (p.d.f.)

expectation

c.f.	characteristic function

d.f.	distribution function

Hotelling

multivariate normal distribution in ‐dimensions with mean (column vector of length ) and covariance matrix

variance–covariance matrix

correlation matrix

Wishart distribution

The terms variance matrix, covariance matrix, and variance–covariance matrix are synonymous.

Matrix Decompositions

• Any symmetric matrix can (by the spectral decomposition theorem) be written as where is a diagonal matrix of eigenvalues of (which are real‐valued), i.e. , and is an orthogonal matrix whose columns are standardized eigenvectors, i.e. and . See Theorem A.6.8.
• Using the above, we define the symmetric square root of a positive definite matrix by
• If is an matrix of rank , then by the singular value decomposition, it can be written as where and are column orthonormal matrices, and is a diagonal matrix with positive elements. See Theorem A.6.8.

Geometry

Table A.5 sets out the basic concepts in ‐dimensional geometry. In particular,

Length of a vector

Euclidean distance between and

Squared Mahalanobis distance – one of the most important distances in multivariate analysis, since it takes account of a covariance, i.e.

Table 14.6 gives a list of various distances.

Main Abbreviations and Commonly Used Notation

approximately equal to

(conditionally) independent of

is distributed as

the set of elements that are members of but not

Euclidean distance between and

transpose of matrix

determinant of matrix

inverse of matrix

‐inverse (generalized inverse)

column vector of 1s

column vector or matrix of 0s

between‐groups sum of squares and products (SSP) matrix

beta variable

normalizing constant for beta distribution (note nonitalic font to distinguish from the above)

BLUE	best linear unbiased estimate

covariance between and

chi‐squared distribution with degrees of freedom

upper α critical value of chi‐squared distribution with degrees of freedom

c.f.	characteristic function

partial derivative – multivariate examples in Appendix A.9

distance matrix

squared Mahalanobis distance

d.f.	distribution function

Kronecker delta

diagonal elements of a square matrix (as column vector) or diagonal matrix created from a vector

(see above)

expectation

distribution with degrees of freedom and

upper α critical value of distribution with degrees of freedom and

cumulative distribution function

probability density function

gamma function

GLS	generalized least squares

centering matrix

identity matrix

ICA	independent component analysis

i.i.d.

independent and identically distributed

Jacobian of transformation (see Table 2.1)

concentration matrix ()

likelihood

log likelihood

LDA	linear discriminant analysis

logarithm to the base (natural logarithm)

LRT	likelihood ratio test

MANOVA

multivariate analysis of variance

MDS	multidimensional scaling

ML	maximum likelihood

m.l.e.

maximum likelihood estimate

mean (population) vector

multivariate normal distribution for...

Produkt-Details

• Erscheint lt. Verlag: 10.6.2024
• Reihe/Serie: Wiley Series in Probability and Statistics
• Reihe/Serie: Wiley Series in Probability and Statistics
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Schlagworte: Angewandte Wahrscheinlichkeitsrechnung u. Statistik • Applied Probability & Statistics • basic multivariate normal theory • cluster analysis • Data Analysis • Datenanalyse • directional data • discriminant analysis • Factor Analysis • Multidimensional Scaling • multiple regression • Multivariate Analyse • multivariate analysis • Principal Component Analysis • Statistics • Statistik • structural equations models
• ISBN-10: 1-118-89251-8 / 1118892518
• ISBN-13: 978-1-118-89251-0 / 9781118892510