Distribution of the canonical correlation matrix

Mathai, A. M.

doi:10.1007/bf02480917

Cited by 11 publications

(3 citation statements)

References 5 publications

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“…it follows from (10.4.2) that the q × q matrix U U has a distribution that is equivalent to that of the p ×p matrix U U , as given in (10.4.4). The distribution of the sample canonical correlation matrix has been derived in Mathai (1981) for a Gaussian population under the assumption that Σ 12 = O.…”

Section: The Sampling Distribution Of the Canonical Correlation Matrixmentioning

confidence: 99%

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Chapter 10: Canonical Correlation Analysis

Mathai

Provost

Haubold

2022

Multivariate Statistical Analysis in the Real and Complex Domains

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We will keep utilizing the same notations in this chapter. More specifically, lower-case letters x, y, … will denote real scalar variables, whether mathematical or random. Capital letters X, Y, … will be used to denote real matrix-variate mathematical or random variables, whether square or rectangular matrices are involved. A tilde will be placed above letters such as $$\tilde {x},\tilde {y},\tilde {X},\tilde {Y}$$ x ~ , y ~ , X ~ , Y ~ to denote variables in the complex domain. Constant matrices will for instance be denoted by A, B, C. A tilde will not be used on constant matrices unless the point is to be stressed that the matrix is in the complex domain. The determinant of a square matrix A will be denoted by |A| or det(A) and, in the complex case, the absolute value or modulus of the determinant of A will be denoted as |det(A)|. When matrices are square, their order will be taken as p × p, unless specified otherwise. When A is a full rank matrix in the complex domain, then AA∗ is Hermitian positive definite where an asterisk designates the complex conjugate transpose of a matrix. Additionally, dX will indicate the wedge product of all the distinct differentials of the elements of the matrix X. Letting the p × q matrix X = (xij) where the xij’s are distinct real scalar variables, $$\mathrm {d}X=\wedge _{i=1}^p\wedge _{j=1}^q\mathrm {d}x_{ij}$$ d X = ∧ i = 1 p ∧ j = 1 q d x i j . For the complex matrix $$\tilde {X}=X_1+iX_2,\ i=\sqrt {(-1)}$$ X ~ = X 1 + i X 2 , i = ( − 1 ) , where X1 and X2 are real, $$\mathrm {d}\tilde {X}=\mathrm {d}X_1\wedge \mathrm {d}X_2$$ d X ~ = d X 1 ∧ d X 2 .

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Section: The Sampling Distribution Of the Canonical Correlation Matrixmentioning

confidence: 99%

“…Note that a function giving rise to a certain M-transform need not be unique. However, by making use of the Laplace transform and its inverse in the real matrix-variate case, Mathai (1981) 10.5. Show that the M-transform in (10.5.5) is available from the density specified in (10.5.10).…”

Section: The Sampling Distribution Of the Multiple Correlation Coeffi...mentioning

confidence: 99%

Chapter 10: Canonical Correlation Analysis

Mathai

Provost

Haubold

2022

Multivariate Statistical Analysis in the Real and Complex Domains

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“…Unbiasedness is proved by Narain (1950) and Anderson and Das Gupta (1964) have also shown that the power increases monotonically when the population canonical correlation coefficient p; increases. Mathai (1981) gives the joint p. d. f. of R by using the generalized Mellin transform for the matrix argument. Sugiura and Fujikoshi (1969) have expressed the exact moment of /\ 3 as a 2FI hypergeometric function.…”

Section: Invariant Polynomials and Related Testsmentioning

confidence: 99%

Invariant Polynomials and Related Tests

Hayakawa¹

1989

American Journal of Mathematical and Management Sciences

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Many test statistics for testing hypotheses on the parameters of one or more multinormal populations are associated with the latent roots of certain positive definite symmetric matrices. Thus the distributions of latent roots play vital roles in some testing problems. One such case is the joint distribution of the latent roots of a Wishart matrix. One representation of this joint density is in terms of zonal polynomials. Zonal polynomials are certain homogeneous polynomials in the latent roots of a symmetric matrix, which are invariant under orthogonal transformations. Such invariant polynomials and related tests are examined in this article. Some applications of these topics such as multivariate calibration, regression models, discriminant analysis, robustness studies and econometric theory are also considered.

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Statistical Modeling and Target Detection of PolSAR Images

Gao

2019

Characterization of SAR Clutter and Its Applications to Land and Ocean Observations

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Distribution of the canonical correlation matrix

Cited by 11 publications

References 5 publications

Chapter 10: Canonical Correlation Analysis

Chapter 10: Canonical Correlation Analysis

Invariant Polynomials and Related Tests

Statistical Modeling and Target Detection of PolSAR Images

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