Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

An asymptotic expansion for large sample size n is derived by a partial differential equation method, up to and including the term of order n^?2, for the ₀F₀ function with two argument matrices which arise in the joint density function of the latent roots of the covariance matrix, when some of the population latent roots are multiple. Then we derive asymptotic expansions for the joint and marginal distributions of the sample roots in the case of one multiple root.     

Asymptotic expansions of the distributions of the latent roots and the latent vector of the Wishart and multivariate F matrices

Asymptotic expansions of the joint distributions of the latent roots of the Wishart matrix and multivariate F matrix are obtained for large degrees of freedom when the population latent roots have arbitrary multiplicity. Asymptotic expansions of the distributions of the latent vectors of the above matrices are also derived when the corresponding population root is simple. The effect of normalizations of the vector is examined.     

Asymptotic expansions for the distributions of some functions of the latent roots of matrices in three situations

In this paper we derive asymptotic expansions for the distributions of some functions of the latent roots of the matrices in three situations in multivariate normal theory, i.e., (i) principal component analysis, (ii) MANOVA model and (iii) canonical correlation analysis. These expansions are obtained by using a perturbation method. Confidence intervals for the functions of the corresponding population roots are also obtained.     

Bayesian Estimation of Discrete Multivariate Latent Structure Models With Structural Zeros

In multivariate categorical data, models based on conditional independence assumptions, such as latent class models, offer efficient estimation of complex dependencies. However, Bayesian versions of latent structure models for categorical data typically do not appropriately handle impossible combinations of variables, also known as structural zeros. Allowing nonzero probability for impossible combinations results in inaccurate estimates of joint and conditional probabilities, even for feasible combinations. We present an approach for estimating posterior distributions in Bayesian latent structure models with potentially many structural zeros. The basic idea is to treat the observed data as a truncated sample from an augmented dataset, thereby allowing us to exploit the conditional independence assumptions for computational expediency. As part of the approach, we develop an algorithm for collapsing a large set of structural zero combinations into a much smaller set of disjoint marginal conditions, which speeds up computation. We apply the approach to sample from a semiparametric version of the latent class model with structural zeros in the context of a key issue faced by national statistical agencies seeking to disseminate confidential data to the public: estimating the number of records in a sample that are unique in the population on a set of publicly available categorical variables. The latent class model offers remarkably accurate estimates of population uniqueness, even in the presence of a large number of structural zeros.     

Normalizing and variance stabilizing transformations of multivariate statistics under an elliptical population

Summary Normalizing and variance stabilizing transformations of a sample correlation, multiple correlation and canonical correlation coefficients are obtained under an elliptical population. It is shown that the Fisher'sz-transformation is efficient for these statistics. A normalizing transformation is also studied for a latent root of a sample covariance matrix in an elliptical sample.     

Multicollinearity

A statistical test for the existence of sufficient conditions for the singularity of a matrix is presented. Various approximations to the distribution of latent roots of determinantal equations and sums of latent roots are discussed. A model of observations afflicted with errors in the variables permits an approximate test for collinearity.     

Asymptotic distributions of the latent roots with multiple population roots in multiple discriminant analysis

Asymptotic expansions are derived for the confluent hypergeometric function₁ F ₁(a; c; R, S) with two argument matrices, which appears in the joint density function of the latent roots in multiple discriminant analysis, whenR is “large” and each of the latent roots ofR assumes the general multiplicity. Laplace's method and a partial differential equation method are utilized in the derivation.     

The conditioning of the maximum entropy covariance matrix and its inverse

The maximum entropy covariance matrix is positive definite even when the number of variables p exceeds the sample size n. However, the inverse of this matrix can have stability problems when p is close to n, although these problems tend to disappear as p increases beyond n. We analyze such problems using the variance of the latent roots in a particular metric as a condition number.     

A new confidence interval for all characteristic roots of a covariance matrix

Confidence intervals for all of the characteristic roots of a sample covariance matrix are derived. Using a perturbation expansion, we obtain a new confidence interval for these roots. Then, we propose another confidence interval based on the results of Monte Carlo simulations. Since it is based on simulations, this new confidence interval is both narrower and more accurate than others when the difference between the largest and smallest characteristic roots of the population covariance matrix is large.     

以代数方法探讨结构系统再设计问题

利用矩阵修改理论探讨结构系统再设计问题,以等惯性转换求解动态劲度矩阵的隐根,并导出将特征值定位的计算方法;继而在隐根为已知下探讨隐向量的特质及解法,并确认修改后结构的振型必须区分成驻留性与非驻留性自然频率等两种状况处理.     

Expressions for some hypergeometric functions of matrix argument with applications

Reasonably simple expressions are given for some hypergeometric functions when the size of the argument matrix or matrices is two. Applications of these expressions in connection with the distributions of the latent roots of a 2 × 2 Wishart matrix are also given.     

Matricial norms and the differences between the zeros of determinants with polynomial elements

In this paper, we give a method for finding upper bounds for the absolute values of the differences between two latent roots of a lambda-matrix, that is to say, for the differences between two zeros of the determinant of a lambda-matrix. We specialize for complex polynomials.     

On the exploration of linear latent effect for multivariate modeling

A data analysis method is proposed to derive a latent structure matrix from a sample covariance matrix. The matrix can be used to explore the linear latent effect between two sets of observed variables. Procedures with which to estimate a set of dependent variables from a set of explanatory variables by using latent structure matrix are also proposed. The proposed method can assist the researchers in improving the effectiveness of the SEM models by exploring the latent structure between two sets of variables. In addition, a structure residual matrix can also be derived as a by-product of the proposed method, with which researchers can conduct experimental procedures for variables combinations and selections to build various models for hypotheses testing. These capabilities of data analysis method can improve the effectiveness of traditional SEM methods in data property characterization and models hypotheses testing. Case studies are provided to demonstrate the procedure of deriving latent structure matrix step by step, and the latent structure estimation results are quite close to the results of PLS regression. A structure coefficient index is suggested to explore the relationships among various combinations of variables and their effects on the variance of the latent structure.     

Computation of the null distribution of the largest or smallest latent roots of a beta matrix

An algorithm is presented for the numerical evaluation of the null distribution of the largest latent root of a beta matrix, based on a finite series recently given by Khatri [6]. The same method can also be used for the distribution of the smallest latent root, and it can be easily adapted to find percentage points. The method is only useful if the size of the matrix and the size of the denominator sample are small, and in this sense it complements some of the large sample approximations.Although the calculation may be fairly lengthy, the algorithm itself is quite short, and a Fortran coding of it will be submitted to the Journal of the Royal Statistical Society, Series C (Applied Statistics) for publication.     

Construction and calculation of reproducing kernel determined by various linear differential operators

We presented a method to construct and calculate the reproducing kernel for the linear differential operator with constant coefficients and a single latent root; further, we gave the formula for calculation. Additionally, by studying the recurrence relation of the reproducing kernel with arithmetic latent roots, we found that the reproducing kernel with a multi-knots interpolation constraint can be concisely represented by one with an initial-value constraint.     

Asymptotic expansions of the distributions of estimators in canonical correlation analysis under nonnormality

Asymptotic expansions of the distributions of typical estimators in canonical correlation analysis under nonnormality are obtained. The expansions include the Edgeworth expansions up to order O(1/n) for the parameter estimators standardized by the population standard errors, and the corresponding expansion by Hall's method with variable transformation. The expansions for the Studentized estimators are also given using the Cornish-Fisher expansion and Hall's method. The parameter estimators are dealt with in the context of estimation for the covariance structure in canonical correlation analysis. The distributions of the associated statistics (the structure of the canonical variables, the scaled log likelihood ratio and Rozeboom's between-set correlation) are also expanded. The robustness of the normal-theory asymptotic variances of the sample canonical correlations and associated statistics are shown when a latent variable model holds. Simulations are performed to see the accuracy of the asymptotic results in finite samples.     

矩阵非中心Г－分布

本文定义了矩阵形式的非中心－分布。它将常见的一元Ｇａｍｍａ分布，非中心Ｘ２－分布，和多元分析中的中心则Ｗｉｓｈａｒｔ分布，非中心Ｆ－分布等纳入一个整体，进而推导了它的特征函数与特征根的分布密度函数，为一些检验奠定了基础．     

The algebraic theory of latent projectors in lambda matrices

Multivariable systems and controls are often formulated in terms of differential equations, which give rise to lambda matrices of the form A(λ) = A₀λⁿ + A₁λ^n-1 + ? + A_n. The inverses of regular lambda matrices can be represented by the latent projectors or matrix residues that have very specific properties. This paper describes the general theory of latent roots, latent vectors, and latent projectors and gives the relationships to eigenvalues, eigenvectors, and eigenprojectors of the companion form.     

Power analysis for the bootstrap likelihood ratio test for the number of classes in latent class models

Latent class (LC) analysis is used to construct empirical evidence on the existence of latent subgroups based on the associations among a set of observed discrete variables. One of the tests used to infer about the number of underlying subgroups is the bootstrap likelihood ratio test (BLRT). Although power analysis is rarely conducted for this test, it is important to identify, clarify, and specify the design issues that influence the statistical inference on the number of latent classes based on the BLRT. This paper proposes a computationally efficient ‘short-cut’ method to evaluate the power of the BLRT, as well as presents a procedure to determine a required sample size to attain a specific power level. Results of our numerical study showed that this short-cut method yields reliable estimates of the power of the BLRT. The numerical study also showed that the sample size required to achieve a specified power level depends on various factors of which the class separation plays a dominant role. In some situations, a sample size of 200 may be enough, while in others 2000 or more subjects are required to achieve the required power.     

Asymptotic expansions of the distributions of the latent roots in MANOVA and the canonical correlations

Asymptotic expansions are given for the density function of the normalized latent roots of S₁S₂^?1 for large n under the assumption of $\text{Ω = O(n)}$, where S₁ and S₂ are independent noncentral and central Wishart matrices having the $\text{W}{}_{\mathrm{p}}\text{(b, Σ; Ω)}$ and W_p(n, Σ) distributions, respectively. The expansions are obtained by using a perturbation method. Asymptotic expansions are also obtained for the density function of the normalized canonical correlations when some of the population canonical correlations are zero.