some results on cononical correlations and measures of multivariate association

The canonical correlations and several orher measures of multivariate association between two sets of variables (x and y) are considered when the covariance matrices are singular. A useful inequality for the canonical correlations when new vari- ables are brought into x or y is obtained for both the nonsingular and singular cases. It is also shown that, under a simple condition, measures of multivariate association equal one if and only if there exists a linear relationship between the sets of variables.     

Influence functions related to eigenvalue problems which appear in multivariate analysis

Tanaka(1988) derived two influence functions related to an ordinary eigenvalue problem (A–λ_s I)v_s = 0 of a real symmetric matrix A and used them for sensitivity analysis in principal component analysis. One of these influence functions was used to develop sensitivity analysis in factor analysis (see e.g. Tanaka and Odaka, 1988a). The present paper derives some additional influence functions related to an ordinary eigenvalue problem and also several influence functions related to a generalized eigenvalue problem (A–θ_s A)u_s = 0, where A and B are real symmetric and real symmetric positive definite matrices, respectively. These influence functions are applicable not only to the case where the eigenvalues of interest are all simple but also to the case where there are some multiple eigenvalues among those of interest.     

四元数矩阵乘积迹的一个不等式的改进

证明了关于四元数矩阵乘积的奇异值的弱控制不等式，推广改进了有关文献的结果     

Categorical information and the singular linear model

C. R. Rao (1978) discusses estimation for the common linear model in the case that the variance matrix σ² Q has known singular form Q . In the more general context of inference, this model exhibits certain special features and illustrates how information concerning unknowns can separate into a categorical component and a statistical component. The categorical component establishes that certain parameters are known in value and thus are not part of the statistical inference.     

Asymptotic expansions for the distributions of statistics based on a correlation matrix

An asymptotic expansion is given for the distribution of the α-th largest latent root of a correlation matrix, when the observations are from a multivariate normal distribution. An asymptotic expansion for the distribution of a test statistic based on a correlation matrix, which is useful in dimensionality reduction in principal component analysis, is also given. These expansions hold when the corresponding latent root of the population correlation matrix is simple. The approach here is based on a perturbation method.     

关于亚正定矩阵的数值半径

给出了亚(半)正定矩阵数值半径计算的一个方法,该结果与文献[1]结果的使用范围互不包含,给出了其Gerschgorin型简单实用的下界估计式,以及其它有关结果。     

Examining rounding error in least absolute values regression computations

Two techniques for detecting inaccuracies in least absolute values (LAV) regression computations are presented and discussed. Examples of the use of the methods are given. The techniques are shown to apply to the more general case of M-estimation.     

Scoring under ordered constraints in contingency tables

Methods are developed for analyzing contingency tables which have ordered categoriesa priori. A correspondence analysis is extended to incorporate this a priori ordering. An exact represention is obtained when differences in the scores of the categories are known.     

Ginzburg—Landan模型动量空间重整化群的微扰图形方法

本文用动量空间重整化群方法处理G—L模型,累积展开用到微扰图形方法,使计算大为减化。得到G—L模型动量空间重整化群累积展开至二级的临界指数。     

Levine势双原子分子体系振转能级的微扰计算

本文从微观的角度,用量子力学的方法定量地讨论具有Levine势的双原子分子体系的振转能级,得到了意义明确的解析结果。