Regressor space outliers in ridge regression

A simple analytical expression is derived for leverage in ridge regression. Leverage is shown to be a monotonically decreasing function of the value of the ridge parameter. This reduction in leverage is greatest for those observations lying substantially in the direction of the minor principal axes. Thus, ridge estimation copes with outliers in regressor space by downweighting their influence. A brief illustration is provided.     

Vec and vech operators for matrices,with some uses in jacobians and multivariate statistics

The vec of a matrix X stacks columns of X one under another in a single column; the vech of a square matrix X does the same thing but starting each column at its diagonal element. The Jacobian of a one-to-one transformation X → Y is then ∣∣?(vecX)/?(vecY) ∣∣ when X and Y each have functionally independent elements; it is ∣∣ ?(vechX)/?(vechY) ∣∣ when X and Y are symmetric; and there is a general form for when X and Y are other patterned matrices. Kronecker product properties of vec(ABC) permit easy evaluation of this determinant in many cases. The vec and vech operators are also very convenient in developing results in multivariate statistics.     

非对称广义特征值问题重特征值的灵敏度分析

以标准特征值问题灵敏度分析的有关结论为基础,证明了单参数非对称广义特征值问题半单重特征值的可微性,给出了特征值导数的表达式和特征向量的级数展开式.以所得结论为基础,定义了广义特征值问题半单重特征值的灵敏度,给出了确定矩阵对中敏感元素的方法.     

关于黄金分割序列及其相伴矩阵的几个性质

本文讨论了黄金分割序列的几个性质及其以相伴矩阵An 为系数的方程组的不稳定性 ,并求出了An 的特征值     

Sensitivity analysis in principal component analysis:influence on the subspace spanned by principal components.

The problem of detecting influential observations in principalcomponent analysis was discussed by several authors. Radhakrishnan and kshirsagar ( 1981 ), Critchley ( 1985 ), jolliffe ( 1986 )among others discussed this topicby using the influence functions I(X;θ_s)and I(X;V_s)of eigenvalues and eigenvectors, which wwere derived under the assumption that the eigenvalues of interest were simple. In this paper we propose the influence functionsI(X;∑^q _s=1θ_sV_sV_s ^T)and I(x;∑^q _s=1V_sV_s ^t)(q<p;p:number of variables) to investigate the influence onthe subspace spanned by principal components. These influence functions are applicable not only to the case where the edigenvalues of interst are all simple but also to the case where there are some multiple eigenvalues among those of interest.     

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.     

Permutational tests for correlation matrices

Permutational tests are proposed for the hypotheses that two population correlation matrices have common eigenvectors, and that two population correlation matrices are equal. The only assumption made in these tests is that the distributional form is the same in the two populations; they should be useful as a prelude either to tests of mean differences in grouped standardised data or to principal component investigation of such data.The performance of the permutational tests is subjected to Monte Carlo investigation, and a comparison is made with the performance of the likelihood-ratio test for equality of covariance matrices applied to standardised data. Bootstrapping is considered as an alternative to permutation, but no particular advantages are found for it. The various tests are applied to several data sets.