Asymptotic distributions of functions of the eigenvalues of some random matrices for nonnormal populations

The authors investigated the asymptotic joint distributions of certain functions of the eigenvalues of the sample covariance matrix, correlation matrix, and canonical correlation matrix in nonnull situations when the population eigenvalues have multiplicities. These results are derived without assuming that the underlying distribution is multivariate normal. In obtaining these expressions, Edgeworth type expansions were used.     

A Local Parameterization of Orthogonal and Semi-Orthogonal Matrices with Applications

This article describes a local parameterization of orthogonal and semi-orthogonal matrices. The parameterization leads to a unified approach for obtaining the asymptotic joint distributions of estimators of singular-values and -vectors, and of eigen-values and -vectors. The singular- or eigen-values can have arbitrary multiplicities. The approach is illustrated on principal components analyzes, canonical correlation analysis, inter-battery factory analysis, and reduced-rank regression.     

两组变量间相关关系的统计分析方法

研究典型相关分析的原理、典型成分的计算方法及计算步骤.把两组变量X与Y转化为具有最大相关性的若干对典型成分,直到两组变量的相关性被分解.通过典型相关系数及其显著性检验.选择典型成分分析两组变量的相关性.实例表明只有第一个典型相关系数能通过显著性检验,而其它两个典型相关系数显著为零,故应选取第一对典型成分F1和G1做分析...     

Robust characterizations of k‐wise independence over product spaces and related testing results

A discrete distribution D over Σ₁ ×··· ×Σ_n is called (non‐uniform) k ‐wise independent if for any subset of k indices {i₁,…,i_k} and for any z₁∈Σ,…,z_k∈Σ, Pr_X～D[X···X = z₁···z_k] = Pr_X～D[X = z₁]···Pr_X～D[X = z_k]. We study the problem of testing (non‐uniform) k ‐wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from k ‐wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0,1}ⁿ. For the non‐uniform case, we give a new characterization of distributions being k ‐wise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. (STOC'07, pp. 496–505) on uniform k ‐wise independence over the Boolean cubes to non‐uniform k ‐wise independence over product spaces. Our results yield natural testing algorithms for k ‐wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

多元统计分析方法的理论研究及应用分析

多元统计分析在多指标实际问题的研究中起着重要的作用.以往文献中,一般采用某一种多元统计方法对一个多指标实际问题进行研究,但其结果存在一定的偏差性.采用四种典型的多元统计方法:主成分分析法、因子分析法、分层聚类和K-均值聚类方法对同一个多指标银行问题进行研究,并对四种方法的不同结果进行比较综合分析,这样可以避免单一方法的偏差性,以便得到的结果更合理、更具有说服力.     

Instability of statistical factor analysis

Factor analysis, a popular method for interpreting multivariate data, models the covariance among variables as being due to a small number (, ) of hidden variables. A factor analysis of can be thought of as an ordered or unordered collection, , of linearly independent lines in . Let be the collection of data sets for which is defined. The ``singularities' of are those data sets, , in the closure, , at which the limit, , does not exist. is unstable near its singularities.

Let be the direct sum of the lines in . determines a -plane bundle, , over a subset, , of . If 1$"> and is rich enough, ordered or, at least if or 3, unordered, must have a singularity at some data set in . The proofs are applications of algebraic topology. Examples are provided.

     

Reduced-rank regression for the multivariate linear model

The problem of estimating the regression coefficient matrix having known (reduced) rank for the multivariate linear model when both sets of variates are jointly stochastic is discussed. We show that this problem is related to the problem of deciding how many principal components or pairs of canonical variates to use in any practical situation. Under the assumption of joint normality of the two sets of variates, we give the asymptotic (large-sample) distributions of the various estimated reduced-rank regression coefficient matrices that are of interest. Approximate confidence bounds on the elements of these matrices are then suggested using either the appropriate asymptotic expressions or the jackknife technique.     

A limit theorem for the eigenvalues of product of two random matrices

The existence of limit spectral distribution of the product of two independent random matrices is proved when the number of variables tends to infinity. One of the above matrices is the Wishart matrix and the other is a symmetric nonnegative definite matrix.     

Tests for independence of two multivariate regression equations with different design matrices

In this paper, the authors considered various procedures for testing for the independence of two multivariate regression equations with different design matrices. Asymptotic null distributions as well as nonnull distributions under local alternatives of the test statistics associated with the above procedures are also derived.     

Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference

From the results of convergence by sampling in linear principal component analysis (of a random function in a separable Hilbert space), the limiting distribution is given for the principal values and the principal factors. These results can be explicitly written in the normal case. Some applications to statistical inference are investigated.     

一种函数型数据的聚类分析方法

分析了函数型数据主成分分析的原理。在此基础上,提出了一种函数型数据的聚类分析方法,以及在低维空间对原始高维数据进行直观表达的方法。给出了函数型数据的距离定义,并分析了这种距离的定义与欧氏距离的关系。提出函数型数据聚类分析的新方法:1)通过变换把离散数据转化为函数数据;2)进行函数型主成分分析;3)利用提取的前几个主成分构成低维空间,在该低维空间中,采用普通的聚类方法进行聚类分析。采用人体肢体多普勒超声血管造影的数据对所提出的方法的合理性进行验证。结果表明该方法可以有效地对函数型数据进行分类,分类结果与专家临床结论相符,因而有助于临床上对样本做客观判断。该方法不依赖专家的经验判断,且计算过程简便,易于计算机实现及临床应用。     

多元分析在分析物价变动主因中的应用

利用多元统计中的典型相关分析、因子分析方法，本文对１９５４年到１９９４年的七种物价指数及影响物价变动的九个因素的环比数列进行了全面分析，并试图找出物价上涨的主要因素，以此治理当前的通货膨胀。     

Properties of analytic Morrey spaces and applications

In this note, we aim to study analytic Morrey spaces . We first give the canonical factorization for . Then by applying p‐Carleson measure, we prove an atomic decomposition theorem of . As an application of the decomposition theorem, the interpolation problem of is solved. Finally, we show the boundedness and compactness of Toeplitz operators on .     

基于主成分分析的伊犁州直县(市)综合实力评价研究

主成分分析就是利用降维的思想,在损失很少信息的前提下把多个指标转化为几个综合指标的多元统计方法.通过对伊犁州直属10个县市的18项指标基于主成分分析进行了定量分析,得到了10个县市的综合实力排名,并指出了各县市经济发展不平衡的原因,给出了相应参考政策.     

Linear relative canonical analysis of Euclidean random variables,asymptotic study and some applications

We introduce the Linear Relative Canonical Analysis (LRCA) of Euclidean random variables. Then similar properties than for usual linear Canonical Analysis are obtained. Furthermore, we develop an asymptotic study of LRCA and apply the obtained results to tests for lack of relative linear association, dimensionality and invariance.     

天津市各区县经济发展水平评价

本文试用主成分分析方法 ,对天津市 18个区县的经济发展情况做出综合评价 ,分析了各区县在天津的经济地位。并提出了利用聚类分析的方法来评价各区县经济地位等级的设想     

2013年春夏季莱州湾海水环境要素的多元统计分析

根据2013年5月(春季)和8月(夏季)莱州湾海水环境要素的调查资料,通过主成分分析提取了3个主成分,累计表达原数据矩阵78.57%的信息,然后分别利用监测要素和主成分综合得分的层次聚类分析筛选出监控指标,并确定了水质污染状况及主要污染区域.结果表明,2013年莱州湾水质春季和夏季较差,导致水质状况恶化的主要要素是溶解无机氮,水质污染程度在时间和空间上存在较大差异,总体上春季海水污染程度高于夏季,莱州湾西侧和南侧处于严重污染或中度污染水平,因此常规监测中应重点关注化学需氧量、无机氮、叶绿素a和石油类这4个指标.     

On the algebraic relation between principal components corresponding to two different sets of weights

In principal component analysis, it is sometimes useful to give weights to observations. When two different sets of weights are attached to the observations of a data table, we derive the algebraic relation between the corresponding two principal component matrices.