有关行列式和迹的极值以及典则变量的最优性的补注

二次型极值已被一些文献诸如[1]和[3]讨论,一些相近的题目可在文献[2]中见到.文献[1]得到与二次型有关的行列式极值和典则变量的最优性质,本文对[1]给出补注.     

CANONICAL CORRELATION EQUATIONS AND THEIR EXTREMAL PROPERTIES

The canonical variables and canonical correlation coefficients satisfy a matrix equation which is called the canonical correlation equation. There are some different forms of the canonical correlation e-quations given in the literature. In this paper, we discuss four different forms of the canonical correlation equations. The purpose of this paper is to give extremal properties of the solutions of the canonical correlation equations. The results show that canonical variables maximize the determinant of the dispersion matrix of the transformed variables.     

Assessing Local Influence in Canonical Correlation Analysis

The first order local influence approach is adopted in this paper to assess the local influence of observations to canonical correlation coefficients, canonical vectors and several relevant test statistics in canonical correlation analysis. This approach can detect different aspects of influence due to different perturbation schemes. In this paper, we consider two different kinds, namely, the additive perturbation scheme and the case-weights perturbation scheme. It is found that, under the additive perturbation scheme, the influence analysis of any canonical correlation coefficient can be simplified to just observing two predicted residuals. To do the influence analysis for canonical vectors, a scale invariant norm is proposed. Furthermore, by choosing proper perturbation scales on different variables, we can compare the different influential effects of perturbations on different variables under the additive perturbation scheme. An example is presented to illustrate the effectiveness of the first order local influence approach.     

Nonnegative determinant of a rectangular matrix: Its definition and applications to multivariate analysis

It is well known that the determinant of a matrix can only be defined for a square matrix. In this paper, we propose a new definition of the determinant of a rectangular matrix and examine its properties. We apply these properties to squared canonical correlation coefficients, and to squared partial canonical correlation coefficients. The proposed definition of the determinant of a rectangular matrix allows an easy and straightforward decomposition of the likelihood ratio when given sets of variables are partitioned into row block matrices. The last section describes a general theorem on redundancies among variables measured in terms of the likelihood ratio of a partitioned matrix.     

Robust canonical correlations: A comparative study

Summary  Several approaches for robust canonical correlation analysis will be presented and discussed. A first method is based on the definition of canonical correlation analysis as looking for linear combinations of two sets of variables having maximal (robust) correlation. A second method is based on alternating robust regressions. These methods are discussed in detail and compared with the more traditional approach to robust canonical correlation via covariance matrix estimates. A simulation study compares the performance of the different estimators under several kinds of sampling schemes. Robustness is studied as well by breakdown plots.     

In Memoriam: Michael (Miki) Neumann (1946–2011)

Kernel canonical correlation analysis (KCCA) is a procedure for assessing the relationship between two sets of random variables when the classical method, canonical correlation analysis (CCA), fails because of the nonlinearity of the data. The KCCA method is mostly used in machine learning, especially for information retrieval and text mining. Because the data is often represented with non-negative numbers, we propose to incorporate the non-negativity restriction directly into the KCCA method. Similar restrictions have been studied in relation to the classical CCA and called restricted canonical correlation analysis (RCCA), so that we call the proposed method restricted kernel canonical correlation analysis (RKCCA). We also provide some possible approaches for solving the optimization problem to which our method translates. The motivation for introducing RKCCA is given in Section 2.     

A Test of Multivariate Independence Based on a Single Factor Model

A test of the independence of two sets of variables is developed to have high power against a special family of dependence. In this each set of variables has the structure of a single factor model and the dependence is solely via the correlation γ between the underlying latent variables. This is a model with only one nonzero canonical correlation. It is shown that a test based on the maximum likelihood estimate of γ is appreciably more powerful than that based on r₁, the largest sample canonical correlation. If, however, the model is used, not just as a family of alternatives but as the basis for interpretation, and if substantial cross-correlation is present then the procedure is essentially equivalent to the use of r₁.     

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.     

A new approach for technical efficiency estimation in multiple output production

The nonparametric technique of Data Envelopment Analysis (DEA) has been used to measure technical efficiency. This approach has proven useful because, unlike regression analyses, it allows multiple outputs and does not require a priori functional form specification. DEA does, however, require correct model specification; inclusion of inappropriate variables or omission of relevant variables leads to distortions. The purpose of this paper is to develop an alternative methodology based on canonical correlation to measure technical efficiency for multiple output production correspondences. Using simulated data, the new methodology is compared with DEA. The results indicate that the canonical regression approach outperforms DEA in most cases.     

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...     

Influence Diagnostics in the Common Canonical Variates Model

As a generalization of the canonical correlation analysis to k random vectors, the common canonical variates model was recently proposed based on the assumption that the canonical variates have the same coefficients in all k sets of variables, and is applicable to many cases. In this article, we apply the local influence method in this model to study the impact of minor perturbations of data. The method is non-standard because of the restrictions imposed on the coefficients. Besides investigating the joint local influence of the observations, we also obtain the elliptical norm of the empirical influence function as a special case of local influence diagnostics. Based on the proposed diagnostics, we find that the results of common canonical variates analysis for the female water striders data set is largely affected by omitting just one single observation.     

A test for additional information in canonical correlation analysis

Summary In canonical correlation analysis a hypothesis concerning the relevance of a subset of variables from each of the two given variable sets is formulated. The likelihood ratio statistic for the hypothesis and an asymptotic expansion for its null distribution are obtained. In discriminant analysis various alternative forms of a hypothesis concerning the relevance of a specified variable subset are also discussed.     

Time-Dependent Correlators of Local Spins of the One-Dimensional XY Heisenberg Chain

Time-dependent temperature correlators of the anisotropic Heisenberg XY chain are calculated by the integration techniques with respect to Grassmann variables. For a chain of length M, the correlators are represented as the determinants of 2M × 2M matrices. In the thermodynamic limit, the correlation functions are expressed in terms of the Fredholm determinants of linear integral operators with matrix kernels. Bibliography: 32 titles.     

Invariance and independence in multivariate distribution theory

Several general results are presented whereby various properties of independence or conditional independence between certain random variables may be deduced from the symmetries enjoyed by their joint distributions. These are applied to the distributions of sample correlation and canonical correlation coefficients when the underlying data-distribution has suitable orthogonal invariance. A typical result is that, for a random sample of observations on three independent normal variables, r₁₂, r₁₃, and r_23.1 are mutually independent.     

Regularized generalized canonical correlation analysis for multiblock or multigroup data analysis

This paper presents an overview of methods for the analysis of data structured in blocks of variables or in groups of individuals. More specifically, regularized generalized canonical correlation analysis (RGCCA), which is a unifying approach for multiblock data analysis, is extended to be also a unifying tool for multigroup data analysis. The versatility and usefulness of our approach is illustrated on two real datasets.     

Functional canonical analysis for square integrable stochastic processes

We study the extension of canonical correlation from pairs of random vectors to the case where a data sample consists of pairs of square integrable stochastic processes. Basic questions concerning the definition and existence of functional canonical correlation are addressed and sufficient criteria for the existence of functional canonical correlation are presented. Various properties of functional canonical analysis are discussed. We consider a canonical decomposition, in which the original processes are approximated by means of their canonical components.     

一类连续最优控制问题的有限维近似

1　引　　言本文考虑具有状态终端约束、控制受限的非线性连续最优控制问题min　h0(x(0))+∫T0f0(x(t),u(t))dt+g0(x(T))(1.1)s.t.　x(t)=f(x(t),u(t)),　　t∈[0,T](1.2)D(x(0))=0,(1.3)E(x(T))=0,(1.4)S(u(t))≤0,　　t∈[0,T](1.5)其中,h0:Rn→R,f0:Rn×Rm→R,f:Rn×Rm→Rn,g0:Rn→R,D:Rn→Rp,E:Rn→Rq,S:Rm→Rr均为二次连续可微函数.T为终端时间(固定),p,q≤n,x(t)∈W1,∞[0,T]n,u(t)∈L∞[0,T]m分别为状态函数和控制函数.U(t)={u:S(u(t))≤0}为紧凸集.问题(1.1)—(1.5)要求寻找最佳控制u(t)使得目标函数(1.1)达到极小.…     

On some properties of the determinants of tensors

We use the Hilbert?s Nullstellensatz (Hilbert?s Zero Point Theorem) to give a direct proof of the formula for the determinants of the products of tensors. By using this determinant formula and using tensor product to represent the transformations of the slices of tensors, we prove some basic properties of the determinants of tensors which are the generalizations of the corresponding properties of the determinants for matrices. We also study the determinants of tensors after two types of transposes. We use the permutational similarity of tensors to discuss the relation between weakly reducible tensors and the triangular block tensors, and give a canonical form of the weakly reducible tensors.     

Distribution of the canonical correlation matrix

Summary Generalized canonical correlation matrix is associated with canonical correlation analysis, multivariate analysis of variance, a large variety of statistical tests and regression problems. In this paper two methods of deriving the distribution are, given and the exact distribution is given in an elegant form. The techniques of derivation are applicable to all versions of the generalized canonical correlation matrices, nonnull distributions in generalized analysis of variance problems and also they give rise to a simpler derivation of the distribution, of the multiple correlation coefficient.     

The influence of sample extremes on the maximum of partial sums of independent random variables

Many papers exist dealing with the distribution of the maximum of partial sums of independent random variables and studying the relationship between the sums and the sample extremes (see [L. Takacs,Combinator Methods in the Theory of Stochastic Processes, Wiley, New York (1967)], [J. Calambos,The Asymptotic Theory of Extreme Order Statistics, Wiley, New York (1978)]). The present paper deals with the effect the sample extremes have on the distribution of the maximum of sums of independent random variables. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993