关于协方差、相关系数与相关性的关系

本文利用最小二乘法,导出了协方差与相关系数这两个概念的由来,说明了人们总是用相关系数而不用协方差来判断两个随机变量线性相关程度的原因,阐明了两个普通变量与两个随机变量存在线性关系时,它们的根本区别,并用直观图形展示了两个随机变量线性相关的概率意义。     

Generalized Variance of Multivariate Omega Functions and Duality

The covariance of probabilistic variables and the geometry of cones in deterministic optimization traditionally belong in distinct domains of study. This paper aims to show a relationship between the generalized variance of multidimensional joint omega functions and the duality of certain linear programs. Omega distributions are ubiquitous, polymorphic, and multifunctional but have been overlooked, partly due to a lack of closed form. However, the covariance/correlation matrix of joint omega functions can be stated. The geometry that links distributional covariance and generalized variance to the volume of dual cones is an exquisitely simple one.     

Robust mean variance optimization problem under Rényi divergence information

In this paper, we consider the robust mean variance optimization problem where the probability distribution of assets’ returns is multivariate normal and the uncertain mean and covariance are controlled by a constraint involving Rényi divergence. We present the closed-form solutions for the robust mean variance optimization problem and find that the choice of order parameter which is related to the Rényi divergence measure will not impact optimal portfolio strategy under the cases that the mean vector and the covariance matrix are uncertain, respectively. Moreover, we obtain the closed-form solution for the robust mean variance optimization problem under the case that the mean vector and the covariance matrix are both uncertain. We illustrate the efficiency of our results with an example.     

Normal correlation coefficient of non-normal variables using piece-wise linear approximation

The correlation coefficient of non-normal variables is expressed as a function of the correlation coefficient of normal variables using piece-wise linear approximation of each univariate transform of normal to anything, and the second order moments of a multiply truncated bivariate normal distribution. For the inverse problem, an algorithm iterates this analytic function in order to assign a normal correlation coefficient to two non-normal variables. The algorithm is applied for the generation of randomized bivariate samples with given correlation coefficient and marginal distributions and used in a randomization test for bivariate nonlinearity. The test correctly does not reject the null hypothesis of linear correlation if the nonlinearity is plausible and due to the sample transform alone.     

From stochastic dominance to mean-risk models: Semideviations as risk measures

Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. The former is based on an axiomatic model of risk-averse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible trade-off analysis, but cannot model all risk-averse preferences. In particular, if variance is used as a measure of risk, the resulting mean–variance (Markowitz) model is, in general, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the mean-risk model consistent with the second degree stochastic dominance, provided that the trade-off coefficient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we use a new tool, the Outcome–Risk (O–R) diagram, which appears to be particularly useful for comparing uncertain outcomes.     

非线性模型中方差和协方差分量的Bayes估计

本文采用Bayes方法从有逆gamma先验信息出发,得到了非张性模型中方差和协方差分量的估计,本文中的方差和协方差分量包含相关系数,而其他学者提出的线性模型中方差和协方差分量的Bayes估计只是本文的特殊情况.     

Eigenanalysis on a bivariate covariance kernel

Certain constructions of copulas can be interpreted as an eigendecomposition of a kernel. We study some properties of the eigenfunctions and their integrals of a covariance kernel related to a bivariate distribution. The covariance between functions of random variables in terms of the cumulative distribution function is used. Some bounds for the trace of the kernel and some inequalities for a continuous random variable concerning a function and its derivative are obtained. We also obtain relations to diagonal expansions and canonical correlation analysis and, as a by-product, series of constants for some particular distributions.     

On estimation in some reduced rank extended growth curve models

The general multivariate analysis of variance model has been extensively studied in the statistical literature and successfully applied in many different fields for analyzing longitudinal data. In this article, we consider the extension of this model having two sets of regressors constituting a growth curve portion and a multivariate analysis of variance portion, respectively. Nowadays, the data collected in empirical studies have relatively complex structures though often demanding a parsimonious modeling. This can be achieved for example through imposing rank constraints on the regression coefficient matrices. The reduced rank regression structure also provides a theoretical interpretation in terms of latent variables. We derive likelihood based estimators for the mean parameters and covariance matrix in this type of models. A numerical example is provided to illustrate the obtained results.     

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.     

一种新的弱相依系数和应用(英文)

本文在积分概率距离意义下提出了两个随机变量之间一种新的弱相依系数,并证明了此系数可获得协方差不等式和强大数定律,而且对于相关随机变量序列,我们还可以进一步研究矩不等式.     

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.     

Latent models for cross-covariance

We consider models for the covariance between two blocks of variables. Such models are often used in situations where latent variables are believed to present. In this paper we characterize exactly the set of distributions given by a class of models with one-dimensional latent variables. These models relate two blocks of observed variables, modeling only the cross-covariance matrix. We describe the relation of this model to the singular value decomposition of the cross-covariance matrix. We show that, although the model is underidentified, useful information may be extracted. We further consider an alternative parameterization in which one latent variable is associated with each block, and we extend the result to models with r-dimensional latent variables.     

The variance and covariance of fuzzy random variables and their applications

The concepts of the variance and covariance of fuzzy random variables and their properties are introduced. Examples show their computation and applications in statistical estimation of parameters when samples or prior information are fuzzy. As their further applications the correlation function and the criterions of mean-square calculus for fuzzy stochastic processes are established.     

Probabilistic inequalities and remarks

The authors use their recently proved integral inequality to obtain bounds for the covariance of two random variables

• 1.in a general setup and
• 2.for a class of special joint distributions.

The same inequality is also used to estimate the difference of the expectations of two random variables. Finally, the authors study the attainability of a related inequality.     

Generation of Over-Dispersed and Under-Dispersed Binomial Variates

Abstract

This article proposes an algorithm for generating over-dispersed and under-dispersed binomial variates with specified mean and variance. The over-dispersed/under-dispersed distributions are derived from correlated binary variables with an underlying continuous multivariate distribution. Different multivariate distributions or different correlation matrices result in different over-dispersed (or under-dispersed) distributions. The over-dispersed binomial distributions that are generated from three different correlation matrices of a multivariate normal are compared with the beta-binomial distribution for various mean and over-dispersion parameters by quantile-quantile (Q-Q) plots. The two distributions appear to be similar. The under-dispersed binomial distribution is simulated to model an example data set that exhibits under-dispersed binomial variation.     

Selection of variables in two-group discriminant analysis by error rate and Akaike's information criteria

This paper deals with two criteria for selection of variables for the discriminant analysis in the case of two multivariate normal populations with different means and a common covariance matrix. One is based on the estimated error rate of misclassification. The other uses Akaike's information criterion. The asymptotic distributions and error rate risks of the criteria are obtained. The result will prove that the two criteria are asymptotically equivalent in the sense of their asymptotic distributions and error rate risks being identical.     

Study and modelling of two apple quality attributes: the soluble solids content and the firmness

Recently, the basic dynamics of fruit characteristics have been modelled using a stochastic approach. The time evolution of apple quality attributes was represented by means of a system of differential equations in which the initial conditions and model parameters are both random. In this work, a complete study of two apple quality attributes, the soluble solids content and the firmness, is carried out. For each of these characteristics, the system of differential equations is linear and the state variables and the parameters are represented as random variables with their statistical properties (mean values, variances, covariances, joint probability density function) known at the initial time. The dynamic behaviour of these statistical properties is analysed. The variance propagation algorithm is used to obtain an analytical expression of the dynamic behaviour of the mean value, the variance, the covariance and the probability density function. A Monte Carlo method and the Latin hypercube method were developed to obtain a numerical expression of the dynamic behaviour of these statistical quantities and particularly to follow the time evolution of joint probability density function which represents one but the best mean to characterize random phenomena linked with fruit quality attributes.     

Generation of random bivariate normal deviates and computation of related integrals

A method which transforms two random variables having rectangular distributions into a pair of bivariate normal deviates with prescribed covariance matrix is described. The same transformation is used for integrating the bivariate normal distribution over areas which are the intersection of the domain outside an equiprobability ellipse and a sector determined by two lines through the point of gravity of the normal distribution.     

Scan Statistic Tail Probability Assessment Based on Process Covariance and Window Size

A scan statistic is examined for the purpose of testing the existence of a global peak in a random process with dependent variables of any distribution. The scan statistic tail probability is obtained based on the covariance of the moving sums process, thereby accounting for the spatial nature of the data as well as the size of the searching window. Exact formulas linking this covariance to the window size and the correlation coefficient are developed under general, common and auto covariance structures of the variables in the original process. The implementation and applicability of the formulas are demonstrated on multiple processes of t-statistics, treating also the case of unknown covariance. A sensitivity analysis provides further insight into the variant interaction of the tail probability with the influence parameters. An R code for the tail probability computation and the data analysis is offered within the supplementary material.     

Time aggregation effect on the correlation coefficient: added-systematically sampled framework

The aggregation of financial and economic time series occurs in a number of ways. Temporal aggregation or systematic sampling is the commonly used approach. In this paper, we investigate the time interval effect of multiple regression models in which the variables are additive or systematically sampled. The correlation coefficient changes with the selected time interval when one is additive and the other is systematically sampled. It is shown that the squared correlation coefficient decreases monotonically as the differencing interval increases, approaching zero in the limit. When two random variables are both added or systematically sampled, the correlation coefficient is invariant with time and equal to the one-period values. We find that the partial regression and correlation coefficients between two additive or systematically sampled variables approach one-period values as n increases. When one of the variables is systematically sampled, they will approach zero in the limit. The time interval for the association analyses between variables is not selected arbitrarily or the statistical results are likely affected.