Pairwise Meta-Modeling of Multivariate Output Computer Models Using Nonseparable Covariance Function

Gaussian process (GP) is a popular method for emulating deterministic computer simulation models. Its natural extension to computer models with multivariate outputs employs a multivariate Gaussian process (MGP) framework. Nevertheless, with significant increase in the number of design points and the number of model parameters, building an MGP model is a very challenging task. Under a general MGP model framework with nonseparable covariance functions, we propose an efficient meta-modeling approach featuring a pairwise model building scheme. The proposed method has excellent scalability even for a large number of output levels. Some properties of the proposed method have been investigated and its performance has been demonstrated through several numerical examples. Supplementary materials for this article are available online.     

多元线性正态SRM参数估计的渐近协方差阵及其估计

针对多元线性正态结构关系模型（SRM）,本文给出了真值向量协方差阵的广义逆构造,并在三种常见误差协方差阵假定下应用标准5法分别给出了模型结构参数约束最小二乘估计的渐近协方差阵及其相合估计。     

Fast Computing for Distance Covariance

Distance covariance and distance correlation have been widely adopted in measuring dependence of a pair of random variables or random vectors. If the computation of distance covariance and distance correlation is implemented directly accordingly to its definition then its computational complexity is O(n²), which is a disadvantage compared to other faster methods. In this article we show that the computation of distance covariance and distance correlation of real-valued random variables can be implemented by an O(nlog?n) algorithm and this is comparable to other computationally efficient algorithms. The new formula we derive for an unbiased estimator for squared distance covariance turns out to be a U-statistic. This fact implies some nice asymptotic properties that were derived before via more complex methods. We apply the fast computing algorithm to some synthetic data. Our work will make distance correlation applicable to a much wider class of problems. A supplementary file to this article, available online, includes a Matlab and C-based software that realizes the proposed algorithm.     

A Fast Algorithm for the Minimum Covariance Determinant Estimator

The minimum covariance determinant (MCD) method of Rousseeuw is a highly robust estimator of multivariate location and scatter. Its objective is to find h observations (out of n) whose covariance matrix has the lowest determinant. Until now, applications of the MCD were hampered by the computation time of existing algorithms, which were limited to a few hundred objects in a few dimensions. We discuss two important applications of larger size, one about a production process at Philips with n = 677 objects and p = 9 variables, and a dataset from astronomy with n = 137,256 objects and p = 27 variables. To deal with such problems we have developed a new algorithm for the MCD, called FAST-MCD. The basic ideas are an inequality involving order statistics and determinants, and techniques which we call “selective iteration” and “nested extensions.” For small datasets, FAST-MCD typically finds the exact MCD, whereas for larger datasets it gives more accurate results than existing algorithms and is faster by orders of magnitude. Moreover, FASTMCD is able to detect an exact fit—that is, a hyperplane containing h or more observations. The new algorithm makes the MCD method available as a routine tool for analyzing multivariate data. We also propose the distance-distance plot (D-D plot), which displays MCD-based robust distances versus Mahalanobis distances, and illustrate it with some examples.     

A Bayesian Analysis of Covariance for Comparing Changes

Behaviormetrika - A Bayesian approach to analysis of covariance is proposed, in which it is assumed that the relevant parameters are exchangeable in the sense of De Finetti, i.e., they still have...     

采用协方差矩阵评定测量不确定度的方法

本文首先评述了测量不确定度估计的研究动向。接着,论述了一种用协方差矩阵来表示和处理测量结果不确定度的方法,详细推导并阐述了其不确定度传播的规律.这种方法以矩阵形式严密、清晰、统一地表达出来,其优点是:评定模型通用性好,应用范围广;概率分布可以是未知的;用矩阵运算,算法简洁,适用于计算机快速测量与计算。本文最后通过两个实例来说明该方法的应用。     

A New Measure of Misfit for Covariance Structure Models

Various fit indices exist in structural equation models. Most of these indices are related to the noncentrality parameter (NCP) of the chi-square distribution that the involved test statistic is implicitly assumed to follow. Existing literature suggests that few statistics can be well approximated by chi-square distributions. The meaning of the NCP is not clear when the behavior of the statistic cannot be described by a chi-square distribution. In this paper we define a new measure of model misfit (MMM) as the difference between the expected values of a statistic under the alternative and null hypotheses. This definition does not need to assume that the population covariance matrix is in the vicinity of the proposed model, nor does it need for the test statistic to follow any distribution of a known form. The MMM does not necessarily equal the discrepancy between the model and the population covariance matrix as has been assumed in existing literature. Bootstrap approaches to estimating the MMM and a related quantity are developed. An algorithm for obtaining bootstrap confidence intervals of the MMM is constructed. Examples with practical data sets contrast several measures of model misfit. The quantile-quantile plot is used to illustrate the unrealistic nature of chi-square distribution assumptions under either the null or an alternative hypothesis in practice.

     

Efficient Bootstrap Tests for the Goodness of Fit in Covariance Structure Analysis

Asymptotic chi-square tests, such as the normal theory likelihood ratio test, are often used to evaluate the goodness-of-fit of a covariance structure analysis model. Another approach is to use the bootstrap test, which is known to have the desired asymptotic level if model restrictions are taken into account in designing a resampling algorithm. The bootstrap test is. however, computationally very tedious and the problem of nonconvergence and improper solutions often arise in bootstrap resampling. In this paper, we propose a bootstrap test which is based on an approximation, by a quadratic form, to the minimum value of a discrepancy function calculated from each bootstrap sample. Hence, the proposed bootstrap test is efficient in the sense of the amount of computing needed and is free from the problem of nonconvergence and improper solutions with resampling. A Monte Carlo experiment is conducted to compare the performance of the proposed method with that of asymptotic chi-square tests for each combination of three distributions and four sample sizes.

     

A Note on the Use of Residuals for Examining the Assumptions of Covariance Analysis

A graphical method for examining the assumptions of covariance analysis is presented. A plot of the response residuals (y – ?) versus the covariate residuals (x – x) provides a useful method for studying the relationship between the covariate and the response. It also aids in the detection of abnormal observations.     

Uniform theory for the diffraction of evanescent plane waves

A uniform diffracted field is obtained in terms of Fresnel functions with complex argument by subtracting the unit step function from the Fresnel integral. The method is applied to the problem of diffraction of inhomogeneous plane waves by a perfectly conducting half-plane and wedge. The results are plotted numerically and compared with results reported in the literature.     

A Control Chart for the Multivariate Coefficient of Variation

Existing charts in the literature usually monitor either the mean or the variance of the process. However, in certain scenarios, the practitioner is not interested in the changes in the mean or the variance but is instead interested in monitoring the relative variability compared with the mean. This relative variability is called the coefficient of variation (CV). In the existing literature, none of the control charts that monitor the CV are applied for multivariate data. To fill this gap in research, this paper proposes a CV chart that monitors the CV for multivariate data. To the best of the authors' knowledge, this proposed chart is the first control chart for this purpose. The distributional properties of the sample CV for multivariate data and the procedures to implement the chart are presented in this paper. Formulae to compute the control limits, the average run length, the standard deviation of the run length, and the expected average run length for the case of unknown shift size are derived. From the numerical examples provided, the effects of the number of variables, the sample size, the shift size and the in‐control value of the CV are studied. Finally, we demonstrate the usefulness and applicability of the proposed chart on real data. Copyright © 2015 John Wiley & Sons, Ltd.     

Variation Charts for Multivariate Processes

Hotelling's T² is customarily used as the control chart for multivariate SPC analysis. This chart responds to changes in both the mean values and the covariance matrix of the responses. In this article, we propose the use of a chart that concentrates on changes in the covariance matrix. The use of this covariance chart in concert with the T² chart enables the user to better determine whether T² points out of control are due to changes in mean values or due to changes in the covariance matrix. Using this chart in conjunction with T² thus furnishes a suite of tools similar to the x-bar and standard deviation charts for univariate processes.     

Multivariate spatial models of excess crash frequency at area level: Case of Costa Rica

Recently, areal models of crash frequency have being used in the analysis of various area-wide factors affecting road crashes. On the other hand, disease mapping methods are commonly used in epidemiology to assess the relative risk of the population at different spatial units. A natural next step is to combine these two approaches to estimate the excess crash frequency at area level as a measure of absolute crash risk. Furthermore, multivariate spatial models of crash severity are explored in order to account for both frequency and severity of crashes and control for the spatial correlation frequently found in crash data.     

Multivariate calibration of non-replicated measurements for the factored noise model

The accuracy of a multivariate calibration (MVC) model for relating concentrations of multicomponent mixtures to their spectral measurements depends on effective handling of errors in the measured data. For the case when error variances vary along only one mode (either along mixtures or along wavelengths), a method to estimate the error variances simultaneously along with the spectral subspace was developed by Narasimhan and Shah (Control Engineering Practice, 16, (2008), 146–155). This method was exploited by Bhatt et al. (Chemom. Intell. Lab. Syst., 85, (2007), 70–81) to develop an iterative principal component regression (IPCR) MVC model, which was shown to be more accurate than models developed using PCR. In this work, the IPCR method is extended to deal with measurement errors whose variances vary along both modes, by using a factored noise model. As a first step, an iterative procedure is developed to estimate the error variance factors along with the spectral subspace, which is subsequently used in developing the regression model. Using simulated and experimental data, it is shown that the quality of the MVC model developed using the proposed method is better than that obtained using PCR, and is as good as the model obtained using Maximum Likelihood PCR, which requires knowledge of the error variances. For dealing with large data sets, a sub-optimal approach is also proposed for estimating the large number of error variances.