Sampling plans based on balanced incomplete block designs for evaluating the importance of computer model inputs

The importance of individual inputs of a computer model is sometimes assessed using indices that reflect the amount of output variation that can be attributed to random variation in each input. We review two such indices, and consider input sampling plans that support estimation of one of them, the variance of conditional expectation or VCE (Mckay, 1995. Los Alamos National Laboratory Report NUREG/CR-6311, LA-12915-MS). Sampling plans suggested by Sobol’, Saltelli, and McKay, are examined and compared to a new sampling plan based on balanced incomplete block designs. The new design offers better sampling efficiency for the VCE than those of Sobol’ and Saltelli, and supports unbiased estimation of the index associated with each input.     

Two group inspection-based control charts for dispersion matrix

In this article, we propose the two control charts, i.e. the ‘VMAX Group Runs’ (VMAX-GR) and ‘VMAX Modified Group Runs’ (VMAX-MGR) control charts based on the bivariate normal processes, for monitoring the covariance matrix. The proposed charts give the faster detection of a process change and have better diagnostic feature. It is verified that the VMAX-GR and the VMAX-MGR charts give a significant reduction in the out-of-control ‘Average Run Length’ (ARL) in the zero state, as well as in the steady state, as compared to the synthetic control chart based on the VMAX statistic and the generalized variance |S| chart.     

A screening approach for non-parametric global sensitivity analysis

Global sensitivity analysis (GSA) can help practitioners focusing on the inputs whose uncertainties have an impact on the model output, which allows reducing the complexity of the model. Screening, as the qualitative method of GSA, is to identify and exclude non- or less-influential input variables in high-dimensional models. However, for non-parametric problems, there remains the challenging problem of finding an efficient screening procedure, as one needs to properly handle the non-parametric high-order interactions among input variables and keep the size of the screening experiment economically feasible. In this study, we design a novel screening approach based on analysis of variance decomposition of the model. This approach combines the virtues of run-size economy and model independence. The core idea is to choose a low-level complete orthogonal array to derive the sensitivity estimates for all input factors and their interactions with low cost, and then develop a statistical process to screen out the non-influential ones without assuming the effect-sparsity of the model. Simulation studies show that the proposed approach performs well in various settings.     

A multivariate non-parametric kernel estimator for global sensitivity analysis

The fact of estimating how a model output is influenced by the variations of inputs has become an important problematic in reliability and sensitivity analysis. This article is interested in estimating sensitivity indices useful to quantify the contribution of inputs to the variance of model output. A multivariate mixed kernel estimator is investigated since, until now, discrete and continuous inputs have been separately considered in kernel estimation of sensitivity indices. To illustrate the differences between the influence of mixed, discrete, and continuous inputs, analytical expressions of Sobol sensitivity indices are expressed in these three cases for the Ishigami test function. Besides, the performance of the mixed kernel estimator is illustrated through simulations in which the Bayesian procedure is applied for bandwidth parameter choice. An application is also realized on a real example. Finally, the use of an appropriate kernel estimator according to the type of inputs is found to be influential on the accuracy of sensitivity indices estimates.     

Inverse regression for ridge recovery: a data-driven approach for parameter reduction in computer experiments

Parameter reduction can enable otherwise infeasible design and uncertainty studies with modern computational science models that contain several input parameters. In statistical regression, techniques for sufficient dimension reduction (SDR) use data to reduce the predictor dimension of a regression problem. A computational scientist hoping to use SDR for parameter reduction encounters a problem: a computer prediction is best represented by a deterministic function of the inputs, so data comprised of computer simulation queries fail to satisfy the SDR assumptions. To address this problem, we interpret SDR methods sliced inverse regression (SIR) and sliced average variance estimation (SAVE) as estimating the directions of a ridge function, which is a composition of a low-dimensional linear transformation with a nonlinear function. Within this interpretation, SIR and SAVE estimate matrices of integrals whose column spaces are contained in the ridge directions’ span; we analyze and numerically verify convergence of these column spaces as the number of computer model queries increases. Moreover, we show example functions that are not ridge functions but whose inverse conditional moment matrices are low-rank. Consequently, the computational scientist should beware when using SIR and SAVE for parameter reduction, since SIR and SAVE may mistakenly suggest that truly important directions are unimportant.

     

中国研究机构的科技资源利用效率评价研究

 本文基于1998-2005年中国1813家研究所构成的均衡面板数据,首次对中国研究机构的科技资源利用效率进行了评价研究。在采用FE和IVE方法控制双向固定效应和潜在的选择性偏误的基础上,估计了研究所投入要素对总收入和科技收入的贡献份额,由大到小依次为：非研究生科技人员、科研业务费、研究生科技人员、科研设备费,研究所的科技投入总体上不存在规模效应。采用GPS方法控制了投入要素的条件概率密度分布,估计了研究所投入要素对边际产出的动态影响,研究发现,随着投入要素规模的不断增大,非研究生科技人员的边际产出呈先升后降的趋势,其它三种投入要素则呈现总体上升趋势。本文的研究结论对研究所提高投入要素的利用效率具有重要的启示和指导意义。     

Importance sampling for Kolmogorov backward equations

The solution of the Kolmogorov backward equation is expressed as a functional integral by means of the Feynman–Kac formula. The expectation value is approximated as a mean over trajectories. In order to reduce the variance of the estimate, importance sampling is utilized. From the optimal importance density, a modified drift function is derived which is used to simulate optimal trajectories from an Itô equation. The method is applied to option pricing and the simulation of transition densities and likelihoods for diffusion processes. The results are compared to known exact solutions and results obtained by numerical integration of the path integral using Euler transition kernels. The importance sampling leads to strong variance reduction, even if the unknown solution appearing in the drift is replaced by known reference solutions. In models with low-dimensional state space, the numerical integration method is more efficient, but in higher dimensions it soon becomes infeasible, whereas the Monte Carlo method still works.     

Deletion residuals in the detection of heterogeneity of variances in linear regression

The heterogeneity of error variance often causes a huge interpretive problem in linear regression analysis. Before taking any remedial measures we first need to detect this problem. A large number of diagnostic plots are now available in the literature for detecting heteroscedasticity of error variances. Among them the ‘residuals’ and ‘fits’ (R–F) plot is very popular and commonly used. In the R–F plot residuals are plotted against the fitted responses, where both these components are obtained using the ordinary least squares (OLS) method. It is now evident that the OLS fits and residuals suffer a huge setback in the presence of unusual observations and hence the R–F plot may not exhibit the real scenario. The deletion residuals based on a data set free from all unusual cases should estimate the true errors in a better way than the OLS residuals. In this paper we propose ‘deletion residuals’ and the ‘deletion fits’ (DR–DF) plot for the detection of the heterogeneity of error variances in a linear regression model to get a more convincing and reliable graphical display. Examples show that this plot locates unusual observations more clearly than the R–F plot. The advantage of using deletion residuals in the detection of heteroscedasticity of error variance is investigated through Monte Carlo simulations under a variety of situations.     

A graphical method for evaluating mixture designs with respect to the slope

Good estimation of the slopes of the mixture response function may be important as well as estimation of mean mixture response. It is possible to evaluate and compare several mixture designs with respect to the slope. A graphical method is proposed that allows us to evaluate a given design's support for the fitted model in terms of slope variance. We can plot variances of slopes along Cox direction or axial direction according to existence of restriction of simplex region or not when comparing several different mixture designs.     

A Geometric Examination of Linear Model Assumptions

From a geometric perspective, linear model theory relies on a single assumption, that (‘corrected’) data vector directions are uniformly distributed in Euclidean space. We use this perspective to explore pictorially the effects of violations of the traditional assumptions (normality, independence and homogeneity of variance) on the Type I error rate. First, for several non‐normal distributions we draw geometric pictures and carry out simulations to show how the effects of non‐normality diminish with increased parent distribution symmetry and continuity, and increased sample size. Second, we explore the effects of dependencies on Type I error rate. Third, we use simulation and geometry to investigate the effect of heterogeneity of variance on Type I error rate. We conclude, in a fresh way, that independence and homogeneity of variance are more important assumptions than normality. The practical implication is that statisticians and authors of statistical computing packages need to pay more attention to the correctness of these assumptions than to normality.     

Objective Bayes Covariate‐Adjusted Sparse Graphical Model Selection

We present an objective Bayes method for covariance selection in Gaussian multivariate regression models having a sparse regression and covariance structure, the latter being Markov with respect to a directed acyclic graph (DAG). Our procedure can be easily complemented with a variable selection step, so that variable and graphical model selection can be performed jointly. In this way, we offer a solution to a problem of growing importance especially in the area of genetical genomics (eQTL analysis). The input of our method is a single default prior, essentially involving no subjective elicitation, while its output is a closed form marginal likelihood for every covariate‐adjusted DAG model, which is constant over each class of Markov equivalent DAGs; our procedure thus naturally encompasses covariate‐adjusted decomposable graphical models. In realistic experimental studies, our method is highly competitive, especially when the number of responses is large relative to the sample size.     

期权定价的蒙特卡罗模拟方差缩减技术研究

蒙特卡罗模拟的方差缩减技术作为模拟效率改进的重要途径,在金融衍生证券的定价分析中得到了广泛的应用和发展,特别是在控制变量、对偶变量、分层抽样、拉丁超立方抽样、矩匹配和重要性抽样技术方面。从方差缩减的效率来看,所有的蒙特卡罗模拟方差缩减技术都能显著地提高期权定价的模拟效率,其中基于最优漂移率的重要性抽样技术与沿着最优分层抽样方向进行的分层抽样技术的组合,要比普通的蒙特卡罗模拟具有极其明显的效率提高效果。     

Graphical evaluation of robust parameter designs based on extended scaled prediction variance and extended spherical average prediction variance

For any response surface design, there are locations in the design region where responses are estimated well and locations where estimation is relatively poor. Consequently, graphical evaluation—such as variance dispersion graphs and the fraction of design space—is used as an alternative to a single-valued criterion. Such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher. In this article, we propose the extended scaled prediction variance and extended spherical average prediction variance as prediction methods. We also illustrate how graphical methods can be employed to evaluate robust parameter designs.     

The robustness of response surface designs with errors in factor levels

ABSRTACT

Since errors in factor levels affect the traditional statistical properties of response surface designs, an important question to consider is robustness of design to errors. However, when the actual design could be observed in the experimental settings, its optimality and prediction are of interest. Various numerical and graphical methods are useful tools for understanding the behavior of the designs. The D- and G-efficiencies and the fraction of design space plot are adapted to assess second-order response surface designs where the predictor variables are disturbed by a random error. Our study shows that the D-efficiencies of the competing designs are considerably low for big variance of the error, while the G-efficiencies are quite good. Fraction of design space plots display the distribution of the scaled prediction variance through the design space with and without errors in factor levels. The robustness of experimental designs against factor errors is explored through comparative study. The construction and use of the D- and G-efficiencies and the fraction of design space plots are demonstrated with several examples of different designs with errors.     

Optimality of quasi-score in the multivariate mean–variance model with an application to the zero-inflated Poisson model with measurement errors

In a multivariate mean–variance model, the class of linear score (LS) estimators based on an unbiased linear estimating function is introduced. A special member of this class is the (extended) quasi-score (QS) estimator. It is ‘extended’ in the sense that it comprises the parameters describing the distribution of the regressor variables. It is shown that QS is (asymptotically) most efficient within the class of LS estimators. An application is the multivariate measurement error model, where the parameters describing the regressor distribution are nuisance parameters. A special case is the zero-inflated Poisson model with measurement errors, which can be treated within this framework.     

A graphical model selection tool for mixed models

ABSTRACT

Model selection can be defined as the task of estimating the performance of different models in order to choose the most parsimonious one, among a potentially very large set of candidate statistical models. We propose a graphical representation to be considered as an extension to the class of mixed models of the deviance plot proposed in the literature within the framework of classical and generalized linear models. This graphical representation allows, once a reduced number of models have been selected, to identify important covariates focusing only on the fixed effects component, assuming the random part properly specified. Nevertheless, we suggest also a standalone figure representing the residual random variance ratio: a cross-evaluation of the two graphical representations will allow to derive some conclusions on the random part specification of the model and a more accurate selection of the final model.     

Sensitivity analysis for ranked data

Sensitivity analysis is to study the influence of a small change in the input data on the output of the analysis. Han and Huh (1995) developed a quantification method for the ranked data. However, the question of stability in the analysis of ranked data has not been considered. Here, we propose a method of sensitivity analysis for ranked data. Our aim is to evaluate perturbations by using a graphical approach suggested by Han and Huh (1995). It extends the results obtained by Tanaka (1984) and Huh (1989) for the sensitivity analysis in Hayashi’s third method of quantification and those by Huh and Park (1990) for the principal component reduction of the case influence derivatives in regression. A numerical example is provided to explain how to conduct sensitivity analysis based on the proposed approach.     

Emulation and interpretation of high-dimensional climate model outputs

Running complex computer models can be expensive in computer time, while learning about the relationships between input and output variables can be difficult. An emulator is a fast approximation to a computationally expensive model that can be used as a surrogate for the model, to quantify uncertainty or to improve process understanding. Here, we examine emulators based on singular value decompositions (SVDs) and use them to emulate global climate and vegetation fields, examining how these fields are affected by changes in the Earth's orbit. The vegetation field may be emulated directly from the orbital variables, but an appealing alternative is to relate it to emulations of the climate fields, which involves high-dimensional input and output. The SVDs radically reduce the dimensionality of the input and output spaces and are shown to clarify the relationships between them. The method could potentially be useful for any complex process with correlated, high-dimensional inputs and/or outputs.     

Tests of marginal homogeneity and special cases

The analysis of two‐way contingency tables is common in clinical studies. In addition to summary counts and percentages, statistical tests or summary measures are often desired. If the data can be viewed as two categorical measurements on the same experimental unit (matched pair data) then a test of marginal homogeneity may be appropriate. The most common clinical example is the so called ‘shift table’ whereby a quantity is tested for change between two time points. The two principal marginal homogeneity tests are the Stuart Maxwell and Bhapkar tests. At present, SAS software does not compute either test directly (for tables with more than two categories) and a programmatic solution is required. Two examples of programmatic SAS code are found in the current literature. Although accurate in most instances, they fail to produce output for certain tables (‘special cases’). After summarizing the mathematics behind the two tests, a SAS macro is presented, which produces correct output for all tables. Finally, several examples are coded and presented with resultant output. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimation and variable selection in nonparametric heteroscedastic regression

The article considers a Gaussian model with the mean and the variance modeled flexibly as functions of the independent variables. The estimation is carried out using a Bayesian approach that allows the identification of significant variables in the variance function, as well as averaging over all possible models in both the mean and the variance functions. The computation is carried out by a simulation method that is carefully constructed to ensure that it converges quickly and produces iterates from the posterior distribution that have low correlation. Real and simulated examples demonstrate that the proposed method works well. The method in this paper is important because (a) it produces more realistic prediction intervals than nonparametric regression estimators that assume a constant variance; (b) variable selection identifies the variables in the variance function that are important; (c) variable selection and model averaging produce more efficient prediction intervals than those obtained by regular nonparametric regression.