Symmetrical Design of Experiment in Global Sensitivity Analysis Based on ANOVA High-dimensional Model Representation

This article reviews symmetrical global sensitivity analysis based on the analysis of variance of high-dimensional model representation. To overcome the computational difficulties and explore the use of symmetrical design of experiment (SDOE), two methods are presented. If the form of the objective function f is known, we use SDOE to estimate the symmetrical global sensitivity indices instead of Monte Carlo or quasi-Monte Carlo simulation. Otherwise, we use the observed values of the experiment to do symmetrical global sensitivity analysis. These methods are easy to implement and can reduce the computational cost. An example is given by symmetrical design of experiment.     

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.     

Orthogonal arrays for estimating global sensitivity indices of non-parametric models based on ANOVA high-dimensional model representation

Global sensitivity indices play important roles in global sensitivity analysis based on ANOVA high-dimensional model representation. However, few effective methods are available for the estimation of the indices when the objective function is a non-parametric model. In this paper, we explore the estimation of global sensitivity indices of non-parametric models. The main result (Theorem 2.1) shows that orthogonal arrays (OAs) are A-optimality designs for the estimation of _ΘM,${\Theta }_M,$ the definition of which can be seen in Section 1. Estimators of global sensitivity indices are proposed based on orthogonal arrays and proved to be accurate for small indices. The performance of the estimators is illustrated by a simulation study.     

A two-stage method for the estimation of global sensitivity indices of non-parametric models

This article aims to propose a method to effectively estimate global sensitivity indices under non-parametric models. The new method involves two stages. First, all the non-influential sensitivity indices are filtered out by an adjustive W-statistic test process with low cost, and then the remaining significant sensitivity indices are precisely estimated by an orthogonal array (OA) with large number of levels and low strength. The method avoids complicated prototype building and shows a much lower experimental cost. The performance of this method as well as comparisons with polynomial regression method, Gaussian Process (GP) method, and component selection and smoothing operator (COSSO) method are tested on three numerical models that are widely used in engineering and statistical areas. Finally, a real data example is analyzed.     

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.     

Reliability of global sensitivity indices

Uncertainty and sensitivity analysis is an essential ingredient of model development and applications. For many uncertainty and sensitivity analysis techniques, sensitivity indices are calculated based on a relatively large sample to measure the importance of parameters in their contributions to uncertainties in model outputs. To statistically compare their importance, it is necessary that uncertainty and sensitivity analysis techniques provide standard errors of estimated sensitivity indices. In this paper, a delta method is used to analytically approximate standard errors of estimated sensitivity indices for a popular sensitivity analysis method, the Fourier amplitude sensitivity test (FAST). Standard errors estimated based on the delta method were compared with those estimated based on 20 sample replicates. We found that the delta method can provide a good approximation for the standard errors of both first-order and higher-order sensitivity indices. Finally, based on the standard error approximation, we also proposed a method to determine a minimum sample size to achieve the desired estimation precision for a specified sensitivity index. The standard error estimation method presented in this paper can make the FAST analysis computationally much more efficient for complex models.     

Note on reduction of dimensionality for second-order response surface design model

To reduce the dimensionality of the second-order response surface design model, variance component indices under imposing and non imposing restrictions on the moment matrix toward the orthogonality are derived and presented and the same is illustrated with suitable examples in this article.     

Modified D-optimal design for logistic model

Asymptotic theory of using the Fisher information matrix may provide poor approximation to the exact variance matrix of maximum likelihood estimation in nonlinear models. This may be due to not obtaining an efficient D-optimal design. In this article, we propose a modified D-optimality criterion, using a more accurate information matrix, based on the Bhattacharyya matrix. The proposed information matrix and its properties are given for two parameters simple logistic model. It is shown that the resulted modified locally D-optimal design is more efficient than the previous one; particularly, for small sample size experiments.     

A sensitivity analysis of probabilistic sensitivity analysis in terms of the density function for the input variables

Probabilistic sensitivity analysis (SA) allows to incorporate background knowledge on the considered input variables more easily than many other existing SA techniques. Incorporation of such knowledge is performed by constructing a joint density function over the input domain. However, it rarely happens that available knowledge directly and uniquely translates into such a density function. A naturally arising question is then to what extent the choice of density function determines the values of the considered sensitivity measures. In this paper we perform simulation studies to address this question. Our empirical analysis suggests some guidelines, but also cautions to practitioners in the field of probabilistic SA.     

A nonparametric procedure for the analysis of balanced crossover designs

The authors propose nonparametric tests for the hypothesis of no direct treatment effects, as well as for the hypothesis of no carryover effects, for balanced crossover designs in which the number of treatments equals the number of periods p, where p ≥ 3. They suppose that the design consists of n replications of balanced crossover designs, each formed by m Latin squares of order p. Their tests are permutation tests which are based on the n vectors of least squares estimators of the parameters of interest obtained from the n replications of the experiment. They obtain both the exact and limiting distribution of the test statistics, and they show that the tests have, asymptotically, the same power as the F‐ratio test.     

Missing data estimators in the general linear model: an evaluation of simulated data as an experimental design

Previous simulations have reported second order missing data estimators to be superior to the more straightforward first order procedures such as mean value replacement. These simulations however were based on deterministic comparisonsbetween regression criteria even though simulated sampling is a random procedure. In this paper a simulation structured asan experimental design allows statistical testing of the various missing data estimators for the various regression criteria as well as different regression specifications. Our results indicate that although no missing data estimator is globally best many of the computationally simpler first order methods perform as well as the more expensive higher order estimators, contrary to some previous findings.     

A gradient approach to efficient design and analysis of multivariate EWMA control charts

Compared to the grid search approach to optimal design of control charts, the gradient-based approach is more computationally efficient as the gradient information indicates the direction to search the optimal design parameters. However, the optimal parameters of multivariate exponentially weighted moving average (MEWMA) control charts are often obtained by using grid search in the existing literature. Note that the average run length (ARL) performance of the MEWMA chart can be calculated based on a Markov chain model, making it feasible to estimate the ARL gradient from it. Motivated by this, this paper develops an ARL gradient-based approach for the optimal design and sensitivity analysis of MEWMA control charts. It is shown that the proposed method is able to provide a fast, accurate, and easy-to-implement algorithm for the design and analysis of MEWMA charts, as compared to the conventional design approach based on grid search.     

Bayesian inference for smooth transition autoregressive (STAR) model: A prior sensitivity analysis

The main aim of this paper is to perform sensitivity analysis to the specification of prior distributions in a Bayesian analysis setting of STAR models. To achieve this aim, the joint posterior distribution of model order, coefficient, and implicit parameters in the logistic STAR model is first being presented. The conditional posterior distributions are then shown, followed by the design of a posterior simulator using a combination of Metropolis-Hastings, Gibbs Sampler, RJMCMC, and Multiple Try Metropolis algorithms, respectively. Following this, simulation studies and a case study on the prior sensitivity for the implicit parameters are being detailed at the end.     

Comparison of designs for the three-fold nested random model

The quality of estimation of variance components depends on the design used as well as on the unknown values of the variance components. In this article, three designs are compared, namely, the balanced, staggered, and inverted nested designs for the three-fold nested random model. The comparison is based on the so-called quantile dispersion graphs using analysis of variance (ANOVA) and maximum likelihood (ML) estimates of the variance components. It is demonstrated that the staggered nested design gives more stable estimates of the variance component for the highest nesting factor than the balanced design. The reverse, however, is true in case of lower nested factors. A comparison between ANOVA and ML estimation of the variance components is also made using each of the aforementioned designs.     

Multivariate analysis for the two-period repeated measures crossover design with application to clinical trials

Multivariate analysis techniques are applied to the two-period repeated measures crossover design. The approach considered in this paper has the advantage over the univariate analysis approach proposed recently by Wallenstein and Fisher (1977) that the former does not require any specific structure on the variance-covariance matrix of the repeated measures factor. (It should be noted that sums and differences of observations over periods are used for all tests. Therefore, there are two matrices under consideration, one for sums and one for differences.) Tests of significance are derived using the Wilks? criterion, and the procedure is illustrated with a numerical example from the area of clinical trials.     

The Bayesian D-optimal design for simple beta regression model in two cases (both known and unknown nuisance parameter)

According to investigated topic in the context of optimal designs, various methods can be used to obtain optimal design, of which Bayesian method is one. In this paper, considering the model and the features of the information matrix, this method (Bayesian optimality criterion) has been used for obtaining optimal designs which due to the variation range of the model parameters, prior distributions such as Uniform, Normal and Exponential have been used and the results analysed.     

ANOVA decomposition of conditional Gaussian processes for sensitivity analysis with dependent inputs

Complex computer codes are widely used in science to model physical systems. Sensitivity analysis aims to measure the contributions of the inputs on the code output variability. An efficient tool to perform such analysis is the variance-based methods which have been recently investigated in the framework of dependent inputs. One of their issue is that they require a large number of runs for the complex simulators. To handle it, a Gaussian process (GP) regression model may be used to approximate the complex code. In this work, we propose to decompose a GP into a high-dimensional representation. This leads to the definition of a variance-based sensitivity measure well tailored for non-independent inputs. We give a methodology to estimate these indices and to quantify their uncertainty. Finally, the approach is illustrated on toy functions and on a river flood model.     

Robust analysis of variance for a randomized block design

In this paper we study the asymptotic theory of M-estimates and their associated tests for a one-factor experiment in a randomized block design. In this case one natural asymptotic theory corresponds to leaving the number of treatments fixed and letting the number of blocks tend to infinity. The classic asymptotic theory of M-estimates does not apply here, because the number of parameters and the number of observations are of the same order. In this paper we prove the consistency and asymptotic normality of the estimators of the treatment effects. It turns out that the asymptotic covariance matrix of the treatment effects estimators differs from the one derived from the classic theory of M-estimates for the linear model with a fixed number of parameters. We also study a test for treatment effects derived from M-estimates and we compare by Monte Carlo simulation the efficiency of this test with respect to the F-test, the Friedman test and the test based on aligned ranks.