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 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.     

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 the Estimation of Global Sensitivity Indices Based on ANOVA High-Dimensional Model Representation

We present an efficient new approach to estimate global sensitivity indices based on ANOVA high-dimensional model representation. The method makes use of the properties of orthogonal arrays and extend the results of Wang et al. (2010 Wang , X. , Tang , Y. , Chen , X. , Zhang , Y. ( 2010 ). Design of experiment in global sensitivity analysis based on ANOVA high-dimensional model representation . Communications in Statistics—Simulation and Computation 39 : 1183 – 1195 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to a more general situation. A theorem is given to illustrate that both the non influential and significant sensitivity indices are well estimated. This new method offers higher sampling efficiency than those of Sobol and Saltelli. We test its performance on two different models, which are widely used in engineering and statistical areas.     

Estimation of Sobol's sensitivity indices under generalized linear models

We derive explicit formulas for Sobol's sensitivity indices (SSIs) under the generalized linear models (GLMs) with independent or multivariate normal inputs. We argue that the main-effect SSIs provide a powerful tool for variable selection under GLMs with identity links under polynomial regressions. We also show via examples that the SSI-based variable selection results are similar to the ones obtained by the random forest algorithm but without the computational burden of data permutation. Finally, applying our results to the problem of gene network discovery, we identify through the SSI analysis of a public microarray dataset several novel higher-order gene–gene interactions missed out by the more standard inference methods. The relevant functions for SSI analysis derived here under GLMs with identity, log, and logit links are implemented and made available in the R package Sobol sensitivity.     

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.     

Symmetrical design for symmetrical global sensitivity analysis of model output

Symmetrical global sensitivity analysis (SGSA) can help practitioners focusing on the symmetrical terms of inputs whose uncertainties have an impact on the model output, which allows reducing the complexity of the model. However, there remains the challenging problem of finding an efficient method to get symmetrical global sensitivity indices (SGSI) when the functional form of the symmetrical terms is unknown, including numerical and non-parametric situations. In this study, we propose a novel sampling plan, called symmetrical design, for SGSA. As a preliminary experiment for model feature extracting, such plan offers the virtue of run-size economy due to its closure respective to the given group. Using the design, we give estimation methods of SGSI as well as their asymptotic properties respectively for numerical model and non-parametrical model directly by the model outputs, and further propose a significance test for SGSI in non-parametric situation. A case study for a benchmark of GSA and a real data analysis show the effectiveness of the proposed design.     

A two-stage estimation for panel data models with grouped fixed effects

ABSTRACT

This paper considers panel data models with fixed effects which have grouped patterns with unknown group membership. A two-stage estimation (TSE) procedure is developed to improve the properties of the GFE estimators of common parameters when the time span is small. Firstly, the common parameters are estimated. Subsequently, the optimal group assignment and the estimators of group effects are obtained by the K-means algorithm. Monte Carlo results reveal that the TSE estimator has a much smaller bias than the GFE estimator when the values of difference between effects are moderately small or at high variance of the idiosyncratic error.     

A non-parametric estimator for the doubly periodic Poisson intensity function

In a series of papers, J. Garrido and Y. Lu have proposed and investigated a doubly periodic Poisson model, and then applied it to analyze hurricane data. The authors have suggested several parametric models for the underlying intensity function. In the present paper we construct and analyze a non-parametric estimator for the doubly periodic intensity function. Assuming that only a single realization of the process is available in a bounded window, we show that the estimator is consistent and asymptotically normal when the window expands indefinitely. In addition, we calculate the asymptotic bias and variance of the estimator, and in this way gain helpful information for optimizing the performance of the estimator.     

A noise estimation method for corrupted correlated data

Data acquisition, both in time and in spatial domains, in many cases yields observations with a measurement error. The identification of such a component, that masks the phenomenon under study (signal), must be carried out before the model of interest is specified. The objective of the paper is to propose an estimator for the parameters of an additive noise and compare it with existing methods by applications to both simulated and real data sets.     

Equivalence of maximum likelihood estimation and iterative two-stage estimation for seemingly unrelated regression models

The seemingly unrelated regression model is viewed in the context of repeated measures analysis. Regression parameters and the variance-covariance matrix of the seemingly unrelated regression model can be estimated by using two-stage Aitken estimation. The first stage is to obtain a consistent estimator of the variance-covariance matrix. The second stage uses this matrix to obtain the generalized least squares estimators of the regression parameters. The maximum likelihood (ML) estimators of the regression parameters can be obtained by performing the two-stage estimation iteratively. The iterative two-stage estimation procedure is shown to be equivalent to the EM algorithm (Dempster, Laird, and Rubin, 1977) proposed by Jennrich and Schluchter (1986) and Laird, Lange, and Stram (1987) for repeated measures data. The equivalence of the iterative two-stage estimator and the ML estimator has been previously demonstrated empirically in a Monte Carlo study by Kmenta and Gilbert (1968). It does not appear to be widely known that the two estimators are equivalent theoretically. This paper demonstrates this equivalence.     

Unbiasedness of two-stage estimation and prediction procedures for mixed linear models

The traditional method for estimating or predicting linear combinations of the fixed effects and realized values of the random effects in mixed linear models is first to estimate the variance components and then to proceed as if the estimated values of the variance components were the true values. This two-stage procedure gives unbiased estimators or predictors of the linear combinations provided the data vector is symmetrically distributed about its expected value and provided the variance component estimators are translation-invariant and are even functions of the data vector. The standard procedures for estimating the variance components yield even, translation-invariant estimators.     

A two-stage algorithm for computation of the exact distribution of sums of spacings

Based on the recursions in Huffer (1988 Huffer, F. (1988). Divided differences and the joint distribution of linear combinations of spacings. Journal of Applied Probability 25:346–354. [Google Scholar]) and Huffer and Lin (2001 Huffer, F. W., Lin, C. T. (2001). Computing the joint distribution of general linear combinations of spacings or exponential variates. Statistica Sinica 11:1141–1157. [Google Scholar]), we present a two-stage algorithm and two specialized methods for evaluating the probabilities involving linear combination of spacings of special forms. The two-stage algorithm combines the advantages of marking algorithm in Huffer and Lin (1997 Huffer, F. W., Lin, C. T. (1997). Computing the exact distribution of the extremes of sums of consecutive spacings. Computational Statistics and Data Analysis 26:117–132. [Google Scholar]) and general algorithm in Huffer and Lin (2001 Huffer, F. W., Lin, C. T. (2001). Computing the joint distribution of general linear combinations of spacings or exponential variates. Statistica Sinica 11:1141–1157. [Google Scholar]). The proposed methods can analytically derive the exact expressions for some specific problems, and efficiently handle problems such as the distribution of the circular scan statistic and multiple coverage probabilities.     

A simple method for the estimation of rational distributed lag models

In an earlier paper we suggested a method for the identification and estimation of linear transfer function models. The method was claimed to be especially suitable for polynomial transfer function models. In this paper we shall consider the case of rational transfer function models (distributed lag models) in more detail. A simple method for the estimation of the parameters of multiple input rational distributed lag models is suggested. The method is based on simple linear identities that the parameters always fulfill. The asymptotic distribution of the proposed estimator is derived. Two illustrative examples of the use of the new method are given.     

A generalized least squares estimation method for VARMA models

In this paper a new generalized least squares procedure for estimating VARMA models is proposed. This method differs from existing ones in explicitly considering the stochastic structure of the approximation error that arises when lagged innovations are replaced with lagged residuals obtained from a long VAR. Simulation results indicate that this method performs better than the Double Regression method and similar to Mauricio's (1995) exact maximum likelihood estimation procedure.     

Efficient estimation of auto-regression parameters and innovation distributions for semiparametric integer-valued AR(p) models

Summary.  Integer-valued auto-regressive (INAR) processes have been introduced to model non-negative integer-valued phenomena that evolve over time. The distribution of an INAR( p ) process is essentially described by two parameters: a vector of auto-regression coefficients and a probability distribution on the non-negative integers, called an immigration or innovation distribution. Traditionally, parametric models are considered where the innovation distribution is assumed to belong to a parametric family. The paper instead considers a more realistic semiparametric INAR( p ) model where there are essentially no restrictions on the innovation distribution. We provide an (semiparametrically) efficient estimator of both the auto-regression parameters and the innovation distribution.     

A two-stage design for multivariate estimation of proportions

ABSTRACT

In this article, the problem of estimation of the several proportions of two inter-dependent sensitive characteristics prevailing in a given population is considered. A new two-stage randomized response model has been proposed by extending Lee et al.’s (2013 Lee, C., Sedory, S., Singh, S (2013). Estimation at least seven measures of qualitative variables from a single sample using randomized response technique. Stat. Probab. Lett. 83(1):399–409.[Crossref], [Web of Science ®] , [Google Scholar]) model. The expressions for biases, variances, and mean square errors of different estimators have been provided. The relative efficiency comparisons of the proposed estimators has been made, numerically by considering the different practicable choices of the design parameters. It was observed that the proposed extended version performs efficiently than simple and crossed models of Lee et al. (2013 Lee, C., Sedory, S., Singh, S (2013). Estimation at least seven measures of qualitative variables from a single sample using randomized response technique. Stat. Probab. Lett. 83(1):399–409.[Crossref], [Web of Science ®] , [Google Scholar]).     

A general algorithm for non-parametric maximum likelihood estimator of stochastically ordered survival functions from case 2 interval-censored data

In this paper, we study an algorithm to compute the non-parametric maximum likelihood estimator of stochastically ordered survival functions from case 2 interval-censored data. The algorithm, simply denoted by SQP (sequential quadratic programming), re-parameterizes the likelihood function to make the order constraints as a set of linear constraints, approximates the log-likelihood function as a quadratic function, and updates the estimate by solving a quadratic programming. We particularly consider two stochastic orderings, simple and uniform orderings, although the algorithm can also be applied to many other stochastic orderings. We illustrate the algorithm using the breast cancer data reported in Finkelstein and Wolfe (1985 Finkelstein, D. M., and R. A. Wolfe. 1985. A semiparametric model for regression analysis of interval-censored failure time data. Biometrics 41:933–45. [Google Scholar]).