NPUA: A new approach for the analysis of computer experiments (original) (raw)
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Assessment of uncertainty in computer experiments from Universal to Bayesian Kriging
Applied Stochastic Models in Business and Industry, 2009
Kriging was first introduced in the field of geostatistics. Nowadays, it is widely used to model computer experiments. Since the results of deterministic computer experiments have no experimental variability, Kriging is appropriate in that it interpolates observations at data points. Moreover, Kriging quantifies prediction uncertainty, which plays a major role in many applications. Among practitioners we can distinguish those who use Universal Kriging where the parameters of the model are estimated and those who use Bayesian Kriging where model parameters are random variables. The aim of this article is to show that the prediction uncertainty has a correct interpretation only in the case of Bayesian Kriging. Different cases of prior distributions have been studied and it is shown that in one specific case, Bayesian Kriging supplies an interpretation as a conditional variance for the prediction variance provided by Universal Kriging. Finally, a simple petroleum engineering case study presents the importance of prior information in the Bayesian approach.
Applied Stochastic Models in Business and Industry, 2009
Our goal in the present work is to give an insight on some important questions to be asked when choosing a Kriging model for the analysis of numerical experiments. We are especially concerned about the cases where the size of the design of experiments is small relatively to the algebraic dimension of the inputs. We first fix the notations and recall some basic properties of Kriging. Then we expose two experimental studies on subjects that are often skipped in the field of computer simulation analysis: the lack of reliability of likelihood maximization with few data, and the consequences of a trend misspecification. We finally propose an example from a porous media application, with the introduction of an original Kriging method in which a non-linear additive model is used as external trend.
Modeling of computer experiments for uncertainty propagation and sensitivity analysis
Statistics and Computing, 2012
Nowadays, computer simulation has undoubtedly become one of the major tools to investigate and predict the behavior of complex systems in more and more disciplinary fields. A crucial question is how to account for uncertainties, tainting both input variables and (possibly) the computer model itself, in the results provided by the numerical code. That is particularly interesting in industrial practice, when results of computer simulations are eventually used to guide decisions, which can involve important financial, societal and safety stakes. For instance, in structural reliability problems, one is interested in assessing the probability for some state variables of a system (e.g. pressure, temperature, ...) to be inside, or outside, a given domain, which is associated to safe operating conditions. While computer simulations are faster and cheaper than physical experiments, computer models generate data (often large amounts) that must be analyzed and care is needed at the design stage to determine more cost-effective and informative simulation settings. Moreover, determining the most influent variables for the numerical code and the relevant parameter ranges within which to set up a computer experimental design is a critical and difficult step in the practical use of any formal statistical experimental planning, be it for screening or optimization purposes.
Variable Selection for Gaussian Process Models in Computer Experiments
Technometrics, 2006
In many situations, simulation of complex phenomena requires a large number of inputs and is computationally expensive. Identifying the inputs which most impact the system so that these factors can be further investigated can be a critical step in the scientific endeavor. In computer experiments, it is common to use a Gaussian spatial process to model the output of the simulator. In this article, we introduce a new, simple method for identifying active factors in computer screening experiments. The approach is Bayesian and only requires the generation of a new inert variable in the analysis; however, in the spirit of frequentist hypothesis testing, the posterior distribution of the inert factor is used as a reference distribution against which the importance of the experimental factors can be assessed. The methodology is demonstrated on an application in material science, a computer experiment from the literature, and simulated examples.
Risk Analysis, 2005
A Monte Carlo method is presented to study the effect of systematic and random errors on computer models mainly dealing with experimental data. It is a common assumption in this type of models (linear and nonlinear regression, and nonregression computer models) involving experimental measurements that the error sources are mainly random and independent with no constant background errors (systematic errors). However, from comparisons of different experimental data sources evidence is often found of significant bias or calibration errors. The uncertainty analysis approach presented in this work is based on the analysis of cumulative probability distributions for output variables of the models involved taking into account the effect of both types of errors. The probability distributions are obtained by performing Monte Carlo simulation coupled with appropriate definitions for the random and systematic errors. The main objectives are to detect the error source with stochastic dominance on the uncertainty propagation and the combined effect on output variables of the models. The results from the case studies analyzed show that the approach is able to distinguish which error type has a more significant effect on the performance of the model. Also, it was found that systematic or calibration errors, if present, cannot be neglected in uncertainty analysis of models dependent on experimental measurements such as chemical and physical properties. The approach can be used to facilitate decision making in fields related to safety factors selection, modeling, experimental data measurement, and experimental design.
Sequential design of computer experiments for the estimation of a probability of failure
2010
This paper deals with the problem of estimating the volume of the excursion set of a function f:mathbbRdtomathbbRf:\mathbb{R}^d \to \mathbb{R}f:mathbbRdtomathbbR above a given threshold, under a probability measure on mathbbRd\mathbb{R}^dmathbbRd that is assumed to be known. In the industrial world, this corresponds to the problem of estimating a probability of failure of a system. When only an expensive-to-simulate model of the system is available, the budget for simulations is usually severely limited and therefore classical Monte Carlo methods ought to be avoided. One of the main contributions of this article is to derive SUR (stepwise uncertainty reduction) strategies from a Bayesian-theoretic formulation of the problem of estimating a probability of failure. These sequential strategies use a Gaussian process model of fff and aim at performing evaluations of fff as efficiently as possible to infer the value of the probability of failure. We compare these strategies to other strategies also based on a Gaussian process model for estimating a probability of failure.
Principles of Bayesian Methods in Data Analysis
Key Engineering Materials, 2010
Bayesian statistics provides a powerful tool for the analysis of data. The methods are flexible enough to permit a realistic modelling of complex measurements. Prior information about the experiment, as well as knowledge from other sources can be used in a natural way. All relevant quantities concerning the measurement, as e. g. the expected values and their associated uncertainties are obtained from probability density functions. Bayesian data analysis strictly follows the rules of probability theory, thus ensuring that the procedure is free of inconsistencies and is in accordance with the Guide to the Expression of Uncertainty in Measurement (GUM).
Recent Advances in Computer Experiment Modeling
2014
OF THE DISSERTATION Recent Advances in Computer Experiment Modeling by YUFAN LIU Dissertation Director: Ying Hung This dissertation develops methodologies for analysis of computer experiments and its related theories. Computer experiments are becoming increasingly important in science and Gaussian process (GP) models are widely used in the analysis of computer experiments. This dissertation focuses on two settings where massive data are observed on irregular grids or quantiles of correlated data are of interests. In this dissertation, we first develop Latin Hypercube Design-based Block Bootstrap method. Then, we investigate quantiles of computer experiments in which correlated data are observed and propose penalized quantile regression with asymmetric Laplace process. The computational issue that hinders GP from broader application is recognized, especially for massive data observed on irregular grids. To overcome the computational issue, we introduce an efficient framework based on...