THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation

We propose an independence criterion based on the eigen-spectrum of covariance operators in reproducing kernel Hilbert spaces (RKHSs), consisting of an empirical estimate of the Hilbert-Schmidt norm of the cross-covariance operator (we term this a Hilbert-Schmidt Independence Criterion, or HSIC). This approach has several advantages, compared with previous kernel-based independence criteria. First, the empirical estimate is simpler than any other kernel dependence test, and requires no user-defined regularisation. Second, there is a clearly defined population quantity which the empirical estimate approaches in the large sample limit, with exponential convergence guaranteed between the two: this ensures that independence tests based on HSIC do not suffer from slow learning rates. Finally, we show in the context of independent component analysis (ICA) that the performance of HSIC is competitive with that of previously published kernel-based criteria, and of other recently published ICA methods.

Measuring statistical dependence with Hilbert-Schmidt norms

This chapter makes a review, in a complete methodological framework, of various global sensitivity analysis methods of model output. Numerous statistical and probabilistic tools (regression, smoothing, tests, statistical learning, Monte Carlo, …) aim at determining the model input variables which mostly contribute to an interest quantity depending on model output. This quantity can be for instance the variance of an output variable. Three kinds of methods are distinguished: the screening (coarse sorting of the most influential inputs among a large number), the measures of importance (quantitative sensitivity indices) and the deep exploration of the model behaviour (measuring the effects of inputs on their all variation range). A progressive application methodology is illustrated on a scholar application. A synthesis is given to place every method according to several axes, mainly the cost in number of model evaluations, the model complexity and the nature of brought information.

A Review on Global Sensitivity Analysis Methods

Sensitivity analysis: A review of recent advances

https://hal.archives-ouvertes.fr/hal-00239494/document

Calculations of sobol indices for the gaussian process metamodel

Global sensitivity analysis of complex numerical models can be performed by calculating variance-based importance measures of the input variables, such as the Sobol indices. However, these techniques, requiring a large number of model evaluations, are often unacceptable for time expensive computer codes. A well known and widely used decision consists in replacing the computer code by a metamodel, predicting the model responses with a negligible computation time and rending straightforward the estimation of Sobol indices. In this paper, we discuss about the Gaussian process model which gives analytical expressions of Sobol indices. Two approaches are studied to compute the Sobol indices: the first based on the predictor of the Gaussian process model and the second based on the global stochastic process model. Comparisons between the two estimates, made on analytical examples, show the superiority of the second approach in terms of convergence and robustness. Moreover, the second approach allows to integrate the modeling error of the Gaussian process model by directly giving some confidence intervals on the Sobol indices. These techniques are finally applied to a real case of hydrogeological modeling.

Calculations of Sobol indices for the Gaussian process metamodel

An efficient methodology for modeling complex computer codes with Gaussian processes

Complex computer codes are often too time expensive to be directly used to perform uncertainty propagation studies, global sensitivity analysis or to solve optimization problems. A well known and widely used method to circumvent this inconvenience consists in replacing the complex computer code by a reduced model, called a metamodel, or a response surface that represents the computer code and requires acceptable calculation time. One particular class of metamodels is studied: the Gaussian process model that is characterized by its mean and covariance functions. A specific estimation procedure is developed to adjust a Gaussian process model in complex cases (non linear relations, highly dispersed or discontinuous output, high dimensional input, inadequate sampling designs, ...). The efficiency of this algorithm is compared to the efficiency of other existing algorithms on an analytical test case. The proposed methodology is also illustrated for the case of a complex hydrogeological computer code, simulating radionuclide transport in groundwater.

The global sensitivity analysis method used to quantify the influence of uncertain input variables on the variability in numerical model responses has already been applied to deterministic computer codes; deterministic means here that the same set of input variables always gives the same output value. This paper proposes a global sensitivity analysis methodology for stochastic computer codes, for which the result of each code run is itself random. The framework of the joint modeling of the mean and dispersion of heteroscedastic data is used. To deal with the complexity of computer experiment outputs, nonparametric joint models are discussed and a new Gaussian process-based joint model is proposed. The relevance of these models is analyzed based upon two case studies. Results show that the joint modeling approach yields accurate sensitivity index estimators even when heteroscedasticity is strong.

/pdf/global-sensitivity-analysis-of-stochastic-computer-models-46o2e9ppf8.pdf

Amandine Marrel

Papers

Calculations of sobol indices for the gaussian process metamodel

Calculations of Sobol indices for the Gaussian process metamodel

An efficient methodology for modeling complex computer codes with Gaussian processes

An efficient methodology for modeling complex computer codes with Gaussian processes

Global sensitivity analysis of stochastic computer models with joint metamodels