http://www.andreasaltelli.eu/file/repository/PUBLISHED_PAPER.pdf

Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index

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

Mathematical modelers from different disciplines and regulatory agencies worldwide agree on the importance of a careful sensitivity analysis (SA) of model-based inference. The most popular SA practice seen in the literature is that of 'one-factor-at-a-time' (OAT). This consists of analyzing the effect of varying one model input factor at a time while keeping all other fixed. While the shortcomings of OAT are known from the statistical literature, its widespread use among modelers raises concern on the quality of the associated sensitivity analyses. The present paper introduces a novel geometric proof of the inefficiency of OAT, with the purpose of providing the modeling community with a convincing and possibly definitive argument against OAT. Alternatives to OAT are indicated which are based on statistical theory, drawing from experimental design, regression analysis and sensitivity analysis proper.

/pdf/how-to-avoid-a-perfunctory-sensitivity-analysis-1xhzjnrms8.pdf

How to avoid a perfunctory sensitivity analysis

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

Global sensitivity analysis with variance-based measures suffers from several theoretical and practical limitations, since they focus only on the variance of the output and handle multivariate variables in a limited way. In this paper, we introduce a new class of sensitivity indices based on dependence measures which overcomes these insufficiencies. Our approach originates from the idea to compare the output distribution with its conditional counterpart when one of the input variables is fixed. We establish that this comparison yields previously proposed indices when it is performed with Csiszar f-divergences, as well as sensitivity indices which are well-known dependence measures between random variables. This leads us to investigate completely new sensitivity indices based on recent state-of-the-art dependence measures, such as distance correlation and the Hilbert–Schmidt independence criterion. We also emphasize the potential of feature selection techniques relying on such dependence measures as altern...

https://hal.archives-ouvertes.fr/hal-00903283/document

Global sensitivity analysis with dependence measures

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

Global sensitivity analysis of stochastic computer models with joint metamodels

Sensitivity indexes when the inputs of a model are not independent are derived from local polynomial techniques. Two original estimators based on local polynomial smoothers are proposed. Both have good theoretical properties, which are illustrated through analytical examples. Comparison with the Bayesian approach developed by Oakley and O’Hagan (2004) is also performed. The two proposed estimators are used to carry out a sensitivity analysis on two real case models with correlated parameters.

https://hal.archives-ouvertes.fr/hal-00266102/document

Local Polynomial Estimation for Sensitivity Analysis on Models With Correlated Inputs

Gaussian process modeling is one of the most popular approaches for building a metamodel in the case of expensive numerical simulators. Frequently, the code outputs correspond to physical quantities with a behavior which is known a priori: Chemical concentrations lie between 0 and 1, the output is increasing with respect to some parameter, etc. Several approaches have been proposed to deal with such information. In this paper, we introduce a new framework for incorporating constraints in Gaussian process modeling, including bound, monotonicity and convexity constraints. We also extend this framework to any type of linear constraint. This new methodology mainly relies on conditional expectations of the truncated multinormal distribution. We propose several approximations based on correlation-free assumptions, numerical integration tools and sampling techniques. From a practical point of view, we illustrate how accuracy of Gaussian process predictions can be enhanced with such constraint knowledge. We finally compare all approximate predictors on bound, monotonicity and convexity examples.

/pdf/gaussian-process-modeling-with-inequality-constraints-4ubj4mwkfp.pdf

Gaussian process modeling with inequality constraints

In this paper, we address the problem of efficient estimation of Sobol sensitivity indices. First, we focus on general functional integrals of conditional moments of the form &#x1d53c;(ψ(&#x1d53c;(ϕ(Y)|X))) where (X, Y) is a random vector with joint density f and ψ and ϕ are functions that are differentiable enough. In particular, we show that asymptotical efficient estimation of this functional boils down to the estimation of crossed quadratic functionals. An efficient estimate of first-order sensitivity indices is then derived as a special case. We investigate its properties on several analytical functions and illustrate its interest on a reservoir engineering case.

https://www.tandfonline.com/doi/pdf/10.1080/10485252.2013.784762

Sébastien Da Veiga

Papers

Global sensitivity analysis with dependence measures

Global sensitivity analysis of stochastic computer models with joint metamodels

Local Polynomial Estimation for Sensitivity Analysis on Models With Correlated Inputs

Gaussian process modeling with inequality constraints

Efficient estimation of sensitivity indices