Global sensitivity analysis for stochastic simulators based on generalized lambda surrogate models
Creators: name: Zhu, Xujia
name: Sudret, Bruno
Publisher: name: ETH Zurich
Year: 2019
Subjects: Stochastic simulators
Stochastic emulators
Global sensitivity analysis
Lambda distributions
Polynomial chaos expansions

description: Global sensitivity analysis aims at quantifying the impact of input variables (taken separately or as a group) onto the variation of the response of a computational model. Classically, such models (also called simulators) are deterministic, in the sense that repeated runs provide the same output quantity of interest. In contrast, stochastic simulators return different results when run twice with the same input values due to additional sources of stochasticity in the code itself. In other words, the output of a stochastic simulator is a random variable for a given vector of input parameters. 
Many sensitivity measures, such as the Sobol’ indices and Borgonovo indices [1], have been developed in the context of deterministic simulators. They can be directly extended to stochastic simulators [2,3], despite the additional randomness of the latter. The calculation of such measures can be carried out through Monte Carlo simulation, which would require many model evaluations though. However, high-fidelity models are often time-consuming: a single model run may require hours or even days. In consequence, direct application of Monte Carlo simulations to calculate sensitivity measures becomes intractable. 
To alleviate the computational burden, surrogate models are constructed so as to mimic the original numerical model at a smaller computational cost though. For deterministic simulators, surrogate models have been successfully developed over the last decade, e.g. polynomial chaos expansions [4]. However, the question of appropriate surrogate modelling for stochastic simulators arose only recently in engineering. 
In this study, we propose to use generalized lambda distributions to flexibly approximate the response of a stochastic simulator. Under this setting, the parameters of the generalized lambda distribution become deterministic functions of the input variables. In this contribution we use sparse polynomial chaos expansions to represent the latter. To construct such a sparse generalized lambda model, we develop an algorithm that combines feasible generalized least-squares with stepwise regression. This method does not require repeated model evaluations for the same input parameters to account for the random nature of the output, and thus it reduces the total number of model runs drastically. 
Once the stochastic emulator is constructed, one can easily evaluate the conditional mean and variance, which is needed for the Sobol’ indices calculation. Because the generalized lambda distribution parametrizes the output quantile function, the surrogate model is expressed as a deterministic function of the input variables and a latent uniform random variable that represents the randomness of the output. As a result, instead of calculating the Sobol’ indices for each input variable through sampling, we can derive analytically the Sobol’ indices by some suitable post-processing. Moreover, the generalized lambda model provides the conditional distribution of the output given any input parameters. Therefore, distribution-based sensitivity measures, such as Borgonovo indices, can also be calculated straightforwardly.

[1] E. Borgonovo, A new uncertainty importance measure. Reliab. Eng. Sys. Safety, 92:771-784, 2007.
[2] A. Marrel, B. Iooss, S. Da Veiga, and M. Ribatet, Global sensitivity analysis of stochastic computer models with joint metamodels, Stat. Comput., 22:833-847, 2012.
[3] M. N. Jimenez, O. P. Le Maître, and O. M. Knio, Nonintrusive polynomial chaos expansions for sensitivity analysis in stochastic differential equations, 5:378-402, 2017
[4] B. Sudret, Global sensitivity analysis using polynomial chaos expansions, Reliab. Eng. Sys. Safety, 93:964-979, 2008