Surrogating stochastic simulators using Karhunen-Loève expansion, sparse PCE and advanced statistical modelling

Abstract

The classical uncertainty quantification approach models all uncertainty about a physical process in the form of input uncertainty to a deterministic computational model. However, this is not always possible: sometimes part of the uncertainty such as high-dimensional environmental variables cannot be easily modelled (e.g., earthquakes, wind fields), or there is intrinsic randomness in the model (e.g., epidemiological SIR models). Then the model is a so-called stochastic simulator: even when holding all input parameters at a fixed value, the model response is still a random variable. A stochastic simulator can also be seen as a random field, where the input space acts as its index set. To simulate random fields, a widely used method is Karhunen-Loève expansion (KLE), which represents the random field as an infinite series involving orthonormal deterministic basis functions and a countable number of uncorrelated random variables. However, for inferring a random field from a small set of model evaluations, two challenges have often limited the applicability of KLE [1,2]: the covariance function, which is needed to compute the KLE, is usually not known; and the joint distribution of KL random variables is in general complicated, non-Gaussian and possibly highly dependent, and therefore difficult to model. Our approach addresses these challenges. Building on the success of sparse polynomial chaos expansions (PCE) as surrogate models for deterministic engineering models, we propose to use them to approximate trajectories from the stochastic simulator of interest, which results in a continuous covariance function. After computing the KLE and the KL random variables associated with the trajectories, we infer a parametric form of their joint distribution by using state-of-the-art probabilistic modelling techniques such as vine copulas [3] and generalized lambda distributions [4]. We demonstrate that our approach results in a stochastic emulator with accurate marginals and covariance function, which furthermore can be sampled to obtain new realizations. [1] Poirion, F., & Zentner, I. (2014). Stochastic model construction of observed random phenomena. Probabilistic Engineering Mechanics, 36, 63-71. [2] Azzi, S., Huang, Y., Sudret, B., & Wiart, J. (2019). Surrogate modeling of stochastic functions - application to computational electromagnetic dosimetry. International Journal for Uncertainty Quantification, 9(4). [3] Torre, E., Marelli, S., Embrechts, P., & Sudret, B. (2019). A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulas. Probabilistic Engineering Mechanics, 55, 1-16. [4] Karian, Z. A., & Dudewicz, E. J. (2000). Fitting statistical distributions: the generalized lambda distribution and generalized bootstrap methods. CRC press.