Abstract
Adding inequality constraints (e.g. positivity, monotonicity, convexity) in Gaussian processes (GPs) leads to more realistic stochastic emulators. Due to the truncated Gaussianity of the posterior, its distribution has to be approximated. In this work, we consider Monte Carlo (MC) and Markov Chain MC (MCMC) methods. However, strictly interpolating the observations may entail expensive computations due to highly restrictive sample spaces. Furthermore, having emulators when data are actually noisy is also of interest for real-world applications. Hence, we introduce a noise term for the relaxation of the interpolation conditions, and we develop the corresponding approximation of GP emulators under linear inequality constraints. We demonstrate on various synthetic examples that the performance of MC and MCMC samplers improves when considering noisy observations. Finally, on 2D and 5D coastal flooding applications, we show that more flexible and realistic emulators are obtained by considering noise effects and by enforcing the inequality constraints.
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References
Azzimonti, D.: profExtrema: Compute and visualize profile extrema functions. R package version 0.2.0 (2018)
Azzimonti, D., Ginsbourger, D., Rohmer, J., Idier, D.: Profile extrema for visualizing and quantifying uncertainties on excursion regions. Application to coastal flooding. Technometrics 0(ja), 1–26 (2019)
Bay, X., Grammont, L., Maatouk, H.: Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation. Electron. J. Stat. 10(1), 1580–1595 (2016)
Botev, Z.I.: The normal law under linear restrictions: simulation and estimation via minimax tilting. J. R. Stat. Soc.: Ser. B 79(1), 125–148 (2017)
Cousin, A., Maatouk, H., Rullière, D.: Kriging of financial term-structures. Eur. J. Oper. Res. 255(2), 631–648 (2016)
Deville, Y., Ginsbourger, D., Durrande, N., Roustant, O.: kergp: Gaussian process laboratory. R package version 0.2.0 (2015)
Dupuy, D., Helbert, C., Franco, J.: DiceDesign and DiceEval: two R packages for design and analysis of computer experiments. J. Stat. Softw. 65(11), 1–38 (2015)
Geyer, C.J.: Practical Markov Chain Monte Carlo. Stat. Sci. 7(4), 473–483 (1992)
Golchi, S., Bingham, D.R., Chipman, H., Campbell, D.A.: Monotone emulation of computer experiments. SIAM/ASA J. Uncertain. Quantif. 3(1), 370–392 (2015)
Goldfarb, D., Idnani, A.: Dual and primal-dual methods for solving strictly convex quadratic programs. Numerical Analysis, pp. 226–239. Springer, New York (1982)
Gong, L., Flegal, J.M.: A practical sequential stopping rule for high-dimensional Markov Chain Monte Carlo. J. Comput. Graph. Stat. 25(3), 684–700 (2016)
Lan, S., Shahbaba, B.: Sampling constrained probability distributions using spherical augmentation. Algorithmic Advances in Riemannian Geometry and Applications, pp. 25–71. Springer, New York (2016)
Larson, M.G., Bengzon, F.: The Finite Element Method: Theory, Implementation, and Applications. Springer, New York (2013)
López-Lopera, A.F.: lineqGPR: Gaussian process regression models with linear inequality constraints. R package version 0.0.3 (2018)
López-Lopera, A.F., Bachoc, F., Durrande, N., Roustant, O.: Finite-dimensional Gaussian approximation with linear inequality constraints. SIAM/ASA J. Uncertain. Quantif. 6(3), 1224–1255 (2018)
Maatouk, H., Bay, X.: A new rejection sampling method for truncated multivariate Gaussian random variables restricted to convex sets. Monte Carlo and Quasi-Monte Carlo Methods, pp. 521–530. Springer, New York (2016)
Maatouk, H., Bay, X.: Gaussian process emulators for computer experiments with inequality constraints. Math. Geosci. 49(5), 557–582 (2017)
Murphy, K.P.: Machine Learning: A Probabilistic Perspective. Adaptive Computation and Machine Learning Series. The MIT Press, Cambridge (2012)
Pakman, A., Paninski, L.: Exact Hamiltonian Monte Carlo for truncated multivariate Gaussians. J. Comput. Graph. Stat. 23(2), 518–542 (2014)
Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. The MIT Press, Cambridge (2005)
Rohmer, J., Idier, D.: A meta-modelling strategy to identify the critical offshore conditions for coastal flooding. Nat. Hazards Earth Syst. Sci. 12(9), 2943–2955 (2012)
Roustant, O., Ginsbourger, D., Deville, Y.: DiceKriging, DiceOptim: two R packages for the analysis of computer experiments by Kriging-based metamodeling and optimization. J. Stat. Softw. 51(1), 1–55 (2012)
Taylor, J., Benjamini, Y.: restrictedMVN: multivariate normal restricted by affine constraints. R package version 1.0 (2016)
Acknowledgements
This research was conducted within the frame of the Chair in Applied Mathematics OQUAIDO, gathering partners in technological research (BRGM, CEA, IFPEN, IRSN, Safran, Storengy) and academia (CNRS, Ecole Centrale de Lyon, Mines Saint-Etienne, Univ. Grenoble, Univ. Nice, Univ. Toulouse) around advanced methods for computer experiments.
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López-Lopera, A.F., Bachoc, F., Durrande, N., Rohmer, J., Idier, D., Roustant, O. (2020). Approximating Gaussian Process Emulators with Linear Inequality Constraints and Noisy Observations via MC and MCMC. In: Tuffin, B., L'Ecuyer, P. (eds) Monte Carlo and Quasi-Monte Carlo Methods. MCQMC 2018. Springer Proceedings in Mathematics & Statistics, vol 324. Springer, Cham. https://doi.org/10.1007/978-3-030-43465-6_18
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