Annals of the Institute of Statistical Mathematics

, Volume 72, Issue 1, pp 123–155

# Estimating quantiles in imperfect simulation models using conditional density estimation

Article

## Abstract

In this article, we consider the problem of estimating quantiles related to the outcome of experiments with a technical system given the distribution of the input together with an (imperfect) simulation model of the technical system and (few) data points from the technical system. The distribution of the outcome of the technical system is estimated in a regression model, where the distribution of the residuals is estimated on the basis of a conditional density estimate. It is shown how Monte Carlo can be used to estimate quantiles of the outcome of the technical system on the basis of the above estimates, and the rate of convergence of the quantile estimate is analyzed. Under suitable assumptions, it is shown that this rate of convergence is faster than the rate of convergence of standard estimates which ignore either the (imperfect) simulation model or the data from the technical system; hence, it is crucial to combine both kinds of information. The results are illustrated by applying the estimates to simulated and real data.

## Keywords

Conditional density estimation Quantile estimation Imperfect models $$L_1$$ error Surrogate models Uncertainty quantification

## Supplementary material

10463_2018_683_MOESM1_ESM.pdf (114 kb)
Supplementary material 1 (pdf 114 KB)

## References

1. Bauer, B., Devroye, L., Kohler, M., Krzy.zak, A., Walk, H. (2017). Nonparametric estimation of a function from noiseless observations at random points. Journal of Multivariate Analysis, 160, 90–104.
2. Bichon, B., Eldred, M., Swiler, M., Mahadevan, S., McFarland, J. (2008). Efficient global reliability analysis for nonlinear implicit performance functions. AIAA Journal, 46, 2459–2468.
3. Bott, A., Kohler, M. (2016). Adaptive estimation of a conditional density. International Statistical Review, 84, 291–316.
4. Bott, A., Kohler, M. (2017). Nonparametric estimation of a conditional density. Annals of the Institute of Statistical Mathematics, 69, 189–214.
5. Bott, A. K., Felber, T., Kohler, M. (2015). Estimation of a density in a simulation model. Journal of Nonparametric Statistics, 27, 271–285.
6. Bourinet, J.-M., Deheeger, F., Lemaire, M. (2011). Assessing small failure probabilities by combined subset simulation and support vector machines. Structural Safety, 33, 343–353.
7. Bucher, C., Bourgund, U. (1990). A fast and efficient response surface approach for structural reliability problems. Structural Safety, 7, 57–66.
8. Das, P.-K., Zheng, Y. (2000). Cumulative formation of response surface and its use in reliability analysis. Probabilistic Engineering Mechanics, 15, 309–315.
9. Deheeger, F., Lemaire, M. (2010). Support vector machines for efficient subset simulations: $$^{2}$$SMART method. In Proceedings of the 10th international conference on applications of statistics and probability in civil engineering (ICASP10), Tokyo, Japan.Google Scholar
10. Devroye, L., Lugosi, G. (2001). Combinatorial methods in density estimation. New York: Springer.
11. Devroye, L., Felber, T., Kohler, M. (2013). Estimation of a density using real and artificial data. IEEE Transactions on Information Theory, 59(3), 1917–1928.
12. Efromovich, S. (2007). Conditional density estimation in a regression setting. Annals of Statistics, 35, 2504–2535.
13. Enss, C., Kohler, M., Krzyżak, A., Platz, R. (2016). Nonparametric quantile estimation based on surrogate models. IEEE Transactions on Information Theory, 62, 5727–5739.
14. Fan, J., Yim, T. H. (2004). A crossvalidation method for estimating conditional densities. Biometrika, 91, 819–834.
15. Fan, J., Yao, Q., Tong, H. (1996). Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika, 83, 189–206.
16. Felber, T., Kohler, M., Krzyżak, A. (2015a). Adaptive density estimation based on real and artificial data. Journal of Nonparametric Statistics, 27, 1–18.
17. Felber, T., Kohler, M., Krzyżak, A. (2015b). Density estimation with small measurement errors. IEEE Transactions on Information Theory, 61, 3446–3456.
18. Gooijer, J. G. D., Zerom, D. (2003). On conditional density estimation. Statistica Neerlandica, 57, 159–176.
19. Györfi, L., Kohler, M., Krzyżak, A., Walk, H. (2002). A distribution-free theory of nonparametric regression. New York: Springer.
20. Hurtado, J. E. (2004). Structural reliability: Statistical learning perspectives. Lecture notes in applied and computational mechanics (Vol. 17). Berlin: Springer.Google Scholar
21. Kaymaz, I. (2005). Application of Kriging method to structural reliability problems. Strutural Safety, 27, 133–151.
22. Kim, S.-H., Na, S.-W. (1997). Response surface method using vector projected sampling points. Structural Safety, 19, 3–19.
23. Kohler, M., Krzyżak, A. (2016). Estimation of a density from an imperfect simulation model (submitted).Google Scholar
24. Kohler, M., Krzyżak, A. (2017). Improving a surrogate model in uncertainty quantification by real data (submitted).Google Scholar
25. Kohler, M., Krzyżak, A. (2018). Adaptive estimation of quantiles in a simulation model. IEEE Transactions on Information Theory, 64, 501–512.
26. Kohler, M., Krzyżak, A., Mallapur, S., Platz, R. (2018). Uncertainty quantification in case of imperfect models: A non-Bayesian approach. Scandinavian Journal of Statistics. .
27. Mallapur, S., Platz, R. (2017). Quantification and evaluation of uncertainty in the mathematical modelling of a suspension strut using bayesian model validation approach. In Proceedings of the international modal analysis conference IMAC-XXXV, Garden Grove, California, USA, Paper 117, 30 January–2 February, 2017.Google Scholar
28. Massart, P. (1990). The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality. Annals of Probability, 18, 1269–1283.
29. Papadrakakis, M., Lagaros, N. (2002). Reliability-based structural optimization using neural networks and Monte Carlo simulation. Computer Methods in Applied Mechanics and Engineering, 191, 3491–3507.
30. Parzen, E. (1962). On the estimation of a probability density function and the mode. Annals of Mathematical Statistics, 33, 1065–1076.
31. Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Annals of Mathematical Statistics, 27, 832–837.
32. Rosenblatt, M. (1969). Conditional probability density and regression estimates. In P. R. Krishnaiah (Ed.), Multivariate analysis II (pp. 25–31). New York: Academic Press.Google Scholar
33. Wong, R. K. W., Storlie, C. B., Lee, T. C. M. (2017). A frequentist approach to computer model calibration. Journal of the Royal Statistical Society, Series B, 79, 635–648.