Data-driven stochastic inversion via functional quantization

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.


In this paper, we propose a new methodology for solving stochastic inversion problems through computer experiments, the stochasticity being driven by a functional random variables. This study is motivated by an automotive application. In this context, the simulator code takes a double set of simulation inputs: deterministic control variables and functional uncertain variables. This framework is characterized by two features. The first one is the high computational cost of simulations. The second is that the probability distribution of the functional input is only known through a finite set of realizations. In our context, the inversion problem is formulated by considering the expectation over the functional random variable. We aim at solving this problem by evaluating the model on a design, whose adaptive construction combines the so-called stepwise uncertainty reduction methodology with a strategy for an efficient expectation estimation. Two greedy strategies are introduced to sequentially estimate the expectation over the functional uncertain variable by adaptively selecting curves from the initial set of realizations. Both of these strategies consider functional principal component analysis as a dimensionality reduction technique assuming that the realizations of the functional input are independent realizations of the same continuous stochastic process. The first strategy is based on a greedy approach for functional data-driven quantization, while the second one is linked to the notion of space-filling design. Functional PCA is used as an intermediate step. For each point of the design built in the reduced space, we select the corresponding curve from the sample of available curves, thus guaranteeing the robustness of the procedure to dimension reduction. The whole methodology is illustrated and calibrated on an analytical example. It is then applied on the automotive industrial test case where we aim at identifying the set of control parameters leading to meet the pollutant emission standards of a vehicle.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16


  1. Abtini, M.: Plans prédictifs à taille fixe et séquentiels pour le krigeage. Ph.D. thesis, Ecole Centrale Lyon (2018)

  2. Bect, J., Ginsbourger, D., Li, L., Picheny, V., Vazquez, E.: Sequential design of computer experiments for the estimation of a probability of failure. Stat. Comput. 22(3), 773–793 (2012)

    MathSciNet  Article  Google Scholar 

  3. Bect, J., Bachoc, F., Ginsbourger, D.: A supermartingale approach to Gaussian process based sequential design of experiments. arXiv preprint arXiv:1608.01118 (2016)

  4. Bolin, D., Lindgren, F.: Excursion and contour uncertainty regions for latent Gaussian models. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 77(1), 85–106 (2015)

    MathSciNet  Article  Google Scholar 

  5. Bonfils, A., Creff, Y., Lepreux, O., Petit, N.: Closed-loop control of a SCR system using a NO\(_{x}\) sensor cross-sensitive to NH\(_3\). IFAC Proc. Vol. 45(15), 738–743 (2012)

    Article  Google Scholar 

  6. Cardot, H., Ferraty, F., Sarda, P.: Functional Linear Model. Stat. Probab. Lett. 45(1), 11–22 (1999)

    MathSciNet  Article  Google Scholar 

  7. Chevalier, C.: Fast uncertainty reduction strategies relying on Gaussian process models. Ph.D. thesis (2013)

  8. Chevalier, C., Ginsbourger, D.: Fast Computation of the multi-points expected improvement with applications in batch selection. In: International Conference on Learning and Intelligent Optimization, pp. 59–69. Springer (2013)

  9. Chevalier, C., Ginsbourger, D., Bect, J., Molchanov, I.: Estimating and quantifying uncertainties on level sets using the Vorob’ev expectation and deviation with Gaussian process models. In: mODa 10–Advances in Model-Oriented Design and Analysis, pp. 35–43. Springer (2013)

  10. Chevalier, C., Picheny, V., Ginsbourger, D.: Kriginv: an efficient and user-friendly implementation of batch-sequential inversion strategies based on kriging. Comput. Stat. Data Anal. 71, 1021–1034 (2014a)

    MathSciNet  Article  Google Scholar 

  11. Chevalier, C., Bect, J., Ginsbourger, D., Vazquez, E., Picheny, V., Richet, Y.: Fast parallel kriging-based stepwise uncertainty reduction with application to the identification of an excursion set. Technometrics 56(4), 455–465 (2014b)

    MathSciNet  Article  Google Scholar 

  12. Chevalier, C., Emery, X., Ginsbourger, D.: Fast update of conditional simulation ensembles. Math. Geosci. 47(7), 771–789 (2015)

    Article  Google Scholar 

  13. Flury, B.A.: Principal points. Biometrika 77(1), 33–41 (1990)

    MathSciNet  Article  Google Scholar 

  14. French, J.P., Sain, S.R., et al.: Spatio-temporal exceedance locations and confidence regions. Ann. Appl. Stat. 7(3), 1421–1449 (2013)

    MathSciNet  Article  Google Scholar 

  15. Jackson, D.A.: Stopping rules in principal components analysis: a comparison of heuristical and statistical approaches. Ecology 74(8), 2204–2214 (1993)

    Article  Google Scholar 

  16. Janusevskis, J., Le Riche, R.: Simultaneous kriging-based estimation and optimization of mean response. J. Glob. Optim. 55(2), 313–336 (2013)

    MathSciNet  Article  Google Scholar 

  17. Jin, R., Chen, W., Sudjianto, A.: An efficient algorithm for constructing optimal design of computer experiments. J. Stat. Plan. Inference 134(1), 268–287 (2005)

    MathSciNet  Article  Google Scholar 

  18. Johnson, M.E., Moore, L.M., Ylvisaker, D.: Minimax and maximin distance designs. J. Stat. Plan. Inference 26(2), 131–148 (1990)

    MathSciNet  Article  Google Scholar 

  19. L’Ecuyer, P., Lemieux, C.: Recent advances in randomized quasi-Monte Carlo methods. In: Dror, M., L’Ecuyer, P., Szidarovszky, F. (eds.) Modeling Uncertainty, pp. 419–474. Springer, Berlin (2005)

    Google Scholar 

  20. L’Ecuyer, P., Owen, A.B.: Monte Carlo and Quasi-Monte Carlo Methods 2008. Springer, Berlin (2009)

    Google Scholar 

  21. Levrard, C.: High-dimensional vector quantization: convergence rates and variable selection. Ph.D. thesis, Universite de Paris 11 (2014)

  22. Luschgy, H., Pagès, G.: Greedy vector quantization. J. Approx. Theory 198, 111–131 (2015)

    MathSciNet  Article  Google Scholar 

  23. Luschgy, H., Pagès, G., Wilbertz, B.: Asymptotically optimal quantization schemes for Gaussian processes on Hilbert spaces. ESAIM Probab. Stat. 14, 93–116 (2010)

    MathSciNet  Article  Google Scholar 

  24. Miranda, M., Bocchini, P.: Functional Quantization of stationary Gaussian and non-Gaussian random processes. In: Deodatis, G., Ellingwood, B.R., Frangopol, D.M. (eds.) Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures, pp. 2785–2792. CRC Press/Balkema, London (2013)

    Google Scholar 

  25. Miranda, M.J., Bocchini, P.: A versatile technique for the optimal approximation of random processes by functional quantization. Appl. Math. Comput. 271, 935–958 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Morris, M.D., Mitchell, T.J.: Exploratory designs for computational experiments. J. Stat. Plan. Inference 43(3), 381–402 (1995)

    Article  Google Scholar 

  27. Nanty, S., Helbert, C., Marrel, A., Pérot, N., Prieur, C.: Sampling, metamodeling, and sensitivity analysis of numerical simulators with functional stochastic inputs. SIAM/ASA J. Uncertain. Quantif. 4(1), 636–659 (2016)

    MathSciNet  Article  Google Scholar 

  28. Pagès, G.: Introduction to optimal vector quantization and its applications for numerics. Tech. rep. (2014).

  29. Pagès, G., Printems, J.: Functional quantization for numerics with an application to option pricing. Monte Carlo Methods Appl. mcma 11(4), 407–446 (2005)

    MathSciNet  Article  Google Scholar 

  30. Pagès, G., Printems, J.: Optimal quantization for finance: from random vectors to stochastic processes. In: Bensoussan, A., Zhang, Q. (eds.) Handbook of Numerical Analysis, vol. 15, pp. 595–648. Elsevier, Amsterdam (2009)

    Google Scholar 

  31. Picheny, V., Ginsbourger, D., Roustant, O., Haftka, R.T., Kim, N.H.: Adaptive designs of experiments for accurate approximation of a target region. J. Mech. Des. 132(7), 071008 (2010)

    Article  Google Scholar 

  32. Pronzato, L., Müller, W.G.: Design of computer experiments: space filling and beyond. Stat. Comput. 22(3), 681–701 (2012)

    MathSciNet  Article  Google Scholar 

  33. Ramsay, J.O.: Functional Data Analysis. Wiley Online Library, New York (2006)

    Google Scholar 

  34. 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 (2013)

  35. Vazquez, E., Bect, J.: A sequential Bayesian algorithm to estimate a probability of failure. IFAC Proc. Vol. 42(10), 546–550 (2009)

    Article  Google Scholar 

  36. Williams, B.J., Santner, T.J., Notz, W.I.: Sequential design of computer experiments to minimize integrated response functions. Stat. Sin. 10, 1133–1152 (2000)

Download references


The authors would like to thank the anonymous reviewers and the associate editor for their helpful comments which substantially improved this paper. We also thank the Inria Associate Team UNcertainty QUantification is ESenTIal for OceaNic & Atmospheric flows proBLEms. This work was supported by IFPEN and the OQUAIDO chair.

Author information



Corresponding author

Correspondence to Mohamed Reda El Amri.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

El Amri, M.R., Helbert, C., Lepreux, O. et al. Data-driven stochastic inversion via functional quantization. Stat Comput 30, 525–541 (2020).

Download citation


  • Functional random variable
  • Karhunen–Loève expansion
  • Data reduction
  • Functional quantization
  • Set estimation
  • Gaussian process models