Computational Statistics

, Volume 32, Issue 2, pp 559–583 | Cite as

Uncertainty quantification for functional dependent random variables

  • Simon NantyEmail author
  • Céline Helbert
  • Amandine Marrel
  • Nadia Pérot
  • Clémentine Prieur
Original Paper


This paper proposes a new methodology to model uncertainties associated with functional random variables. This methodology allows to deal simultaneously with several dependent functional variables and to address the specific case where these variables are linked to a vectorial variable, called covariate. In this case, the proposed uncertainty modelling methodology has two objectives: to retain both the most important features of the functional variables and their features which are the most correlated to the covariate. This methodology is composed of two steps. First, the functional variables are decomposed on a functional basis. To deal simultaneously with several dependent functional variables, a Simultaneous Partial Least Squares algorithm is proposed to estimate this basis. Second, the joint probability density function of the coefficients selected in the decomposition is modelled by a Gaussian mixture model. A new sparse method based on a Lasso penalization algorithm is proposed to estimate the Gaussian mixture model parameters and reduce their number. Several criteria are introduced to assess the methodology performance: its ability to approximate the functional variables probability distribution, their dependence structure and their features which explain the covariate. Finally, the whole methodology is applied on a simulated example and on a nuclear reliability test case.


Probability Density Function Partial Little Square Functional Variable Gaussian Mixture Model Partial Little Square Regression 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Anstett-Collin F, Goffart J, Mara T, Denis-Vidal L (2015) Sensitivity analysis of complex models: coping with dynamic and static inputs. Reliab Eng Syst Saf 134:268–275CrossRefGoogle Scholar
  2. Bien J, Tibshirani RJ (2011) Sparse estimation of a covariance matrix. Biometrika 98:807–820MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bongiorno E, Goia A (2015) Some insights about the small ball probability factorization for hilbert random elements.
  4. Bongiorno E, Goia A (2016) Classification methods for hilbert data based on surrogate density. Comput Stat Data Anal, 99, 204–222.
  5. Bongiorno EG, Salinelli E, Goia A, Vieu P (2014) Contributions in infinite-dimensional statistics and related topics. Societa Editrice Esculapio. doi: 10.15651/978-88-748-8763-7
  6. Conover WJ (1971) Practical Nonparametric Statistics. Wiley, New YorkGoogle Scholar
  7. De Rocquigny E, Devictor N, Tarantola S (2008) Uncertainty in industrial practice. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  8. Delaigle A, Hall P (2010) Defining probability density for a distribution of random functions. Ann Stat 38(2):1171–1193Google Scholar
  9. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Series B (Methodological) 39:1–38MathSciNetzbMATHGoogle Scholar
  10. Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer Science & Business Media, BerlinzbMATHGoogle Scholar
  11. Friedman J, Hastie T, Tibshirani R (2008) Sparse inverse covariance estimation with the graphical Lasso. Biostatistics 9:432–441CrossRefzbMATHGoogle Scholar
  12. Fromont M, Laurent B, Lerasle M, Reynaud-Bouret P (2012) Kernels based tests with non-asymptotic bootstrap approaches for two-sample problem. In: 25th annual conference on learning theory 23:1–22Google Scholar
  13. Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, BerlinCrossRefzbMATHGoogle Scholar
  14. Goia A, Vieu P (2016) An introduction to recent advances in high/infinite dimensional statistics. J Multivar Anal, 146, 1–6. Special issue on statistical models and methods for high or infinite dimensional spacesGoogle Scholar
  15. Hastie T, Tibshirani R (1990) Generalized additive models. Chapman and Hall/CRC, Boca RatonzbMATHGoogle Scholar
  16. Helton J, Johnson J, Sallaberry C, Storlie C (2006) Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliab Eng Syst Saf 91(10–11):1175–1209CrossRefGoogle Scholar
  17. Horváth L, Kokoszka P (2012) Inference for functional data with applications. Springer, BerlinCrossRefzbMATHGoogle Scholar
  18. Höskuldsson A (1988) PLS regression methods. J Chemom 2:211–228CrossRefGoogle Scholar
  19. Hyndman RJ, Shang HL (2010) Rainbow plots, bagplots, and boxplots for functional data. J Computat Graph Stat 19:29–45MathSciNetCrossRefGoogle Scholar
  20. Jacques J, Preda C (2014a) Functional data clustering: a survey. Adv Data Anal Classif 8(3):231–255MathSciNetCrossRefGoogle Scholar
  21. Jacques J, Preda C (2014b) Model-based clustering for multivariate functional data. Computat Stat Data Anal 71:92–106MathSciNetCrossRefGoogle Scholar
  22. Loève M (1955) Probability theory. Springer, BerlinzbMATHGoogle Scholar
  23. Ma X, Zabaras N (2011) Kernel principal component analysis for stochastic input model generation. J Comput Phys 230(19):7311–7331MathSciNetCrossRefzbMATHGoogle Scholar
  24. Marrel A, Iooss B, Van Dorpe F, Volkova E (2008) An efficient methodology for modeling complex computer codes with gaussian processes. Comput Stat Data Anal 52(10):4731–4744MathSciNetCrossRefzbMATHGoogle Scholar
  25. Mclachlan J, Krishnan T (1997) The EM algorithm and extension. Wiley inter-science, New YorkzbMATHGoogle Scholar
  26. Oakley J, O’Hagan A (2002) Bayesian inference for the uncertainty distribution of computer model outputs. Biometrika 89(4):769–784CrossRefGoogle Scholar
  27. Pearson K (1901) On lines and planes of closest fit to systems of points in space. London Edinburgh Dublin Philos Mag J Sci 2:559–572CrossRefzbMATHGoogle Scholar
  28. Popelin AL, Iooss B (2013) Visualization tools for uncertainty and sensitivity analyses on thermal-hydraulic transients. In: SNA+ MC 2013—Joint international conference on supercomputing in nuclear applications and Monte CarloGoogle Scholar
  29. Ramsay JO, Silverman BW (2005) Functional data analysis. Springer, Springer Series in Statistics, BerlinCrossRefzbMATHGoogle Scholar
  30. Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. MIT Press, CambridgezbMATHGoogle Scholar
  31. Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Stat 27(3):832–837MathSciNetCrossRefzbMATHGoogle Scholar
  32. Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–423. doi: 10.2307/2245858 MathSciNetCrossRefzbMATHGoogle Scholar
  33. Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464MathSciNetCrossRefzbMATHGoogle Scholar
  34. Scott DW (2009) Multivariate density estimation: theory, practice, and visualization. John Wiley & Sons, HobokenGoogle Scholar
  35. Van Deun K, Smilde A, van der Werf M, Kiers H, Van Mechelen I (2009) A structured overview of simultaneous component based data integration. BMC Bioinform 10:246–261CrossRefGoogle Scholar
  36. Wan J, Zabaras N (2014) A probabilistic graphical model based stochastic input model construction. J Comput Phys 272:664–685MathSciNetCrossRefzbMATHGoogle Scholar
  37. Wang H (2013) Coordinate descent algorithm for covariance graphical Lasso. Stat Comput 6:1–9Google Scholar
  38. Welch WJ, Buck RJ, Sacks J, Wynn HP, Mitchell TJ, Morris MD (1992) Screening, predicting, and computer experiments. Technometrics 34(1):15–25CrossRefGoogle Scholar
  39. Wold H (1966) Estimation of principal components and related models by iterative least squares. Academic Press, CambridgezbMATHGoogle Scholar
  40. Wu CFJ (1983) On the convergence properties of the EM algorithm. Ann Stat 11:95–103MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Simon Nanty
    • 1
    • 3
    Email author
  • Céline Helbert
    • 2
  • Amandine Marrel
    • 1
  • Nadia Pérot
    • 1
  • Clémentine Prieur
    • 3
  1. 1.CEA, DENSaint-Paul-lez-DuranceFrance
  2. 2.UMR 5208, Ecole Centrale de Lyon, Institut Camille JordanUniversité de LyonLyonFrance
  3. 3.Université Joseph Fourier and INRIAGrenobleFrance

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