State Dependent Regressions: From Sensitivity Analysis to Meta-modeling


State Dependent Parameter (SDP) modelling has been developed by Professor Peter Young in the 1990s to identify non-linearities in the context of dynamic transfer function models. SDP is a very efficient approach and it is based on recursive filtering and Fixed Interval Smoothing (FIS) algorithms. It has been applied successfully in many applications, especially to identify Data-Based Mechanistic models from observed time series data in environmental sciences. In this paper we highlight the role played by the SDP ideas, namely in the simplified State-Dependent Regression (SDR) form, in the context of sensitivity analysis and meta-modelling. Fruitful joint co-operation with Peter Young has led to a series of papers, where SDR has been applied to perform sensitivity analysis, to reduce model’s complexity and to build meta-models (or emulators) capable to reproduce the main features of large simulation models. Finally, we will describe how SDR algorithms can be effectively used in the context of the identification and estimation of tensor product smoothing splines ANOVA models, improving their performances.


Monte Carlo Reproduce Kernel Hilbert Space Global Sensitivity Analysis Generalize Maximum Likelihood State Dependent Parameter 
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. 1.
    Borgonovo, E.: Measuring uncertainty importance: investigation and comparison of alternative approaches. Risk Anal. 26, 1349–1361 (2006) CrossRefGoogle Scholar
  2. 2.
    Borgonovo, E.: A new uncertainty importance measure. Reliab. Eng. Syst. Saf. 92, 771–784 (2007) CrossRefGoogle Scholar
  3. 3.
    Lophaven, S., Nielsen, H., Sondergaard, J.: DACE a Matlab kriging toolbox, version 2.0. Technical Report IMM-TR-2002-12, Informatics and Mathematical Modelling, Technical University of Denmark (2002).
  4. 4.
    Gu, C.: Smoothing Spline ANOVA Models. Springer, Berlin (2002) MATHGoogle Scholar
  5. 5.
    Kalman, R.: A new approach to linear filtering and prediction problems. J. Basic Eng. D 82, 35–45 (1960) Google Scholar
  6. 6.
    Lin, Y., Zhang, H.: Component selection and smoothing in smoothing spline analysis of variance models. Ann. Stat. 34, 2272–2297 (2006) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Ng, C., Young, P.C.: Recursive estimation and forecasting of non-stationary time series. J. Forecast. 9, 173–204 (1990) CrossRefGoogle Scholar
  8. 8.
    Priestley, M.B.: Nonlinear and Nonstationary Time Series Analysis. Academic Press, New York (1988) Google Scholar
  9. 9.
    Ratto, M., Pagano, A., Young, P.C.: Non-parametric estimation of conditional moments for sensitivity analysis. Reliab. Eng. Syst. Saf. 94, 237–243 (2009) CrossRefGoogle Scholar
  10. 10.
    Ratto, M., Pagano, A.: Using recursive algorithms for the efficient identification of smoothing spline ANOVA models. AStA Adv. Stat. Anal. 94(4), 367–388 (2010) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ratto, M., Pagano, A., Young, P.C.: State dependent parameter metamodelling and sensitivity analysis. Comput. Phys. Commun. 177, 863–876 (2007) CrossRefGoogle Scholar
  12. 12.
    Sadeghi, J., Tych, W., Chotai, A., Young, P.C.: Multi-state dependent parameter model identification and estimation for nonlinear dynamic systems. Electron. Lett. 46(18), 1265–1266 (2011) CrossRefGoogle Scholar
  13. 13.
    Saltelli, A., Chan, K., Scott, M. (eds.): Sensitivity Analysis. Wiley, New York (2000) MATHGoogle Scholar
  14. 14.
    Storlie, C., Bondell, H., Reich, B., Zhang, H.: Surface estimation, variable selection, and the nonparametric oracle property. Stat. Sin. 21(2), 679–705 (2011) MATHCrossRefGoogle Scholar
  15. 15.
    Schweppe, F.: Evaluation of likelihood functions for Gaussian signals. IEEE Trans. Inf. Theory 11, 61–70 (1965) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Tibshirani, R.: Regression shrinkage and selection via the LASSO. J. R. Stat. Soc. B 58(1), 267–288 (1996) MathSciNetMATHGoogle Scholar
  17. 17.
    Wahba, G.: Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics (1990) MATHCrossRefGoogle Scholar
  18. 18.
    Wecker, W.E., Ansley, C.F.: The signal extraction approach to non linear regression and spline smoothing. J. Am. Stat. Assoc. 78, 81–89 (1983) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Weinert, H., Byrd, R., Sidhu, G.: A stochastic framework for recursive computation of spline functions: Part II, smoothing splines. J. Optim. Theory Appl. 30, 255–268 (1983) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Young, P.C.: Time variable and state dependent modelling of nonstationary and nonlinear time series. In: Rao, T.S. (ed.) Developments in Time Series Analysis, pp. 374–413. Chapman and Hall, London (1993) Google Scholar
  21. 21.
    Young, P.C.: Data-based mechanistic modeling of environmental, ecological, economic and engineering systems. Environ. Model. Softw. 13, 105–122 (1998) CrossRefGoogle Scholar
  22. 22.
    Young, P.C.: Nonstationary time series analysis and forecasting. Progr. Environ. Sci. 1, 3–48 (1999) Google Scholar
  23. 23.
    Young, P.C.: Stochastic, dynamic modelling and signal processing: Time variable and state dependent parameter estimation. In: Fitzgerald, W.J., Smith, R.L., Walden, A.T., Young, P.C. (eds.) Nonlinear and Nonstationary Signal Processing, pp. 74–114. Cambridge University Press, Cambridge (2000) Google Scholar
  24. 24.
    Young, P.C.: The identification and estimation of nonlinear stochastic systems. In: Mees, F.A.I. (ed.) Nonlinear Dynamics and Statistics. Birkhäuser, Boston (2001) Google Scholar
  25. 25.
    Young, P.C.: Data-based mechanistic modelling: natural philosophy revisited? (in this book) Google Scholar
  26. 26.
    Young, P.C., McKenna, P., Bruun, J.: The identification and estimation of nonlinear stochastic systems. Int. J. Control 74, 1837–1857 (2001) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Young, P.C., Pedregal, D.J.: Recursive fixed interval smoothing and the evaluation of Lidar measurements. Environmetrics 7, 417–427 (1996) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.JRC, Joint Research CentreThe European CommissionIspra (VA)Italy

Personalised recommendations