Statistics for Model Calibration

Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 9)


Mathematical models of dynamic processes contain parameters which have to be estimated based on time-resolved experimental data. This task is often approached by optimization of a suitably chosen objective function. Maximization of the likelihood, i.e. maximum likelihood estimation, has several beneficial theoretical properties ensuring efficient and accurate statistical analyses and is therefore often performed for identification of model parameters.

For nonlinear models, optimization is challenging and advanced numerical techniques have been established to approach this issue. However, the statistical methodology typically applied to interpret the optimization outcomes often still rely on linear approximations of the likelihood.

In this review, we summarize the maximum likelihood methodology and focus on nonlinear models like ordinary differential equations. The profile likelihood methodology is utilized to derive confidence intervals and for performing identifiability and observability analyses.



The authors acknowledge financial support provided by the BMBF-grants 0315766-VirtualLiver, 0316042GLungSysII.


  1. 1.
    Atkinson, A.: Likelihood ratios, posterior odds and information criteria. J. Econ. 16, 15–20 (1981)CrossRefGoogle Scholar
  2. 2.
    Azzalini, A.: Statistical Inference: Based on the Likelihood. Chapman and Hall, London (1996)MATHGoogle Scholar
  3. 3.
    Baake, E., Baake, M., Bock, H., Briggs, K.M.: Fitting ordinary differential equations to chaotic data. Phys. Rev. A 45(8), 5524–5529 (1992)CrossRefGoogle Scholar
  4. 4.
    Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)MATHGoogle Scholar
  5. 5.
    Bock, H.: Numerical treatment of inverse problems in chemical reaction kinetics. In: Ebert, K., Deuflhard, P., Jäger, W. (eds.) Modeling of Chemical Reaction Systems, vol. 18, pp. 102–125. Springer, New York (1981)CrossRefGoogle Scholar
  6. 6.
    Bock, H.: Recent advances in parameter identification for ordinary differential equations. In: Deuflhard, P., Hairer, E. (eds.) Progress in Scientific Computing, vol. 2, pp. 95–121. Birkhäuser, Boston (1983)Google Scholar
  7. 7.
    Box, G.E.P., Hill, W.J.: Discrimination among mechanistic models. Technometrics 9, 57–71 (1967)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Coleman, T., Li, Y.: An interior, trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6, 418–445 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cox, D., Hinkley, D.: Theoretical Statistics. Chapman & Hall, London (1994)Google Scholar
  10. 10.
    Feder, P.I.: On the distribution of the log likelihood ratio test statistic when the true parameter is “near” the boundaries of the hypothesis regions. Ann. Math. Stat. 39(6), 2044–2055 (1968)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hand, D.J.: Understanding the new statistics: effect sizes, confidence intervals, and meta-analysis by geoff cumming. Int. Stat. Rev. 80(2), 344–345 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Honerkamp, J.: Statistical Physics: An Advanced Approach with Applications. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Kirkpatrick, S., Gelatt, C., Jr., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Koch, O., Weinmüller, E.B.: The convergende of shooting methods for singular bondary value problems. Math. Comput. 72(241), 289–305 (2001)CrossRefGoogle Scholar
  15. 15.
    Kreutz, C., Timmer, J.: Systems biology: experimental design. FEBS J. 276(4), 923–942 (2009)CrossRefGoogle Scholar
  16. 16.
    Kreutz, C., Raue, A., Timmer, J.: Likelihood based observability analysis and confidence intervals for predictions of dynamic models. BMC Syst. Biol. 6, 120 (2012)CrossRefGoogle Scholar
  17. 17.
    Kreutz, C., Raue, A., Kaschek, D., Timmer, J.: Profile likelihood in systems biology. FEBS J. 280(11), 2564–2571 (2013)CrossRefGoogle Scholar
  18. 18.
    Kronfeld, H.P.M., Zell, A.: The EvA2 optimization framework. Learn. Intell. Optim. 6073, 247–250 (2010)CrossRefGoogle Scholar
  19. 19.
    Leis, J., Kramer, M.: The simultaneous solution and sensitivity analysis of systems described by ordinary differential equations. ACM Trans. Math. Softw. 14(1), 45–60 (1988)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lory, P.: Enlarging the domain of convergence for mutiple shooting by homotopy method. Numer. Math. 35, 231–240 (1980)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Marquardt, D.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11(2), 431–441 (1963)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Meeker, W., Escobar, L.: Teaching about approximate confidence regions based on maximum likelihood estimation. Am. Stat. 49(1), 48–53 (1995)Google Scholar
  23. 23.
    Neyman, L., Pearson, E.: On the problem of the most efficient tests of statistical hypotheses. Phil. Trans. Roy. Soc. A 231, 289–337 (1933)CrossRefGoogle Scholar
  24. 24.
    Peifer, M., Timmer, J.: Parameter estimation in ordinary differential equations for biochemical processes using the method of multiple shooting. IET Syst. Biol. 1, 78–88 (2007)CrossRefGoogle Scholar
  25. 25.
    Pinheiro, J.C., Bates, D.M.: Mixed-effects models in S and S-plus. In: Statistics and Computing. Springer, New York (2000)CrossRefMATHGoogle Scholar
  26. 26.
    Poli, R., Kennedy, J., Blackwell, T.: Particle swarm optimization: an overview. Swarm Intell. J. 1(1), 33–57 (2007)CrossRefGoogle Scholar
  27. 27.
    Press, W., Flannery, B., Saul, S., Vetterling, W.: Numerical Recipes. Cambridge University Press, Cambridge (1992)Google Scholar
  28. 28.
    Puntanen, S.: Projection matrices, generalized inverse matrices, and singular value decomposition by haruo yanai, kei takeuchi, yoshio takane. Int. Stat. Rev. 79(3), 503–504 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Quinn, G.P., Keough, M.J.: Experimental Design and Data Analysis for Biologists. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  30. 30.
    Raue, A., Kreutz, C., Maiwald, T., Bachmann, J., Schilling, M., Klingmüller, U., Timmer, J.: Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 1923–1929 (2009)CrossRefGoogle Scholar
  31. 31.
    Raue, A., Kreutz, C., Maiwald, T., Klingmüller, U., Timmer, J.: Addressing parameter identifiability by model-based experimentation. IET Syst. Biol. 5(2), 120–130 (2011)CrossRefGoogle Scholar
  32. 32.
    Raue, A., Schilling, M., Bachmann, J., Matteson, A., Schelker, M., et al.: Lessons learned from quantitative dynamical modeling in systems biology. PLoS One 8(9), e74335 (2013). doi:10.1371/journal.pone.0074335CrossRefGoogle Scholar
  33. 33.
    Raue, A., Steiert, B., Schelker, M., Kreutz, C., Maiwald, T., Hass, H., Vanlier, J., Tönsing, C., Adlung, L., Engesser, R., Mader, W., Heinemann, T., Hasenauer, J., Schilling, M., Höfer, T., Klipp, E., Theis, F., Klingmüller, U., Schöberl, B., Timmer, J.: Data2Dynamics: a modeling environment tailored to parameter estimation in dynamical systems. Bioinformatics (2015). doi:10.1093/bioinformatics/btv405. First published online 3 July 2015Google Scholar
  34. 34.
    Reid, N., Fraser, D.: Likelihood inference in the presence of nuisance parameters. arXiv:physics/0312079 (2003)Google Scholar
  35. 35.
    Seber, G., Wild, C.: Nonlinear Regression. Wiley, New York (1989)CrossRefMATHGoogle Scholar
  36. 36.
    Steiert, B., Raue, A., Timmer, J., Kreutz, C.: Experimental design for parameter estimation of gene regulatory networks. PLoS One 7(7), e40052 (2012)CrossRefGoogle Scholar
  37. 37.
    Steward, W.E., Henson, T.L., Box, G.E.P.: Model discrimination and criticism with single-response data. AIChE J. 42, 3055–3062 (1996)CrossRefGoogle Scholar
  38. 38.
    Steward, W.E., Shon, Y., Box, G.E.P.: Discrimination and goodness of fit of multiresponse mechanistic models. AIChE J. 66, 1404–1412 (1998)CrossRefGoogle Scholar
  39. 39.
    Wald, A.: Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans. Am. Math. Soc. 54(3), 426–482 (1943)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of PhysicsUniversity of FreiburgFreiburgGermany
  2. 2.BIOSS Centre for Biological Signalling Studies and Institute of PhysicsUniversity of FreiburgFreiburgGermany

Personalised recommendations