Computational Statistics

, Volume 32, Issue 4, pp 1621–1643 | Cite as

Fast derivatives of likelihood functionals for ODE based models using adjoint-state method

  • Valdemar Melicher
  • Tom Haber
  • Wim Vanroose
Original Paper


We consider time series data modeled by ordinary differential equations (ODEs), widespread models in physics, chemistry, biology and science in general. The sensitivity analysis of such dynamical systems usually requires calculation of various derivatives with respect to the model parameters. We employ the adjoint state method (ASM) for efficient computation of the first and the second derivatives of likelihood functionals constrained by ODEs with respect to the parameters of the underlying ODE model. Essentially, the gradient can be computed with a cost (measured by model evaluations) that is independent of the number of the ODE model parameters and the Hessian with a linear cost in the number of the parameters instead of the quadratic one. The sensitivity analysis becomes feasible even if the parametric space is high-dimensional. The main contributions are derivation and rigorous analysis of the ASM in the statistical context, when the discrete data are coupled with the continuous ODE model. Further, we present a highly optimized implementation of the results and its benchmarks on a number of problems. The results are directly applicable in (e.g.) maximum-likelihood estimation or Bayesian sampling of ODE based statistical models, allowing for faster, more stable estimation of parameters of the underlying ODE model.


Sensitivity analysis Ordinary differential equations Gradient Hessian Statistical computing Mathematical statistics Algorithm 



We would like to thank Xavier Woot de Trixhe from Janssen Pharmaceutica for numerous very interesting discussions on PK/PD, virology, biological pathways modeling, NLMEMs and on life in general. They were an important source of motivation and provided a view from a different perspective. And we would like to thank the anonymous reviewers for their substantial input, enhancing the quality of the paper.

Supplementary material

180_2017_765_MOESM1_ESM.pdf (109 kb)
Supplementary material 1 (pdf 108 KB)
180_2017_765_MOESM2_ESM.r (4 kb)
Supplementary material 2 (R 4 KB)


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of AntwerpAntwerpBelgium
  2. 2.Expertise Centre for Digital MediaHasselt UniversityDiepenbeekBelgium

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