Correlation Integral Likelihood for Stochastic Differential Equations

  • Heikki Haario
  • Janne Hakkarainen
  • Ramona MaraiaEmail author
  • Sebastian Springer
Part of the MATRIX Book Series book series (MXBS, volume 2)


A new approach was recently introduced for the task of estimation of parameters of chaotic dynamical systems. Here we apply the method for stochastic differential equation (SDE) systems. It turns out that the basic version of the approach does not identify such systems. However, a modification is presented that enables efficient parameter estimation of SDE models. We test the approach with basic SDE examples, compare the results to those obtained by usual state-space filtering methods, and apply it to more complex cases where the more traditional methods are no more available.


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This work was supported by the Centre of Excellence of Inverse Problems, Academy of Finland.


  1. 1.
    Borovkova, S., Burton, R., Dehling, H.: Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation. Trans. Am. Math. Soc. 353(11), 4261–4318 (2001). MathSciNetCrossRefGoogle Scholar
  2. 2.
    Durbin, J., Koopman, S.J.: Time Series Analysis by State Space Methods. Oxford University Press, Oxford (2012)CrossRefGoogle Scholar
  3. 3.
    Haario, H., Laine, M., Mira, A., Saksman, E.: DRAM: Efficient adaptive MCMC. Stat. Comput. 16(4), 339–354 (2006). MathSciNetCrossRefGoogle Scholar
  4. 4.
    Haario, H., Kalachev, L., Hakkarainen, J.: Generalized correlation integral vectors: a distance concept for chaotic dynamical systems. Chaos: Interdiscipl. J. Nonlinear Sci. 25(6), 063102 (2015). MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hakkarainen, J., Ilin, A., Solonen, A., Laine, M., Haario, H., Tamminen, J., Oja, E., Järvinen, H.: On closure parameter estimation in chaotic systems. Nonlinear Process. Geophys. 19(1), 127–143 (2012). Scholar
  6. 6.
    Hakkarainen, J., Solonen, A., Ilin, A., Susiluoto, J., Laine, M., Haario, H., Järvinen, H.: A dilemma of the uniqueness of weather and climate model closure parameters. Tellus A Dyn. Meteorol. Oceanogr. 65(1), 20147 (2013). Scholar
  7. 7.
    Laine, M., Latva-Pukkila, N., Kyrölä, E.: Analysing time-varying trends in stratospheric ozone time series using the state space approach. Atmos. Chem. Phys. 14(18), 9707–9725 (2014).
  8. 8.
    Mbalawata, I.S., Särkkä, S., Haario, H.: Parameter estimation in stochastic differential equations with Markov chain Monte Carlo and non-linear Kalman filtering. Comput. Stat. 28(3), 1195–1223 (2013). MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ollinaho, P., Lock, S.J., Leutbecher, M., Bechtold, P., Beljaars, A., Bozzo, A., Forbes, R.M., Haiden, T., Hogan, R.J., Sandu, I.: Towards process-level representation of model uncertainties: stochastically perturbed parametrizations in the ECMWF ensemble. Quart. J. R. Meteorol. Soc. 143(702), 408–422 (2017). CrossRefGoogle Scholar
  10. 10.
    Rougier, J.: ‘Intractable and unsolved’: some thoughts on statistical data assimilation with uncertain static parameters. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci/ 371(1991) (2013).
  11. 11.
    Särkkä, S.: Bayesian Filtering and Smoothing. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
  12. 12.
    Solonen, A., Järvinen, H.: An approach for tuning ensemble prediction systems. Tellus A Dyn Meteorol Oceanogr 65(1), 20594 (2013). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Heikki Haario
    • 1
  • Janne Hakkarainen
    • 2
    • 3
  • Ramona Maraia
    • 1
    Email author
  • Sebastian Springer
    • 1
  1. 1.School of Engineering ScienceLappeenranta University of TechnologyLappeenrantaFinland
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  3. 3.Earth Observation, Finnish Meteorological InstituteEarth Observation, Finnish Meteorological InstituteHelsinkiFinland

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