Abstract
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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References
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). https://doi.org/10.1090/S0002-9947-01-02819-7
Durbin, J., Koopman, S.J.: Time Series Analysis by State Space Methods. Oxford University Press, Oxford (2012)
Haario, H., Laine, M., Mira, A., Saksman, E.: DRAM: Efficient adaptive MCMC. Stat. Comput. 16(4), 339–354 (2006). https://doi.org/10.1007/s11222-006-9438-0
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). http://dx.doi.org/10.1063/1.4921939
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). http://dx.doi.org/10.5194/npg-19-127-2012
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). http://dx.doi.org/10.3402/tellusa.v65i0.20147
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). https://doi.org/10.5194/acp-14-9707-2014. https://www.atmos-chem-phys.net/14/9707/2014/
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). https://doi.org/10.1007/s00180-012-0352-y
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). http://dx.doi.org/10.1002/qj.2931
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). https://doi.org/10.1098/rsta.2012.0297. http://rsta.royalsocietypublishing.org/content/371/1991/20120297
Särkkä, S.: Bayesian Filtering and Smoothing. Cambridge University Press, Cambridge (2013)
Solonen, A., Järvinen, H.: An approach for tuning ensemble prediction systems. Tellus A Dyn Meteorol Oceanogr 65(1), 20594 (2013). http://dx.doi.org/10.3402/tellusa.v65i0.20594
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This work was supported by the Centre of Excellence of Inverse Problems, Academy of Finland.
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Haario, H., Hakkarainen, J., Maraia, R., Springer, S. (2019). Correlation Integral Likelihood for Stochastic Differential Equations. In: de Gier, J., Praeger, C., Tao, T. (eds) 2017 MATRIX Annals. MATRIX Book Series, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-04161-8_3
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DOI: https://doi.org/10.1007/978-3-030-04161-8_3
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