# Berlin 2008 – wissenschaftliches Programm

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# DY: Fachverband Dynamik und Statistische Physik

## DY 30: Nonlinear stochastic systems

### DY 30.10: Vortrag

### Freitag, 29. Februar 2008, 12:30–12:45, MA 001

**Reconstruction of nonlinear dynamics from discrete observations** — •Andreas Ruttor and Manfred Opper — Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Franklinstr. 28/29, 10587 Berlin, Germany

Nonlinear dynamical systems are often described by diffusion models based on stochastic differential equations (SDEs). As long as the distance between observations is small, drift and diffusion can be calculated directly and used to determine unknown parameters. Otherwise, both the state of the system between observations and the parameters of the SDEs have to be estimated. However, Markov chain Monte Carlo based methods used for that purpose can be very time consuming. As an alternative a fast approximate approach is proposed. By solving the backward Fokker-Planck equation of the diffusion model in the weak noise limit, it is possible to obtain the drift of the posterior SDE directly. Afterwards the posterior statistics can be computed either by applying the weak noise limit again (leading to an approximate Gaussian posterior process), or more simply by simulating many samples of the posterior SDE. Parameter estimation is based on the negative log-likelihood of the data. Minimizing an upper bound of this quantity, which can also be calculated in the state inference algorithm, leads to type II maximum likelihood estimates of unknown system parameters. Results obtained in the case of reaction systems indicate that this approach works well.