Incollection,
Estimation of Drift and Diffusion function in presence of measurement noise
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
The understanding of complex systems greatly has benefited from the
concept of order parameters, that obey stochastic partial differential
equations 1. Recently, a method for the direct
estimation of these equation from measured data sets has been proposed
2. However, this procedure involves the estimation of
the moments of the transition probability density functions (pdfs) in
the limit of infinitesimal small increments in time, that frequently
are not accessible from discrete measurements. Moreover, measurement
noise seriously impacts the transition pdfs at small time increments
and, therefore, tampers the results of the estimation procedure.
This contribution addresses the progress of two recent works with
respect to this shortcoming. First, an iterative method was proposed,
that avoids the limiting procedure and, therefore, is less sensitive
to measurement noise 3. It is based on the iterative
optimisation of the transition pdfs in reference to the pdfs, that
directly can be obtained form the measured data set. Recently, the
conformance of this procedure with maximum likelihood methods could be
demonstrated 4.
Second, the former method could be extended for noisy data
5. Thereby, the increasing impacts of measurement
noise on the transition pdfs at small time increments can be utilised
for the simultaneous estimation of the noise amplitude and the
process' dynamics. For the Ornstein-Uhlenbeck process, closed
expressions for the estimation procedure could be derived, that permit
the proper reconstruction even in case of high noise amplitudes.\\
1) H.~Haken. Synergetics.
Springer Series in Synergetics. Springer-Verlag, Berlin,
2004. Introduction and advanced topics, Reprint of the
third (1983) edition Synergetics and the first (1983)
edition Advanced synergetics.\\
2) S.~Siegert, R.~Friedrich, and J.~Peinke.
Analysis of datasets of stochastic systems. \newblock
Physics Letters A, 243:275--280, 1998.\\
3) D.~Kleinhans, R.~Friedrich, A.~Nawroth, and
J.~Peinke. An iterative procedure for the estimation of
drift and diffusion coefficients of langevin processes. \newblock
Phys Lett A, 346:42--46, 2005.\\
4) D.~Kleinhans and R.~Friedrich. \newblock
Maximum likelihood estimation of drift and diffusion functions.
(to be published in Phys. Lett. A), preprint
available at http://arxiv.org/abs/physics/0611102.\\
5) F.~Boettcher, J.~Peinke, D.~Kleinhans,
R.~Friedrich, P.G.~Lind, and M.~Haase. Reconstruction
of complex dynamical systems affected by strong measurement noise.
Phys. Rev. Lett., 97:090603, 2006.
