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Blind model reduction for high-dimensional time-dependent data

Please always quote using this URN:urn:nbn:de:0296-matheon-3171
  • We consider the problem of automatically extracting simplified models out of complex high-dimensional and time-dependent data. The simplified model is given by a linear Langevin equation with time-varying coefficients. The reduced model may still be high-dimensional, but it is physically intuitive and much easier to interpret than the original data. In particular we can distinguish whether certain dynamical effects are influenced by friction, noise, or systematic drift. The parameters for the reduced model are obtained by a robust and efficient numerical predictor-corrector scheme which relies on analytical solutions to a maximum-likelihood problem provided the time steps between successive observations are not too large. Our approach emphasizes the specific hypoelliptic structure of the Langevin equation given high-dimensional observation data, and therefore can be considered as complemetary to the procedure recently proposed in \emph{Horenko et al. (submitted SIAM MMS, 2007)} by one of the authors, or to the problem of incomplete (one-dimensional) observations \emph{Pokern et al. (submitted to JRSSB, 2007)}. If the data set is very heterogeneous the time series is better described not by a single model, but by a collection of reduced models. This scenario is accounted for by embedding the parameter estimation procedure into the framework of hidden Markov models which it is particularly suited to treat high-dimensional data. That is, we decompose the data into several subsets, each of which gives rise to an appropriate linear Langevin model, where the switching between the local model is done by a Markov jump process. The optimal decomposition into submodels can then be regarded as one global Langevin model with piecewise constant coefficients. We illustrate the performance of the algorithm by means of several examples. Especially we focus on the numerical error as a function of the time step of the observation sequence.

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Metadaten
Author:Illia Horenko, Carsten Hartmann
URN:urn:nbn:de:0296-matheon-3171
Referee:Christof Schütte
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2008/03/13
Release Date:2007/08/14
Institute:Freie Universität Berlin
Zuse Institute Berlin (ZIB)
Preprint Number:478
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