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We consider the problem of automatically extracting simplified models out of complex high--dimensional and t
ime--dependent data. The simplified model is given by a linear Langevin equation with time--varying coeffici
ents. The reduced model may still be high--dimensional, but it is physically intuitive and much easier to in
terpret than the original data. In particular we can distinguish whether dynamical effects are influenced b
y friction, noise, or deterministic motion. The parameters for the reduced model are obtained by a robust an
d efficient numerical predictor--corrector scheme which relies on analytical solutions to a maximum-likeliho
od problem provided the time steps between successive observations are not too large. If the data set is ver
y heterogeneous the time series is better described not by a single model, but by a collection of reduced mo
dels. This scenario is accounted for by embedding the parameter estimation procedure into the framework of h
idden Markov models, i.e., we decompose the data into several subsets, each of which gives rise to an approp
riate linear Langevin model. The switching between the local model is done by a Markov jump process. The opt
imal 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 foc
us on the numerical error as a function of the time step of the observation sequence.
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.
he generalized Langevin equation is useful for modeling a wide
range of physical processes. Unfortunately its parameters,
especially the memory function, are difficult to determine for
nontrivial processes. In this paper, relations between a
time-discrete generalized Langevin model and discrete multivariate
autoregressive (AR) or autoregressive moving average models (ARMA)
are established. This allows a wide range of discrete linear
methods known from time series analysis to be applied. In
particular, the determination of the memory function {\it via} the
order of the respective AR or ARMA model is addressed. The method
is illustrated on a one-dimensional test system and subsequently
applied to the molecular dynamics of a biomolecule which exhibits
an interesting relationship between the solvent method used, the
molecular conformation and the depth of the memory.
We report on a novel approach to the automatic identification
of metastable states from long term simulation of complex
molecular systems. The new approach is based on a hierarchical concept
of metastability: metastable states are understood as subsets of
state or configuration space from which the dynamics exits only very rarely;
subsets with the smallest exit probabilities are of most interest, their
further decomposition then may reveal subsets from which exiting
is less but comparably difficult for the system under investigation.
The article gives a survey of the theoretical foundation of
the approach and its algorithmic realization that generalizes
the well-known concept of Hidden Markov Models.
The performance of the resulting algorithm are illustrated by
application to a 100 ns simulation of penta-alanine with explicit water.
We demonstrate the resulting metastable states allow to
reveal the conformation dynamics of the moelcule.