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On Selecting Models for Nonlinear Time Series
 Physica D
, 1995
"... Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintainin ..."
Abstract

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Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series. 1 The Model Selection Problem As our understanding of chaotic and other nonlinear phenomena has grown, it has become apparent that linear models are inadequate to model most dynamical processes. Nevertheless, linear models...
Triangulating Noisy Dynamical Systems
"... Triangulation and tesselation methods have proven successful for reconstructing dynamical systems which have approximately the same dynamics as given trajectories or time series. Such reconstructions can produce other trajectories with similar dynamics, so giving a system on which one can conduct ..."
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Triangulation and tesselation methods have proven successful for reconstructing dynamical systems which have approximately the same dynamics as given trajectories or time series. Such reconstructions can produce other trajectories with similar dynamics, so giving a system on which one can conduct experiments, but can also be used to locate equilibria and determine their types, carry out bifurcation studies, estimate state manifolds and so on. In the past, reconstruction by triangulation and tesselation methods has been restricted to lownoise or nonoise cases. This paper shows how to construct triangulation models for noisy systems; the models can then be used in the same ways as models of noisefree systems. 1 Introduction An important part of controlling systems is modeling them. The system identification and modeling problem is wellstudied for linear systems, but there seem to be few results for strongly nonlinear systems. In this paper we present an approach that gener...