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NONLINEAR TIME SERIES ANALYSIS |  |
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The theory of nonlinear dynamical systems provide new ways and methods for the characterization of irregular time series data. The basis of the new technique is based on the important Taken's Embedding Theorem[1] which allows reconstruction of the attractor in the time delayed embedded space and preserving its topological characteristics. The reconstruction of the attractor is done from a finite time series of the observation of a single variable. In what follows, an attempt is made to give a simple description of the method of delay time embedding (basically explaining the jargon and a mind boggling number of methods) is given. The method of nonlinear time series analysis using delay time embedding, it is important to realize that method relies on a choice of good delay time and the embedding dimension. The embedding theorem is silent on both issues (actually there is an upper bound provided for the embedding dimension) and presentation here is based on the algorithms for the estimation of delay time(autocorrelation function, mutual information) and embedding dimension(false nearest neighbor). At no time any claim to the originality of the algorithm is even implied. Check the few references that are given here and the references therein.
Most observational data reflect just a few of the many physical variables of a system and measurements of all variables are rarely possible. However, this difficulty can be overcome if the variables are nonlinearly coupled, in which case the time delay embedding technique [1,2] can be used to reconstruct the phase or state space from the time series data. In this technique a multi-dimensional embedding space is constructed from the time series data, and a point in it represents the state of the system at a given time.
The time-delay coordinate vector x is
constructed from single scalar measurement y(t) according to
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