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The correlation integral

We start with a brief description of the correlation integral method, since it is the basis of our maximum likelihood approach. To characterize a dynamical system from a time series $x(t)$, first the phase space of the system is reconstructed with the method of time delayed coordinates [Takens, 1981,Packard et al., 1980], i.e. with $M$ vectors
\begin{displaymath}
\vec{x}(t_{i}) = [x(t_{i}), \; x(t_{i} + l \Delta t), \; \ldots ,\; x(t_{i} + (d-1) l \Delta t)]
\end{displaymath} (1)

where $\Delta t$ is the sample time, $d$ is the embedding dimension and $l$ is an appropriate lag. The correlation integral $C(r)$ is defined as
\begin{displaymath}
C(r) = M^{-2} \; ({\mbox{number of pairs }} (i,j) \; \mbox{w...
...\parallel \vec{x}(t_{i}) - \vec{x}(t_{j}) \parallel \; \leq r)
\end{displaymath} (2)

where $\parallel \cdots \parallel$ denotes a norm. It is assumed that distances $r$ between two randomly chosen points in phase space obey the cumulative probability distribution function [Grassberger and Procaccia, 1983b]
\begin{displaymath}
P(r) = \phi r^{\nu} = \tilde{\phi} \exp (-d l \Delta t K_{2}) r^{\nu}
\end{displaymath} (3)

and that $C(r) \sim P(r)$ for sufficiently small $r$ and large $M$. We also assume that $\phi$ and $\tilde{\phi}$ do not depend on $r$ (see, however, Theiler, 1988) and that $\tilde{\phi}$ does not depend on the embedding dimension. The correlation dimension $\nu$ is now given by [Eckmann and Ruelle, 1985]
\begin{displaymath}
\nu = \lim_{r \rightarrow 0} \lim_{M \rightarrow \infty}
\frac{\ln C(r)} {\ln r}
\end{displaymath} (4)

if $d$ is large enough [Takens, 1981]. The correlation entropy $K_2$ is given by
\begin{displaymath}
\lim_{r \rightarrow 0} \lim_{d \rightarrow \infty} \frac{1}{d} \lim_{M \rightarrow \infty}
\ln C(r) = - l \Delta t K_{2}
\end{displaymath} (5)

With experimental data, a ``scaling region'' $]r_{l},r_{u}]$ must be identified, for which the distribution (3) is valid. The dimension can be estimated by the slope of a straight line through a number of points $( \ln r_{i}, \ln C(r_{i}) )$ in the scaling region. The entropy can be estimated by observing the behaviour of $C(r)$ as a function of $d$ but using equation (5) directly may result in slow convergence. Therefore one usually studies the quotient of two correlation integrals, at embedding dimensions $d$ and $d+e$ [Grassberger and Procaccia, 1983a] :

\begin{displaymath}
\ln \left( \frac{C_{d}(r)} {C_{d+e}(r)} \right) = e l \Delta t K_{2}
\end{displaymath} (6)

We will now show how to obtain maximum likelihood estimators for both $\nu$ and $K_2$, together with their asymptotic variances, under the assumption that the finite number of distances is the only source of error.


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Next: Maximum likelihood methods Up: mlbmb Previous: Introduction
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