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The original definition and some modifications

Grassberger and Procaccia defined the correlation integral as [Grassberger and Procaccia, 1983b]:
\begin{displaymath}
C_{d}(r) = \lim_{N \rightarrow \infty} N^{-2} \times
\left\{...
...l \vec{x}(t_{i}) - \vec{x}(t_{j}) \parallel \; \leq r \right\}
\end{displaymath} (4.1)

where $\vec{x}(t)$ denote $d$-dimensional vectors (points in phase space) constructed from a time series using the method of delays (see eq. (2.6)), $N$ denotes the number of points and $\parallel \cdots \parallel$ denotes a norm.

The correlation dimension (usually denoted by $\nu$ instead of $D_2$) is then defined by the limit [Eckmann and Ruelle, 1985]:

\begin{displaymath}
\nu = \lim_{r \rightarrow 0}
\frac{\ln C(r)} {\ln r}
\end{displaymath} (4.2)

for a sufficiently large embedding dimension (see section 2.5). The Renyi dimension spectrum can be assessed for integer $q$ by using $q$-tuples of points [Grassberger and Procaccia, 1984]. For any $q$ it can be assessed by averaging, using reference points [Pawelzik and Schuster, 1987,Schuster, 1988]. This generalization can be written as:
\begin{displaymath}
C_{d}^{q}(r,N_{ref},N_{w}) =
{\left[ \frac{1}{N_{ref}} \sum_...
...c{x}(t_{j}) \parallel)
\right]}^{q-1} \right]}^{\frac{1}{q-1}}
\end{displaymath} (4.3)

where the counting of the distances is symbolized by the Heaviside step function $\theta$, $N_{ref}$ is the number of reference points and $N_{w}$ is the number of distances calculated for every reference point. The Heaviside step function is defined by:
\begin{displaymath}
\theta(x) = \left\{ \begin{array}{ll}
1 & \mbox{for } x \ge...
...nd{array}\right. %% Well, with probability 1 this is wrong...!
\end{displaymath} (4.4)

To avoid spurious correlations, two points should not be too close in time [Theiler, 1986]. Therefore, let $\left\vert j-i \right\vert > W$, where the number $W$ is called the Theiler correction (see also section 6.5).


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Next: Methods for estimating the Up: THE CORRELATION INTEGRAL Previous: Introduction   Contents
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