next up previous contents
Next: Two analytical examples Up: THE CORRELATION INTEGRAL Previous: The original definition and   Contents

Methods for estimating the dimension and entropy spectra

It is assumed that [Pawelzik and Schuster, 1987]
\begin{displaymath}
\lim_{r \rightarrow 0} \lim_{d \rightarrow \infty} \lim_{N \...
...rrow \infty}
C_{d}^{q}(r,N) \sim r^{D_{q}} \exp(-d \tau K_{q})
\end{displaymath} (4.5)

where $C_d^q(r,N)$ is $C_d^q(r,N_{ref}=N,N_{w}=N)$ and $\tau = l \Delta t$ (see section 2.5). For example, $C_d^2(r,N)$ is an estimate of $\sum p_i^2$ in eqs (2.3) and (2.5). From these definitions we see that the correlation integral should scale as both $r^{D_2}$ and $\exp(-d \tau K_2)$. For more details, see [Schuster, 1988].

If we calculate correlation integrals for different values of $d$ and $q$, we are able to estimate both the dimension spectrum $D_{q}$ and the entropy spectrum $K_{q}$. We assume that the scaling law eq. (4.5) is also valid for finite $r$, $d$ and $N$.

In practice, a log-log plot of the correlation integral vs. the distances for which it is calculated usually shows three regions: a linear region where eq. (4.5) holds, the ``scaling region''; the lower region is distorted by fluctuations due to noise or the small number of points and the upper region is distorted due to the finite size of the attractor. The slope of the scaling region will equal $D_q$ if the embedding dimension is large enough (see section 2.5). The dimension spectrum is usually estimated by fitting straight lines using a linear squares method. However, the standard expression for the variance of the slope is invalid since $C(r)$ and $C(r + \Delta r)$ are correlated so that the variance will be underestimated [Theiler, 1990,Holzfuss and Mayer-Kress, 1986]. The entropy spectrum is given by [Eckmann and Ruelle, 1985]

\begin{displaymath}
K_q = - \lim_{r \rightarrow 0} \lim_{d \rightarrow \infty}
\...
...1} \frac{1}{d\tau}
\lim_{N \rightarrow \infty} \ln C_d^q(r,N)
\end{displaymath} (4.6)

The entropy $K_q$ can be estimated by observing the behaviour of $C_d^q(r)$ as a function of $d$ but using eq. (4.6) directly may result in slow convergence. However, note that successive (as a function of $d$) log-log plots of the correlation integrals are displaced by a factor $\tau K_q$. Hence, the $K_q$ can be estimated by
\begin{displaymath}
\hat{K}_{q} = \frac{1}{e \tau} \ln \left( \frac{C_{d}^{q}(r)}{C_{d+e}^q(r)} \right)
\end{displaymath} (4.7)

for some $r$ within the scaling region. By increasing $e$, fluctuations in the displacement factor can be reduced. For example, Grassberger and Procaccia [Grassberger and Procaccia, 1983a,Grassberger and Procaccia, 1984] use $e=4$. It would also be possible to fit straight lines through a number of correlation integrals under the condition that these lines are displaced by the factor containing the entropy.


next up previous contents
Next: Two analytical examples Up: THE CORRELATION INTEGRAL Previous: The original definition and   Contents
webmaster@rullf2.xs4all.nl