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Effects of correlations between distances

For the derivation of the maximum likelihood estimators of the correlation dimension and entropy and their variances, we had to assume that the distances used are independent. The question now arises how many distances are independent if we have $P$ independent points in phase space. Ellner [Ellner, 1988] states that these distances must be calculated from non-overlapping collections of points, so that
\begin{displaymath}
N \leq P/2
\end{displaymath} (E.1)

where $P \leq L-(d-1)l$. However, Theiler [Theiler, 1990] states that $N \leq P$ can be used. We performed Monte Carlo simulations to investigate the effects of correlated distances, using time series obtained from the Hénon map. For different values of $N$ (500, 1000, 2000, 3000, 7000, 10000 and 20000), the dimension and entropy were estimated one thousand times. The embedding dimensions were chosen rather small ($d=3$ and $e=1$), because for higher values also the number of distances and the length of the time series should be increased, making this experiment very expensive with regard to computer time. Other parameter values are: $L=1000$, $L_s=1000$, $r_l = 0.01$ and $r_u = 0.07$.

The estimated dimensions, entropies, $CR$ values (sample variances times $N$) and their confidence intervals were plotted vs. $N$ in figure E.1 (figure E.1b). We estimated 95% confidence intervals for the mean dimension estimates in two ways:

  1. From the sample variance of the $M = 1000$ estimates (narrow bars):
    \begin{displaymath}
\sqrt{ \hat{\mbox{VAR}} \left\{ \hat{\nu} \right\} } \cdot
\frac{ Z_{\alpha/2} } {\sqrt M}
\end{displaymath} (E.2)

    where $Z_\alpha$ is the $\mbox{probability-}\alpha$ critical value of the standard normal distribution ( $Z_\alpha \approx 1.96$ with $\alpha = 0.05$).
  2. From the expressions of the asymptotic variances, but with averaged estimates of $\nu$ and $ N_l + N_s $ since we do not know $\tilde\phi$ (wide bars):
    \begin{displaymath}
\sqrt{ \left( \frac{ {\left( \overline{\hat{\nu}} \right)} ^...
...\nu}}}
\right) } \right) } \cdot \frac{Z_{\alpha/2}}{\sqrt M}
\end{displaymath} (E.3)

Confidence intervals for the mean entropy estimates can be written in a similar way. Confidence intervals for the $CR$ values were obtained by using eq. (5.67). We can compare the estimated dimensions, entropies and $CR$ values with the literature values and the estimated (since we do not know $\phi$) asymptotic variances (times $N$) respectively. We see that the dimension and entropy estimates have a systematic error (as expected, cf. figure 5.4). However, they seem to become more precise as the number of distances increases.

Increasing $N$ does decrease the fluctuations of the dimension and entropy estimates, but not as much as predicted by the formulas because the distances are becoming more and more correlated. From the figure we see that the variances are correct for values of $N$ between 1000 and 10000. The lower bound arises from the fact that $ N_l + N_s $ is getting below 50.


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Next: Bibliography Up: The identification of strange Previous: Estimation of the scaling   Contents
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