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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 independent points in phase space. Ellner (1988) states that these distances must be calculated from non-overlapping collections of points, so that
 (18)

where . However, Theiler (1990) states that can be used. We performed Monte Carlo simulations to investigate the effects of correlated distances, using time series obtained from the Hénon map. Pairs of random indices for the vectors (equation (1)) were drawn and the distances between these vectors calculated with the supremum norm. For different values of , the dimension and entropy were estimated one thousand times. The embedding dimensions were chosen rather small ( and ), 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.

The averaged estimated dimension and entropies were plotted vs. in figure 1a and figure 1b with 95% confidence intervals.

We see that the entropy estimates have large systematic errors, due to the low value of the embedding dimension. This however, is of no consequence for our purpose to study the behaviour of the variances. We estimated 95% confidence intervals in two ways: 1) from the sample variance of the 1000 estimates (narrow bars), and 2) from the expressions of the asymptotic variances, but with averaged estimates of and (wide bars) since we do not know . Increasing 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 between 1000 and 10000. The lower bound arises from the fact that is getting below 50.

Correlations due to the deterministic nature of the dynamics are assumed to be of no consequence since for long time series the invariant probability measure on the attractor is approached. Furthermore, dynamical correlations in short time series can be suppressed by a method due to Theiler (1986) .

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