where . However, Theiler [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. For different values of (500, 1000, 2000, 3000, 7000, 10000 and 20000), 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. Other parameter values are: , , and .

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

- From the sample
variance of the estimates (narrow bars):

(E.2) - From the expressions of the
asymptotic variances, but with averaged estimates of and
since we do not know (wide bars):

(E.3)

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.