next up previous contents
Next: Simulations using time series Up: Monte Carlo simulations Previous: Monte Carlo simulations   Contents

Simulations using ``ideal'' data

We first applied the derived estimators to distances drawn from the proposed distribution (5.1), with $\nu=1.22$, $K_{2}=0.325$ and $\tilde{\phi}=1.67$. These are literature values for the Hénon map (see the next subsection), except for $\tilde{\phi}$, which we estimated (see appendix D). Random numbers from the distribution were obtained from uniformly distributed ones (see section 5.7). We chose $r_l = 0.01$, and $r_u = 0.07$ for embedding dimensions 4 and 5 ($d=4$ and $e=1$); the number of distances drawn was between 800 and 20000. The correlation dimensions and their variances were estimated by evaluating eqs (5.26) and (5.37) (``single'' estimates, denoted by $\nu_d$ and $\nu_{d+e}$), (5.47) and (5.53) (``double'' estimates), the correlation entropies and their variances by eqs (5.49) and (5.55). For convenience we computed the variances times $N$ (denoted by $CR$) since the asymptotic variances times $N$ are constants. This was done for $M = 1000$ Monte Carlo trials, using the MLDK2SIM program (see section 5.7). The estimated dimensions and entropies were averaged and sample variances and confidence intervals estimated. These quantities are presented as a function of the number of distances in table 5.2 and figure 5.1 (figure 5.1b).



N $\overline{\nu_d}$ $CR(\nu_d)$ $\overline{(N_l+N_s)}_d$ $\overline{\nu_{d+e}}$ $CR(\nu_{d+e})$ $\overline{(N_l+N_s)}_{d+e}$ $\overline{\nu}$ $CR(\nu)$ $\overline{K_2}$ $CR(K_2)$
800 1.307 151.0 14.178 1.373 444.8 10.251 1.266 74.1 0.332 146.5
1000 1.283 116.0 17.750 1.310 244.0 12.808 1.252 63.5 0.339 159.9
2000 1.250 108.0 35.797 1.260 150.3 25.395 1.235 54.2 0.347 135.0
3000 1.243 94.4 53.437 1.254 137.5 38.442 1.235 49.6 0.334 135.2
4000 1.235 98.4 71.059 1.242 128.7 51.118 1.231 55.6 0.333 140.3
5000 1.232 97.6 89.084 1.233 137.5 64.556 1.226 54.2 0.324 121.9
6000 1.234 94.3 106.380 1.235 146.1 76.058 1.228 54.5 0.337 135.5
7000 1.231 98.8 124.845 1.236 128.2 89.536 1.229 54.0 0.334 130.6
8000 1.228 96.8 141.964 1.231 122.3 102.895 1.225 56.5 0.323 130.0
9000 1.228 102.7 160.058 1.228 131.4 115.423 1.224 54.0 0.328 143.4
10000 1.230 95.4 177.884 1.221 124.6 128.684 1.223 55.0 0.325 136.0
20000 1.225 99.4 354.566 1.222 137.5 257.099 1.222 56.2 0.322 137.1

Table 5.2: Simulations using ideal data (see text). All quantities (except $N$) are estimated ones. Theoretical values for $CR$ can be obtained using the expressions for the asymptotic variances: $CR(\nu_d) \approx 92.5$, $CR(\nu_{d+e}) \approx 128.0$, $CR(\nu) \approx 53.7$ and $CR(K_2) \approx 132.3$).

In figure 5.1 (figure 5.1b) we see that the expressions for the variances are accurate for rather small values of $N$, while the bias, in particular of the estimator of the dimension, persists up to the highest values of $N$ (cf. eq. (5.13)). From the figure we conclude that for $N=3000$ the sample variance is sufficiently close to the asymptotic variance. However, the number of $N_{l}+N_{s}$ is more useful since it appears in the expressions for the variances with the estimators of the parameters substituted. (the factors $1/N_d$ and $1/N_{d+e}$ in eq. (5.57) can usually be neglected). Therefore, using table 5.2, we conclude that the expressions of the variances are reasonably accurate if $N_{l}+N_{s}$ is larger than about 50. We did similar experiments with non-equal scaling regions (since for equal values of $r_{l,d}$ and $r_{l,d+e}$ the factor $\ln^2$ in eq. (5.55) becomes zero). We concluded that the expression for the variance of the estimator of the entropy is also valid in this case.

In the next subsection, we perform Monte Carlo simulations using time series from the Hénon map for different embedding dimensions. To compare these results to the case where we draw distances from the distribution, we used the MLDK2MCE program (see section 5.7) with the same parameters as above, but for embedding dimensions $d = 1,\;2,\;,\ldots\;20$, $e=1$, 2, and 3, $N=100,000$ and $M=100$ (figure 5.2 and table 5.3). As the embedding dimension increases, the number $ N_l + N_s $ decreases so that the bias (especially for the dimension estimators) and the variances increase.


next up previous contents
Next: Simulations using time series Up: Monte Carlo simulations Previous: Monte Carlo simulations   Contents
webmaster@rullf2.xs4all.nl