N | ||||||||||

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 ) are estimated ones. Theoretical values for can be
obtained using the expressions for the asymptotic variances:
,
,
and
).

In figure 5.1 (figure 5.1b) we see that the expressions for the variances are accurate for rather small values of , while the bias, in particular of the estimator of the dimension, persists up to the highest values of (cf. eq. (5.13)). From the figure we conclude that for the sample variance is sufficiently close to the asymptotic variance. However, the number of is more useful since it appears in the expressions for the variances with the estimators of the parameters substituted. (the factors and 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 is larger than about 50. We did similar experiments with non-equal scaling regions (since for equal values of and the factor 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 , , 2, and 3, and (figure 5.2 and table 5.3). As the embedding dimension increases, the number decreases so that the bias (especially for the dimension estimators) and the variances increase.