(19) | |||

contain the true values. Here denotes a dimension or entropy estimate, the square root of its (estimated) asymptotic variance, the critical value of the standard normal distribution and the size of the test. For the size of a test we always used . The asymptotic variances were estimated by substitution of the estimated values of , and in equations (14), (11) and (12). This procedure was repeated 100 times. The scaling regions were chosen by visual inspection, and justified by goodness-of-fit tests. Due to the supremum norm, the scaling regions hardly move to higher distance ranges as the embedding dimension increases, so we used the same and for every embedding dimension. Furthermore, we checked that there were no significant differences in successive ``single'' dimension estimates (equation (13)).

Table 1a: Averaged ``single'' dimension (
),
standard deviation (
)
and coverage frequency (CF) estimates for the Hénon map
as functions of the embedding dimension.
, averaged number of distances below
(, , , , and
).

two-sided CF | lower CF | upper CF | ||||

1 | 0.942 | 0.009 | 0.00 | 1.00 | 0.00 | 15241.2 |

2 | 1.195 | 0.018 | 0.66 | 0.99 | 0.48 | 6587.9 |

3 | 1.256 | 0.021 | 0.64 | 0.45 | 1.00 | 4227.4 |

4 | 1.203 | 0.026 | 0.89 | 1.00 | 0.80 | 2547.1 |

5 | 1.186 | 0.028 | 0.77 | 1.00 | 0.69 | 1711.0 |

6 | 1.201 | 0.035 | 0.94 | 0.98 | 0.91 | 1145.6 |

7 | 1.199 | 0.044 | 0.93 | 0.99 | 0.88 | 789.8 |

8 | 1.205 | 0.050 | 0.97 | 0.98 | 0.94 | 553.6 |

9 | 1.201 | 0.065 | 0.95 | 0.98 | 0.92 | 388.8 |

10 | 1.231 | 0.078 | 0.95 | 0.96 | 0.96 | 281.8 |

11 | 1.235 | 0.095 | 0.96 | 0.93 | 0.95 | 194.8 |

12 | 1.229 | 0.128 | 0.91 | 0.95 | 0.90 | 138.9 |

13 | 1.199 | 0.150 | 0.89 | 0.94 | 0.89 | 95.4 |

14 | 1.171 | 0.141 | 0.90 | 0.99 | 0.88 | 70.9 |

15 | 1.206 | 0.177 | 0.96 | 0.97 | 0.93 | 51.0 |

16 | 1.185 | 0.240 | 0.90 | 0.97 | 0.87 | 36.4 |

17 | 1.213 | 0.244 | 0.93 | 1.00 | 0.88 | 26.1 |

18 | 1.292 | 0.361 | 0.95 | 0.98 | 0.92 | 18.8 |

19 | 1.272 | 0.415 | 0.93 | 0.98 | 0.87 | 14.1 |

20 | 1.229 | 0.454 | 0.91 | 1.00 | 0.88 | 10.5 |

Table 1b: Averaged ``double'' dimension, standard deviation
and coverage frequency estimates for the Hénon map;
.

two-sided CF | lower CF | upper CF | |||

4 | 1.202 | 0.020 | 0.87 | 0.99 | 0.81 |

6 | 1.202 | 0.028 | 0.92 | 1.00 | 0.88 |

8 | 1.213 | 0.043 | 0.95 | 0.98 | 0.92 |

10 | 1.227 | 0.070 | 0.90 | 0.96 | 0.90 |

12 | 1.204 | 0.093 | 0.89 | 0.97 | 0.88 |

14 | 1.164 | 0.109 | 0.90 | 1.00 | 0.84 |

In table 1 the results for the Hénon map are summarized.
Noting that we have to divide the standard deviation by the square
root of the number of Monte Carlo trials
to obtain the standard error of the mean,
we observe that
the dimension estimates (tables 1a and 1b) generally show small but statistically
significant deviations from the literature value.
Since is large enough, these
systematic errors are not due to the use of the maximum likelihood
estimator per sé,
but to another source, probably lacunarity (non-constant
[Theiler, 1988]).
In general, the two-sided coverage frequencies are too low
and the lower and upper coverage frequencies are asymmetric;
these should all be .
Since the asymptotic variances are accurate,
these results are most probably
due to systematic errors in the estimated correlation
dimension.
This is further illustrated by the fact that
if we use the mean dimension estimate
from the Monte Carlo trials
as the ``true'' value, the
coverage frequencies fluctuate closely around 0.95.
We emphasize that in contrast to Ellner's numerical results
[Ellner, 1988], our
computed confidence intervals are not ``conservative''.
The entropy estimates converge to the literature value
rather slowly (tables 1c and 1d).

Table 1c: Averaged entropy (
), standard deviation
and coverage frequency estimates for the Hénon map;
.

two-sided CF | lower CF | upper CF | |||

4 | 0.3980 | 0.0311 | 0.33 | 0.23 | 1.00 |

6 | 0.3718 | 0.0496 | 0.79 | 0.70 | 0.98 |

8 | 0.3535 | 0.0728 | 0.90 | 0.87 | 0.95 |

10 | 0.3709 | 0.1061 | 0.89 | 0.86 | 0.98 |

12 | 0.3785 | 0.1551 | 0.90 | 0.87 | 0.95 |

14 | 0.3370 | 0.2034 | 0.96 | 0.94 | 0.95 |

Table 1d: Averaged entropy, standard deviation
and coverage frequency estimates for the Hénon map;
.

two-sided CF | lower CF | upper CF | |||

4 | 0.3905 | 0.0131 | 0.00 | 0.00 | 1.00 |

6 | 0.3604 | 0.0210 | 0.53 | 0.45 | 0.99 |

8 | 0.3490 | 0.0325 | 0.87 | 0.78 | 0.97 |

10 | 0.3627 | 0.0450 | 0.86 | 0.79 | 1.00 |

In table 2a we present the results for the logistic map.
Note that the number of distances used did not satisfy equation (18).
Nevertheless the results for the coverage frequencies indicate that
the variances are precise.
Similar simulations, in which equation (18)
was satisfied, do not yield
better coverage frequencies as can be seen from table 2b.
Table 3 shows the results for the sine wave: the estimated entropies
do not significantly differ from zero, except at embedding dimension 4.

Table 2a: Averaged entropy, standard deviation
and coverage frequency estimates for the logistic map
(, , , , ,
and ).

two-sided CF | lower CF | upper CF | |||

4 | 0.6774 | 0.0455 | 0.92 | 0.98 | 0.88 |

6 | 0.6743 | 0.0915 | 0.94 | 0.97 | 0.92 |

8 | 0.7172 | 0.1902 | 0.95 | 0.93 | 0.95 |

Table 2b: Averaged entropy, standard deviation
and coverage frequency estimates for the logistic map
(, , , , ,
and ).

two-sided CF | lower CF | upper CF | |||

4 | 0.6787 | 0.0449 | 0.95 | 0.97 | 0.90 |

6 | 0.6891 | 0.0780 | 0.97 | 0.98 | 0.97 |

8 | 0.7063 | 0.1990 | 0.91 | 0.91 | 0.96 |

Table 3: Averaged entropy, standard deviation
and coverage frequency estimates for the sine wave
(, , (with Theiler
correction 3 (Theiler, 1986)), , ,
and ).

two-sided CF | lower CF | upper CF | |||

4 | 0.0056 | 0.0211 | 0.96 | 0.92 | 0.97 |

6 | 0.0002 | 0.0218 | 0.93 | 0.94 | 0.93 |

8 | -0.0042 | 0.0212 | 0.96 | 0.95 | 0.94 |