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Next: An application Up: Simulations using time series Previous: Effects of correlations between

Coverage frequencies

In this section we present results from Monte Carlo simulations, using time series from the given models, for different values of the embedding dimension. Furthermore, following Ellner, we computed coverage frequencies, i.e. the fractions of trials for which the confidence intervals
$\displaystyle \mbox{two-sided:}$   $\displaystyle [\hat{q}-Z_{\alpha/2}\sigma, \;
\hat{q}+Z_{\alpha/2}\sigma]$ (19)
$\displaystyle \mbox{lower:}$   $\displaystyle [\hat{q}-Z_{\alpha}\sigma, \; \infty[$  
$\displaystyle \mbox{upper:}$   $\displaystyle ]-\infty, \; \hat{q}+Z_{\alpha}\sigma]$  

contain the true values. Here $\hat{q}$ denotes a dimension or entropy estimate, $\sigma$ the square root of its (estimated) asymptotic variance, $Z_{\alpha}$ the $\mbox{probability-}\alpha$ critical value of the standard normal distribution and $\alpha$ the size of the test. For the size of a test we always used $\alpha=0.05$. The asymptotic variances were estimated by substitution of the estimated values of $\nu$, $\rho_{d}$ and $\rho_{d+e}$ in equations (14), (11) and (12). This procedure was repeated 100 times. The scaling regions were chosen by visual inspection, and justified by $X^{2}$ 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 $r_{l}$ and $r_{u}$ 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 ( $\overline{\hat{\nu}}$), standard deviation ( $\hat{\mbox{sd}}$) and coverage frequency (CF) estimates for the Hénon map as functions of the embedding dimension. $\overline{N_l + N_s}$, averaged number of distances below $r_u$ ($L = 10000$, $L_{s} = 1000$, $l = 1$, $r_{l} = 0.01$, $r_{u} = 0.07$ and $N = 100000$).

$d$ $\overline{\hat{\nu}}$ $\hat{\mbox{sd}}(\hat{\nu})$ two-sided CF lower CF upper CF $\overline{N_l + N_s}$
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; $e=1$.

$d$ $\overline{\hat{\nu}}$ $\hat{\mbox{sd}}(\hat{\nu})$ 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 $N_{l} + N_{s}$ 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 $\tilde{\phi}$ [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 $0.95 \pm 0.02$. 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 ( $\overline{\hat{K_2}}$), standard deviation and coverage frequency estimates for the Hénon map; $e=1$.

$d$ $\overline{\hat{K_2}}$ $\hat{\mbox{sd}}(\hat{K_2})$ 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; $e=3$.

$d$ $\overline{\hat{K_2}}$ $\hat{\mbox{sd}}(\hat{K_2})$ 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 ($L = 2000000$, $L_{s} = 1000$, $l = 1$, $r_{l} = 0.001$, $r_{u} = 0.005$, $N = 1000000$ and $e=1$).

$d$ $\overline{\hat{K_2}}$ $\hat{\mbox{sd}}(\hat{K_2})$ 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 ($L = 100000$, $L_{s} = 1000$, $l = 1$, $r_{l} = 0.001$, $r_{u} = 0.005$, $N = 1000000$ and $e=1$).

$d$ $\overline{\hat{K_2}}$ $\hat{\mbox{sd}}(\hat{K_2})$ 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 ($L = 10000$, $L_{s} = 0$, $l = 3$ (with Theiler correction 3 (Theiler, 1986)), $r_{l} = 0.01$, $r_{u} = 0.07$, $N = 10000$ and $e=1$).

$d$ $\overline{\hat{K_2}}$ $\hat{\mbox{sd}}(\hat{K_2})$ 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


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