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Estimation of the scaling distribution constant

In section 5.6.1 we need an estimate of $\tilde{\phi}$, the unknown constant in the distribution eq. (5.1). It can be estimated using eq. (5.25):
\begin{displaymath}
\hat{\tilde{\phi}} = \frac{N_l+N_s}{Nr_u^{\nu}\exp(-dK_2)}
\end{displaymath} (D.1)

For $\nu$ and $K_2$ we substituted literature values instead of the estimated values. For $d$ and $ N_l + N_s $ we used results from Monte Carlo simulations using time series generated using the Hénon map as described in section 5.6.2, i.e. for $d = 1,\; 2,\; \ldots,\; 20$ we substituted $\overline{N_l+N_s}$ (as in table 5.5). In that Monte Carlo experiment, we draw 100,000 distances from a time series which was 10,000 points in length. Later on, we draw 100,000 distances from a time series which was 200,000 in length to satisy the independence assumption. The $\overline{N_l+N_s}$ did not differ much; moreover, the exact value of $\tilde{\phi}$ is not very important, as long as we can compare the experiments with the ``ideal data'' and the time series. As can be seen in figure D.1, $\tilde{\phi}$ converges to a value of about 1.67.















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