next up previous contents
Next: Simulations using ``ideal'' data Up: MAXIMUM LIKELIHOOD METHODS Previous: The goodness-of-fit test and   Contents

Monte Carlo simulations

The expression for the variance of the estimator of the correlation entropy is a new result; we therefore wish to verify its validity. Monte Carlo methods can be used to investigate properties of the distributions of random variables (like estimators) by use of simulated random numbers [Kotz and Johnson, 1985]. From a sample of $M$ Monte Carlo trials of some estimated quantity $q$ we can estimate
\overline{\hat{q}} = \frac{1}{M} \sum_{i=1}^{M} \hat{q}_i
\end{displaymath} (5.64)

\hat{\mbox{VAR}}\left\{\hat{q}\right\} =
\frac{1}{M-1} \left...
...rac{1}{M} {\left( \sum_{i=1}^{M} \hat{q}_i \right) }^2 \right)
\end{displaymath} (5.65)

These estimators of the sample mean and variance have variances themselves; if $M$ is large enough, we have [Hogg and Craig, 1978, Ch. 6], [Kendall and Stuart, 1979, exam. 17.10]
\hat{\mbox{VAR}}\left\{\overline{\hat{q}}\right\} =
\end{displaymath} (5.66)

\hat{\mbox{VAR}} \left\{ \hat{\mbox{VAR}} \left\{\hat{q}\rig...\} =
\frac{2}{M} {\hat{\mbox{VAR}}}^2 \left\{\hat{q}\right\}
\end{displaymath} (5.67)

The quantity of which we wish to study its properties is one of the following: the ``doubly censored'' estimator at one and two embedding dimensions and the entropy estimator, in the sequel denoted by $\nu$ ``single'' (eq. (5.26)), $\nu$ ``double'' (eq. (5.47)) and $K_{2}$ (eq. (5.49)) respectively. These can be estimated in two interesting situations:
  1. From distances drawn directly from the distribution (5.1); we have to specify $\nu$, $K_{2}$, $\tau$ and $\tilde{\phi}$. For an increasing number of distances, the estimated sample mean and variance should converge to the specified value of $q$ and its asymptotic variance respectively. By observing this convergence, with confidence intervals constructed using eqs (5.66) and (5.67), we can determine a minimum sample size for which the expressions for the asymptotic variances are sufficiently accurate.
  2. From distances between points in reconstructed phase space, using time series of a (simulated) chaotic system for which the dimension and entropy are either analytically derived or for which literature estimates are available. However, $\tilde{\phi}$ is in general unknown. Fortunately, if the sample size is large enough we may substitute estimated values. Systematic errors may occur, for example due to the fact that we cannot let $r \rightarrow 0$ (which is required according to the definitions of the correlation dimension and entropy in sections 2.3 and 2.4), lacunarity or the finite length of the time series.

Confidence intervals for an asymptotically normal (eq. (5.2)) estimator $\hat{q}$ are given by

$\displaystyle \mbox{two-sided:}$   $\displaystyle [\hat{q}-Z_{\alpha/2}\sigma, \;
$\displaystyle \mbox{lower:}$   $\displaystyle [\hat{q}-Z_{\alpha}\sigma, \; \infty[$ (5.68)
$\displaystyle \mbox{upper:}$   $\displaystyle ]-\infty, \; \hat{q}+Z_{\alpha}]$  

where $\sigma$ is the square root of the asymptotic variance of $\hat{q}$ and $Z_{\alpha}$ is the $\mbox{probability-}\alpha$ critical value of the standard normal distribution, with
\int_{-\infty}^{Z_{\alpha}} p_N(0,1) dx = 1-\alpha \quad \mb...
\int_{-Z_{\alpha/2}}^{Z_{\alpha/2}} p_N(0,1) dx = 1-\alpha
\end{displaymath} (5.69)

We will use 95% confidence intervals, so $\alpha = 0.05$ (and $Z_{\alpha} \approx 1.65$, $Z_{\alpha/2} \approx 1.96$). Coverage frequencies, i.e. the fraction of trials for which the confidence intervals contain the ``true'' value of $q$, should equal $1-\alpha$. They are governed by a binomial distribution, so their variances are given by
\mbox{VAR}\left\{\hat{p}\right\} = \frac{p(1-p)}{M}
\end{displaymath} (5.70)

Deviations from asymptotic behaviour can be observed by statistically significant deviations of the coverage frequencies from $1-\alpha$.

In the following two subsections we present numerical results for the two situations.

next up previous contents
Next: Simulations using ``ideal'' data Up: MAXIMUM LIKELIHOOD METHODS Previous: The goodness-of-fit test and   Contents