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Methods to estimate the dimension and entropy functions

An expression for nearest neighbour distances including the ``entropy function'' $K(\gamma)$ can be found in [Grassberger, 1986,Broggi, 1988]:
\begin{displaymath}
\ln <r_{k}^{\gamma}> \sim
\frac{\gamma}{D(\gamma)} \left[ d \tau K(\gamma) - \ln N \right]
\end{displaymath} (3.11)

where $d$ is the embedding dimension reconstruction of phase space and $\tau = l \Delta t$ (see section 2.5). This expression, however, does not include the correction function3.1. The dimension and entropy functions are related to the generalized dimensions and entropies as introduced in Chapter 2 by the implicit relations [Badii and Politi, 1985,van de Water and Schram, 1988]:
$\displaystyle D_q$ $\textstyle =$ $\displaystyle D(\gamma=(1-q)D_q)$  
$\displaystyle K_q$ $\textstyle =$ $\displaystyle K(\gamma=(1-q)D_q)$ (3.12)

By first estimating the moments of the probability density function of nearest neighbour distances, and then using eqs (3.9) and (3.11), it is possible to estimate the dimension and entropy spectra. In practice, the weighted distances ${<r_{k}^{\gamma}>}^{1/\gamma}$ are calculated for some values of $k$ and $N$.

There are three possibilities:

N-method
: Badii and Politi [Badii and Politi, 1985] use a fixed value of $k$ (in which case the correction function is a constant) and study the behaviour of eq. (3.9) as a function of $N$.
k-method
: Termonia and Alexandrowicz [Termonia and Alexandrowicz, 1983] fix $N$ at its maximum value and study the behaviour of eq. (3.9) as a function of $k$.
k(N)-method
: There is a third possibility: consider the well-known estimator of the local density
\begin{displaymath}
\hat{p}(x) = \frac{k}{NV}
\end{displaymath} (3.13)

where $V$ is the volume of a $D$-dimensional hypersphere. For a given value of $N$, there exists an optimal value of $k$ (optimal in the sense that $\hat{p}(x)$ has the fastest convergence) given by [Somorjai, 1986]
\begin{displaymath}
k = \alpha N^{\beta}
\end{displaymath} (3.14)

So we can observe the scaling behaviour of eq. (3.9) where $k$ and $N$ vary together according to eq. (3.14).

If we calculate the distances ${<r_{k}^{\gamma}>}^{1/\gamma}$ for different values of $d$ and $\gamma$, we are able to estimate both the dimension function $D(\gamma)$ and the entropy function $K(\gamma)$. However, eq. (3.11) is only valid for fixed $k$.


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Next: An example Up: NEAREST NEIGHBOUR METHODS Previous: The expected value of   Contents
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