An Algorithm based on the Calculation of the Correlation Dimension

One of the many quantities which are usually used to characterize a strange attractor is the correlation dimension. It was introduced by Grassberger and Procaccia as a measure of strangeness which is easier to handle (especially numerically) than the measures used until then [5,6]. For instance, if the dynamical laws of a system are not known then the Hausdorff dimension is usually computed using box-counting algorithms (see for example [14], chapter 3.3). These are very time-consuming, especially for higher-dimensional systems, because the number of calculations grows exponentially with the dimension [5,6,8]. The problems we encounter for our special case of analyzing a dynamical system where the only information about the system is given in the form of a time series are even larger, since we have to go through the embedding process first. Grassberger and Procaccia suggested to use, instead of the Hausdorff dimension, the correlation dimension $\nu$, which can be computed directly from the time series without greater difficulties. We will see that calculation of $\nu$ will provide us with a method to find a proper embedding dimension ([8] and [14], chapter 5.3), as well. To determine the correlation dimension we first calculate the correlation integral:

$$\quad C_N(r) = \frac{1}{N^2} \sum_{i=1}^{N} \sum_{j=1}^{N} H \left( r- \left\Vert x_i-x_j \right\Vert \right) \quad,$$ (25)

where $r$ is the correlation length and $H$ the Heaviside function. For $N$ sufficiently large and $r$ sufficiently small the logarithm of $C_N$ as a function of the logarithm of $r$ will have a linear region, the scaling region, and the slope in this region is the correlation dimension $\nu$:

$$\quad \log\left(C_N(r)\right) = \nu\log r + \cdots \quad.$$ (26)

Because $\nu$ is closely related to the Hausdorff-dimension¹¹ which itself is a generalization of the intuitive concept of dimensionality [17], it seems sensible to apply a procedure similar to the one presented in the previous section [8]: We calculate $C_N$ for a sequence of embedding dimensions $n$ and infer the respective $\nu(n)$'s. We expect that if the embedding dimension is too small then $\nu(n)$ equals $n$. (In this case $\nu(n)$ is not the true correlation dimension of the attractor but only an artefact, due to the embedding dimension being too small.) So increasing $n$ yields an increasing $\nu(n)$. But for $n$ sufficiently large $\nu(n)$ will be smaller than $n$ and equal to the true correlation dimension. A further increase should not change the value of $\nu$ any more. Thus we have a method to compute the proper embedding dimension: we simply use the correlation dimension as the embeding dimension.

Footnotes

... Hausdorff-dimension¹¹

For the relationship of $\nu$ to the Hausdorff and information dimensions see [5,6]; Schuster [14], chapter 5.3, shows that these three types of dimensions can be organized within the framework of generalized dimensions $D_q$: $D_{q=2}=\nu$, for example. It seems to be true that in most cases all three dimensions do not differ very much and often they are even the same, e.g. when the attractor is covered uniformly by the $x_i$ [6].