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## 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 , which can be computed directly from the time series without greater difficulties. We will see that calculation of 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:
 (25)

where is the correlation length and the Heaviside function. For sufficiently large and sufficiently small the logarithm of as a function of the logarithm of will have a linear region, the scaling region, and the slope in this region is the correlation dimension :
 (26)

Because 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 for a sequence of embedding dimensions and infer the respective 's. We expect that if the embedding dimension is too small then equals . (In this case is not the true correlation dimension of the attractor but only an artefact, due to the embedding dimension being too small.) So increasing yields an increasing . But for sufficiently large will be smaller than and equal to the true correlation dimension. A further increase should not change the value of 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 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 : , 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 [6].

Next: Singular System Analysis Up: Analysis of Real-World Data Previous: Topological Considerations   Contents
Martin_Engel 2000-05-25