# Chaotic Data: Correlation Dimension

Chaotic Data: Correlation Dimension

This Demonstration shows how to infer the so-called correlation dimension for four chaotic datasets (each of length 4000). The data is derived from the logistic, Hénon and Lorenz models and NMR laser data. The log-log figure on the left contains the so-called correlation sums for some embedding dimensions (as functions of a distance ). They are calculated at discrete points, which are connected with lines. (The correlation sums were calculated outside of this Demonstration because the computation takes too long; for these computations, see "Analysis of Chaotic Data with Mathematica" in the Related Links.)

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From this figure we first search for a so-called scaling region where the correlation sums develop approximately linearly, with approximately the same slope for several values of . To help see this, we show the two blue vertical lines. A linear fit is calculated to the values of the correlation sum in between the vertical lines (endpoints included) for each . The slopes of these lines are shown in the right-hand plot for each value of . If, for some scaling region, we get slopes that are approximately constant for several values of , this constant is an estimate of the correlation dimension. We mark these slopes with the two purple vertical lines. The figure then shows the average slope between these lines (endpoints included); this average is an estimate of the so-called correlation dimension. The red line can be used to check the constancy of the selected slopes.

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