Correlation Dimension


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Correlation Dimension

An estimate of the Fractal Dimension which measures the probability that two points chosen at random will be within a certain distance of each other, and examines how this probability changes as the distance is increased. White noise will fill its space since its components are uncorrelated, and its correlation dimension is equal to whatever dimension it is placed in. A dependent system will be held together by its correlations and retain its dimension whatever embedding dimension it is placed in, as long as it is greater than its fractal dimension.
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Scheinkman, 1987, "A Test of Independence Based on the Correlation Dimension," Unpublished Manuscript, University of Wisconsin, Madison, University of Houston, and University of Chicago.
To determine the chaotic dynamics within the stream flows, the correlation dimension method was applied.
We can now understand why the correlation dimension have a role in our problem [24].
Values of correlation dimension vary with electroencephalograph epoch length, lead position, method of calculation, level of arousal, and age (12).
If the correlation dimension does not explode with the embedding dimension, then evidence of nonlinearity in the data exists.
The intuition that lies behind the search for convergence of the GP correlation dimension ([v.
A popular approach to approximate the correlation dimension in the face of limited data is to estimate the statistic
While the estimation of the correlation dimension is straightforward and has been applied by physicists, its application to economic and financial data raises a number of issues.
An investigation of seismotectonic parameters in and around Turkey involved using the b-value of the G-R relationship (Equation 1) and fractal correlation dimension Dc-value of Turkish earthquake distribution from 1970 to 2011; Turkey was divided into 55 seismic source zones in an attempt to find an empirical relationship between b-value and Dc-value.
The JEI and KLEI series are subjected to Correlation Dimension and BDS tests.
Finally, the three moments test and the GP correlation dimension test is used to determine if the observed nonlinearity is, indeed, mean nonlinearity to provide conclusive evidence of the existence of chaotic determinism.
A related measure, a lower bound for the Hausdorff dimension useful for time series data, is the correlation dimension developed by Grassberger and Procaccia (1983).

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