Correlation Dimension
Define the correlation integral as
where H is the Heaviside step function. When the below limit exists, the correlation dimension is then defined as
![画像: D_2=d_(cor)=lim_(epsilon,epsilon^'->0^+)(ln[(C(epsilon))/(C(epsilon^'))])/(ln(epsilon/(epsilon^'))).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/CorrelationDimension/NumberedEquation2.svg){kind=link}
If nu is the correlation exponent, then
| lim_(epsilon->0)nu->D_2. |
(3)
|
It satisfies
| d_(correlation)<=d_(information)<=d_(capacity) |
(4)
|
where d_(capacity) is the capacity dimension and d_(information) is the information dimension (correcting the typo in Baker and Gollub 1996), and is conjectured to be equal to the Lyapunov dimension.
To estimate the correlation dimension of an M-dimensional system with accuracy (1-Q) requires N_(min) data points, where
![画像: N_(min)>=[(R(2-Q))/(2(1-Q))]^M,](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/CorrelationDimension/NumberedEquation5.svg){kind=link}
where R>=1 is the length of the "plateau region." If an attractor exists, then an estimate of D_2 saturates above some M given by
| M>=2D+1, |
(6)
|
which is sometimes known as the fractal Whitney embedding prevalence theorem.
See also
Capacity Dimension, Correlation Exponent, Information Dimension, q-DimensionExplore with Wolfram|Alpha
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References
Baker, G. L. and Gollub, J. B. Chaotic Dynamics: An Introduction, 2nd ed. Cambridge, England: Cambridge University Press, 1996.Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 547-548, 1995.Referenced on Wolfram|Alpha
Correlation DimensionCite this as:
Weisstein, Eric W. "Correlation Dimension." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CorrelationDimension.html