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Correlation Dimension


Define the correlation integral as

 C(epsilon)=lim_(N->infty)1/(N^2)sum_(i,j=1; i!=j)^inftyH(epsilon-|x_i-x_j|),
(1)

where H is the Heaviside step function. When the below limit exists, the correlation dimension is then defined as

 D_2=d_(cor)=lim_(epsilon,epsilon^'->0^+)(ln[(C(epsilon))/(C(epsilon^'))])/(ln(epsilon/(epsilon^'))).
(2)

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

 N_(min)>=[(R(2-Q))/(2(1-Q))]^M,
(5)

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-Dimension

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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 Dimension

Cite this as:

Weisstein, Eric W. "Correlation Dimension." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CorrelationDimension.html

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