Define the correlation integral as
 |
(1)
|
where 
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^'))).](/images/equations/CorrelationDimension/NumberedEquation2.svg) |
(2)
|
If 
is the correlation exponent, then
 |
(3)
|
It satisfies
 |
(4)
|
where 
is the capacity dimension and 
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 
-dimensional system with accuracy 
requires 
data points, where
![N_(min)>=[(R(2-Q))/(2(1-Q))]^M,](/images/equations/CorrelationDimension/NumberedEquation5.svg) |
(5)
|
where 
is the length of the "plateau region." If an attractor
exists, then an estimate of 
saturates above some 
given by
 |
(6)
|
which is sometimes known as the fractal Whitney embedding prevalence theorem.
