Correlation Dimension

Correlation dimension can be calculated using the distances between each pair of points in the set of N number of points,

$$s(i,j) \;=\; \vert X_{i}- X_{j}\vert.$$

A correlation function, C(r), is then calculated using,

$$C(r) \;=\; \frac{1}{N^2}\;\times (number\;of\;pairs\;(i,j)\;with\;s(i,j)\;<\;r).$$

C(r) has been found to follow a power law similar to the one seen in the capacity dimension: $C(r)\; =\; k \;r^D$. Therefore, we can find D_corr with estimation techniques derived from the formula:

$$D_{corr}\; =\; \mathop{\lim}_{r \rightarrow 0} \frac{\log(C(r))}{\log(r)}.$$

C(r) can be written in a more mathematical form as

$$C(r)\; =\; \mathop{\lim}_{N \rightarrow \infty} \frac{1}{N^2}... ... \mathop{\sum}_{i=j+1}^{N} \theta (r - \vert X_{i}-X_{j}\vert),$$

where $\theta$ is the Heaviside step function described as,

$$\theta(r - \vert X_{i}-X_{j}\vert)= \left\{ \begin{array}{lc}... ... 0 & 0 > (r - \vert X_{i}-X_{j}\vert) \\ \end{array} \right..$$