Correlation Dimension
Definition 1 2
In a metric space $\left( X , d \right)$, let’s express elements of a set $S = \left\{ x_{1} , \cdots , x_{n} \right\} \subset X$ as $x \in S$. The number of elements in an open ball $B \left( x ; \varepsilon \right)$ with center $x$ and radius $\varepsilon > 0$ is $N_{x} ( \varepsilon )$: $$ N_{x} ( \varepsilon ) := \left| B \left( x ; \varepsilon \right) \cap S \right| $$ If the following limit exists, it is called the pointwise dimension at $x$. $$ \lim_{\varepsilon \to 0} {\frac{ \log N_{x} (\varepsilon) }{ \log \varepsilon }} $$ Given $\varepsilon$, define $C ( \varepsilon )$ as follows: $$ \begin{align*} C (\varepsilon) =& {\frac{ \left| \left\{ (u, v) \in S \times S : d (u, v) < \varepsilon \right| \right\} }{ \left| S \times S \right| }} \\ =& {\frac{ 1 }{ n }} \sum_{x \in S} {\frac{ N_{x} ( \varepsilon ) }{ n }} \end{align*} $$ If the following limit $\operatorname{cordim} (S)$ exists, it is called the correlation dimension of $S$. $$ \operatorname{cordim} (S) = \lim_{\varepsilon \to 0} {\frac{ \log C (\varepsilon) }{ \log \varepsilon }} $$
- $\left| \cdot \right|$ is the cardinality of a set.
Explanation
The correlation dimension is advantageous in computations as it does not suffer from the curse of dimensionality compared to other fractal dimensions like the box-counting dimension. As the dimension of $X$ increases, the number of axes one must manage grows, making partitioning boxes increasingly difficult.
Although the definition itself does not specify conditions that set $S$ must satisfy, from the perspective of dynamical systems, one could consider a trajectory $S = \left\{ x_{1} , \cdots , x_{n} \right\}$ obtained through $n$ iterations around a system often represented by a map $x_{t+1} = f \left( x_{t} \right)$. In this context, $N_{x} (\varepsilon)$ represents how frequently the system state visits near $x$, while $N_{x} (\varepsilon) / n$ indicates how long points included in the entire trajectory of length $n$ remain near $x$, expressed as a percentage. Furthermore, $C (\varepsilon)$, which is defined as the average over the entire $S$, takes values $1 / n$ when $\varepsilon = 0$, and $1$ when $\varepsilon = \infty$: $$ \begin{align*} C ( 0 ) =& {\frac{ 1 }{ n }} \to 0 \text{ as } n \to \infty \\ C ( \infty ) =& 1 \end{align*} $$
Typically, such $C \left( \varepsilon \right)$ is said to follow a power law with respect to the radius $\varepsilon$: $$ C ( \varepsilon ) \approx \varepsilon^{d} $$ Accordingly, we call $d$ the correlation dimension and numerically determine it by comparing values of $\log C$ and $\log \varepsilon$ to decide on the slope.