Correlation Dimension | Nonlinear Dynamics & Chaos

Learning objectives

By the end of this lesson, you should be able to:

Define the correlation sum C(ε).
Interpret a log–log correlation-sum scaling plot.
Estimate a correlation dimension from a scaling slope.
Distinguish correlation dimension from box-counting dimension.

Prerequisites: Phase-Space Reconstruction and Takens Embedding

EXPERIMENT Interactive dynamics laboratory

Reconstruct → Count → Estimate

Change one parameter at a time. Predict the qualitative behavior before running or moving the control.

From geometry to statistics

A phase-space reconstruction gives a geometric representation of the dynamics, but visual appearance alone does not quantify its complexity. Correlation dimension uses distances among reconstructed state vectors to estimate how densely the trajectory occupies state space.

Counting nearby states

For a neighborhood radius \(\varepsilon\), the correlation sum \(C(\varepsilon)\) measures the fraction of admissible pairs of reconstructed states separated by less than \(\varepsilon\). Small radii count only very close neighbors; larger radii include progressively more pairs.

Scaling and dimension

If the attractor has scale-dependent fractal structure, then over an intermediate range \(C(\varepsilon)\sim\varepsilon^{D_2}\). Therefore a plot of \(\log C(\varepsilon)\) against \(\log\varepsilon\) can contain an approximately linear scaling region whose slope estimates the correlation dimension \(D_2\).

Why the scaling region matters

Very small radii contain too few pairs for stable statistics, while very large radii approach saturation. A useful estimate therefore comes from an intermediate range. Temporally adjacent points should also be excluded so that trivial time-neighbor correlations do not dominate the smallest distances.

Finite-data interpretation

Correlation dimension is an empirical finite-data estimate. Its value depends on trajectory length, sampling, noise, delay, embedding dimension, temporal-neighbor exclusion, and the selected scaling range. Unlike box counting, it is based on pairwise neighborhood statistics rather than occupied grid boxes.

Key takeaways

What should stay with you

Correlation dimension is based on pairwise neighborhood statistics.
A usable intermediate scaling region is needed before interpreting the slope.
Finite sampling, temporal correlations, and reconstruction choices affect the estimate.

Knowledge check