Correlation Dimension

Correlation dimension is a fractal measure of intrinsic dimensionality, introduced by Grassberger and Procaccia in 1983. It estimates how the number of point pairs within distance r scales as r approaches zero. For a d-dimensional manifold, the scaling follows a power law with exponent d; for fractal structures, the exponent is non-integer, revealing a geometry that classical manifold assumptions cannot capture.

The method is among the most robust estimators of intrinsic dimensionality because it does not assume a parametric form and can reveal scale-dependent structure. It has been applied to chaotic dynamical systems, neural population activity, and financial time series.

Its limitation is sensitivity to noise and finite-sample effects: at very small scales, noise dominates; at large scales, the scaling breaks down. The art lies in identifying the scaling regime where the power law genuinely holds.