Correlation Dimension

A fractal-dimension statistic given by the small-radius scaling exponent of the probability or normalized count that two sampled points lie within distance epsilon.

Version

v3 · 2026-09-06 · History

Domain-specific #

1567

Origin domain

chaos theory

Subdomain

strange-attractor characterization

Aliases

Correlation exponent, Grassberger–Procaccia dimension

Core Idea

The correlation dimension is a fractal dimension defined by how the probability that two independently sampled points lie within distance r scales as r becomes small. For a probability measure μ on a metric space, define a correlation integral.

C(r) = ∫ μ(B(x,r)) dμ(x) = P(d(X,Y) < r)

for independent X,Y ~ μ, subject to boundary and convention details. When a stable limit exists,

D_2 = lim_{r→0} log C(r) / log r.

For finite data, the Grassberger–Procaccia estimator counts close pairs over a range of radii and estimates the slope of log C_N(r) against log r in a defensible scaling region.

Scope of Application

Correlation dimension is used in nonlinear dynamics, fractal geometry, experimental time-series analysis, recurrence analysis, and characterization of invariant measures. Its computational accessibility made it a prominent tool for assessing low-dimensional attractor hypotheses from observations.

For points sampled independently from a smooth k-dimensional region with regular density, close-pair probability typically scales like r^k away from boundaries. Multifractal measures can yield a correlation dimension reflecting order-two mass concentration and differing from support dimensions.

Clarity

A common empirical estimator for N distinct sample indices is

C_N(r) = [2/(N(N−1))] Σ_{i<j} 1{d(x_i,x_j)<r}.

Some conventions include ordered pairs or self-pairs with a different normalization. Self-pairs are dangerous because their zero distance dominates as r shrinks. For trajectories, pairs with |i−j| below a chosen exclusion lag should often be omitted.

Manages Complexity

A strange attractor may occupy a complicated subset of a high-dimensional phase space. Correlation dimension compresses its mass distribution into one exponent that can distinguish a line-like, surface-like, or fractional concentration pattern. Pair counting avoids explicit construction of an optimal cover.

That efficiency trades away spatial detail. Different measures can share D_2; local dimensions may vary; topology and dynamical equations are not recovered.

Abstract Reasoning

Start from probability, not the graph. Interpret C(r) as close-pair mass before fitting a line.

Locate scale boundaries. Reject radii dominated by instrument resolution, noise, self-pairs, or finite diameter.

Check slope stability. A candidate exponent should persist over a meaningful interval and across increasing data size.

Knowledge Transfer

The portable skeleton is scale exponent from pairwise neighborhoods: count how rapidly the probability of local coincidence grows with radius and infer an effective dimension. This can organize analyses of point clouds, spatial events, and representation manifolds.

Literal transfer requires a metric, a sampling measure, and a defensible asymptotic or scaling regime. A log–log slope in network statistics or market data is only analogous until those elements and dependence controls are supplied.

Relationships to Other Abstractions

Current abstraction Correlation Dimension Domain-specific

Parents (3) — more general patterns this builds on

• Correlation Dimension is a kind of Dimension Prime

  Dimension is instantiated as the sought scaling exponent.

• Correlation Dimension presupposes Allometry and Scaling Law Prime

  Dimension is instantiated as the sought scaling exponent.

• Correlation Dimension presupposes Metric Prime

  Dimension is instantiated as the sought scaling exponent.

Hierarchy paths (3) — routes to 3 parentless roots

• Correlation Dimension → Dimension

Show alternative paths (2)

Neighborhood in Abstraction Space

Correlation Dimension sits in a sparse region of the domain-specific corpus (70^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

• Empirical Measure — 0.88
• Benford's Law — 0.85
• Variogram — 0.84
• Particle Filter — 0.83
• Least-Squares Adjustment — 0.83

Computed from structural-signature embeddings · 2026-09-08