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04 / Scaling across radii

Correlation dimension

Count close pairs. Watch how space fills.

Preparing the geometry lab…

Behind the estimate

A little geometry. A clearer picture.

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How it works

As a ball grows, more pairs fit inside it. The scaling of that count gives a view of the geometry across distance scales.

C(r)=2N(N−1)∑i<j1[‖xi−xj‖≤r]

In a scaling region, C(r) behaves approximately like r raised to D₂. The slope of log C(r) against log r estimates D₂.

  1. Count distinct pairs.For each radius, find the fraction of unordered pairs separated by at most that radius. Self-pairs are excluded.
  2. Find a useful scale interval.The highlighted portion of the plot is the interval used in the fit.
  3. Read the slope.Fit a straight line in log–log space. Compare intervals before trusting a single number.

The plot is a view of the data. Distances, angles and covariance are computed in all generated coordinates. Noise, sampling and scale can change an estimate; the underlying noiseless manifold is only a reference.