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05 / Variance & linear structure

PCA participation ratio

How many directions carry the variance?

Preparing the geometry lab…

Behind the estimate

A little geometry. A clearer picture.

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How it works

Read the covariance spectrum as a soft count of important axes. This measures effective linear dimension, which can differ from manifold dimension.

DPR=(∑iλi)2∑iλi2

λᵢ are the eigenvalues of the centred sample covariance. Equal variance across D axes gives a participation ratio of D.

  1. Centre the data.Subtract the sample mean and compute the covariance in the full coordinate space.
  2. Inspect the spectrum.Each eigenvalue measures variance along a principal axis.
  3. Count effective directions.The participation ratio weights axes by variance. The 95% rank instead counts axes needed for 95% of total variance.

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.