ML Prep

PCA direction and explained variance

Problem

A centered two-dimensional dataset has covariance diag⁡(9,1)\operatorname{diag}(9,1). Give the first principal direction, the fraction of variance it explains, and the reconstruction-error variance after projection to one dimension. Explain why PCA should first center raw observations.

Reveal answer or reference solution

The first direction is (1,0)(1,0), it explains

99+1=0.9\frac{9}{9+1}=0.9

of the variance, and the discarded, or reconstruction-error, variance is 11. Centering removes the mean offset so that the principal directions describe variation around the mean rather than the mean location itself.

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