PCA direction and explained variance

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Problem

A centered two-dimensional dataset has covariance $\operatorname{diag}(9,1)$ . 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 $(1,0)$ , it explains

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

of the variance, and the discarded, or reconstruction-error, variance is $1$ . 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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Appears in

Dimensionality Reduction