Center data and find PCA direction

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Problem

For the points $(1,0)$ , $(3,0)$ , and $(5,0)$ , compute the mean, centered data, covariance with factor $\frac{1}{n}$ , first principal direction, and its explained-variance fraction.

Reveal answer or reference solution

The mean is $(3,0)$ , and the centered points are $(-2,0)$ , $(0,0)$ , and $(2,0)$ . The covariance is

\begin{bmatrix}\frac{8}{3}&0\\0&0\end{bmatrix}.

A first principal direction is $(1,0)$ , with sign reversal equivalent. It captures all nonzero variance, so the explained fraction is $1$ .

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Center data and find PCA direction

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