Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / E8 / d / Solution

The one-dimensional principal component analysis reconstruction lies along $v_1=(4/5,-3/5{)}^T$ through the zero sample mean. Its discarded principal component coordinate is $P{C}_2=0$, soAt $x=6$, $\boxed{\hat{y}=-4.5}$. This intersects the retained principal component line with the vertical line at the supplied $x$. It is a reconstruction from that line, rather than an ordinary linear regression of $y$ on $x$.