Question $1$: Principal components analysis $($based on the Matlab

laboratories $2$ and $3)$

$($Note: any prior knowledge of the principal component analysis is not required in this question. The

description of the question is self$-$contained. All necessary information is provided.$)$

A patch of a forest consists of $50$ trees planted in a random fashion, as indicated by circles in Fig$\mathrm{.}1.$ The

location of each tree is given by two coordinates $($xi; yi$),($i $=1$; $2$; : : : ; $50)$ stored in the le "trees.dat" in

such a way that the rst column contains all xi values and the second column contains the corresponding

yi values. The le "trees.dat" is available on Canvas. A camping ground is to be built in the centre of

the forest $(0$; $0).$ As a chief engineer of the construction project, you are planning to build two straight

roads beginning in the origin $(0$; $0)$ and leading out of the forest. The rst road gives the direction of

the fastest escape and the second road gives the direction of the slowest escape out of the forest. It is

known that these two directions can be identied with the eigenvectors of the so$-$called covariance matrix

C $=$ bT b$,$ where b is a $502$ matrix that contains the coordinates of all $50$ trees