$5.$ Principal Component Analysis $(15$ marks$)$

For part $($a$)$ and part $($b$)$ below, we consider an unlabelled dataset D with $40000$

datapoints, each datapoint is in R

$2023$

$.$

$($a$)$ One application of Principal Component Analysis $($PCA$)$ is to plot datapoints

from a high$-$dimensional space on a $2-$D graph.

Describe the main steps of using PCA to plot the datapoints in D on a $2-$D

graph. You should describe all pre$-$processing steps, the outputs from PCA

you need, and the projection step. $[7$ marks$]$

$($b$)$ When performing PCA on D$,$ a key step is to compute the scatter matrix.

What is the dimension of the scatter matrix? $[2$ marks$]$

$($c$)$ PCA is performed on another data set. The scatter matrix is

S $=$

$$

$$

$$

$$

$$

$$

$1\mathrm{.}10\mathrm{.}3000$

$0\mathrm{.}31\mathrm{.}9000$

$000\mathrm{.}800$

$0000\mathrm{.}60$

$00000\mathrm{.}6$

$$

$$

$$

$$

$$

$$

$.$

Verify that the vector

$$

$$

$$

$$

$$

$$

$1$

$3$

$0$

$0$

$0$

$$

$$

$$

$$

$$

$$

is an eigenvector of S$.$ What is the corresponding

eigenvalue? $[4$ marks$]$

$($d$)$ Write down all the other eigenvectors of S$.[2$ marks$]$