Hiotammon and $K$ NF with $h-1,$ Histogram and KDE with $h=0\mathrm{.}1$

Figure $1$: Estimates $($histogram and KDE $)$ of the probability density function of data taking

values in $R($shown as circle on abscissa$)$ for bandwidth $h=1($a$)$ and $h=0\mathrm{.}1($b$).$

$($a$)$ Explain what is a kernel density estimate $($KDE$).$ Comment on the impact of the band$-$

width $h$ when computing a KDE $($cf$.$ Fig $1).$

$[5$ marks$]$

$($b$)$ Explain the similarities and differences between histograms and KDEs $($cf$.$ Fig $1).$

$[5$ marks$]$

$($c$)$ Explain what is the Mean Shift algorithm and how it works.

$[5$ marks$]$

$($d$)$ Consider the data point of value equals to $2$ in Fig. $1($a$)$ as the starting value for the Mean

Shift algorithm: what is the final value $($approximately$)$ using this Mean Shift algorithm

till convergence $($using the KDE shown in Fig. $1($a$))?$ If the Mean Shift algorithm was

instead computed with the KDE shown in Fig$\mathrm{.}1($b$),$ would the final value be the same?

Explain.

$[5$ marks$]$

$($e$)$ Explain how the Mean Shift algorithm can be used for segmentation of colour images. Please do not use ChatGTP and explain how you got the answer for each question