$(30$ points$)$ Given the following data points in $1$D: $x_1=4,x_2=6,x_3=7,x_4=8,x_5=$

$10,x_6=12,x_7=14,x_8=16,$ perform k$-$means clustering algorithm for $K=2.$

$($a$)$ Start from initial cluster centers $c_1=3,c_2=12.$ Show your steps for all iterations:

$(1)$ the cluster assignments $y_1,$cdots,$y_9$;$(2)$ the updated cluster centers at the end

of that iteration.

$($b$)$ How many iterations does it take for k$-$means algorithm to converge $($i$.$e$.,$ number

of iterations includes all iterations you perform to find convergence$)?$ What is

the reconstruction error $($i$.$e$.,$ distortion measure $J,$ equation $9\mathrm{.}1$ of the Bishop's

textbook$)$ at the end of that iteration?

$($c$)$ Repeat the above steps with initial cluster centers $c_1=8,c_2=14.$

$($d$)$ How many iterations does it take for k$-$means algorithm to converge in this case?

What is the reconstruction error at the end of that iteration?

$($e$)$ Comparing $($a$)$ with $($c$),$ which solution is better? Why?