Q$5.$ Clustering $($Total $10$pt$)$

Consider the following set of one$-$dimensional data points: $\backslash (\backslash \{0\mathrm{.}1,0\mathrm{.}25,0\mathrm{.}45,0\mathrm{.}55,0\mathrm{.}8,0\mathrm{.}9\backslash \}\backslash ).$

All the points are located in the range between $\backslash ([0,1]\backslash ).$

$($a$)(8$pt$)$ Suppose we apply k$-$means clustering to obtain three clusters, A$,$ B$,$ and C$.$ If the initial centroids are located at $\backslash (\backslash \{0,0\mathrm{.}4,1\backslash \}\backslash ),$ respectively, we have the following cluster assignment of the data points.

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$multirow$\{2\}\{*\}\{$ Iter $\}$ & $\backslash$multicolumn$\{4\}\{|$c$|\}\{$ Cluster assignment of data points $\}$ & $\backslash$multicolumn$\{3\}\{|$c$|\}\{$ Centroid Locations $\}\backslash \backslash$

$\backslash$cline $\{2-10\}$ & $0\mathrm{.}10$ & $0\mathrm{.}25$ & $0\mathrm{.}45$ & $0\mathrm{.}55$ & $0\mathrm{.}80$ & $0\mathrm{.}90$ & A & B & C $\backslash \backslash$

$\backslash$hline $0$ & A & B & B & B & C & C & $0\mathrm{.}00$ & $0\mathrm{.}40$ & $1\mathrm{.}00\backslash \backslash$

$\backslash$hline $1$ & A & A & B & B & C & C & $0\mathrm{.}1$ & $0\mathrm{.}42$ & $0\mathrm{.}85\backslash \backslash$

$\backslash$hline $2$ & A & A & B & B & C & C & $0\mathrm{.}18$ & $0\mathrm{.}5$ & $0\mathrm{.}85\backslash \backslash$

$\backslash$hline $3$ & A & A & B & B & C & C & $0\mathrm{.}18$ & $0\mathrm{.}5$ & $0\mathrm{.}85\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

Find the sum$-$of$-$squared errors $($SSE$)$ of the clustering after the third iteration. And find the silhouette coefficient of data point $0\mathrm{.}25$ after the third iteration.

$($b$)(2$pt$)$ For the dataset given in part $($a$),$ is it possible to obtain empty clusters? Why?