Suppose we have five observations, x

$(1)$

$,$ x

$(2)$

$,$ x

$(3)$

$,$ x

$(4)$ and x

$(5),$ for which we compute the

following dissimilarity matrix

$.$

$($a$)$ Based on the given dissimilarity matrix, hierarchically cluster the observations using

complete linkage. Sketch the dendrogram, clearly illustrating the height at which

each cluster fusion occurs. $[10$ marks$]$

$($b$)$ Assume that a clustering algorithm produces the following two clusters: C$1=$

$($x

$(1)$

$,$ x

$(2))$ and C$2=($x

$(3)$

$,$ x

$(4)$

$,$ x

$(5)).$ For each of the observations in cluster C$1,$

compute the Silhouette coefficient using the information from the dissimilarity matrix. Comment on the suitability of their assignment to cluster C$1.$

Note: The Silhouette coefficient of an observation x is defined as:

SC$($x$)=$ b$($x$)$ a$($x$)$

max$\{$b$($x$),$ a$($x$)\}$

$,$

where a$($x$)$ is the average dissimilarity of x with respect to other observations in

its cluster, and b$($x$)$ is the minimum average dissimilarity of x with respect to all

clusters to which it does not belong. $[10$ marks$]$