I Let $d_{IJ}^{single},d_{IJ}^{\text{c}\text{o}\text{m}\text{p}\text{l}\text{e}\text{t}\text{e}}$ be the dissimilarities between a generic pair of clusters I,$J.$ As the names indicate, these dissimilarities are

built using single linkage and complete linkage, respectively. Briefly ellaborate whether the following inequality holds or not:

$d_{IJ}^{single}{d}_{IJ}^{\text{c}\text{o}\text{m}\text{p}\text{l}\text{e}\text{t}\text{e}}$

II The following data are acidity measurements $($in $pH$ scale$)$ of $n=5$ soil samples. The variables measured are results obtained with

litmus paper $(x_1)$ and with electric probe $(x_2),$ and the samples $($individuals$)$ are labelled as usual $1$ to $5$ from top to bottom.

With these data, perform agglomerative clustering using the manhattan metric and complete linkage. In particular,

a$)$ Compute and write the distance matrix between individuals, then

b$)$ report the steps of agglomerative clustering. At each stage of clustering, state the dissimilarity matrix and the new cluster formed.

c$)$ Sketch the dendrogram for this analysis.

d$)$ If the tree is cut at a height $($distance$)h=0\mathrm{.}51,$ how many clusters are formed at that value $h$ and which sizes do they have?

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