I Let $d_{IJ}^{single},d_{IJ}^{\text{a}\text{v}\text{e}\text{r}\text{a}\text{g}\text{e}}$ be the dissimilarities between a generic pair of clusters I,$J.$ As the names indicate, these

dissimilarities are built using single linkage and average linkage, respectively. Briefly ellaborate whether the

following inequality holds or not:

$d_{IJ}^{single}{d}_{IJ}^{\text{a}\text{v}\text{e}\text{r}\text{a}\text{g}\text{e}}$

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 single 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$)h=0\mathrm{.}14,$ how many clusters are formed at that value $h$ and which sizes do

they have?