The researchers carried out a cluster analysis on the penguins data, using

bill.len.mm$,$

bill.dep.mm$,$

flipper.len.mm$,$ and body.mass.g as the variables for clustering, and

implemented model$-$based clustering $($MBC$).$ Some $R$ code and output is given

below:

$($a$)$ With reference to the relevant summary outputs, briefly explain why the re$-$

searchers might have chosen to use MBC versus another clustering technique.

$($b$)$ What is the optimum model and number of clusters? What does the chosen

model imply for the cluster structure?

$($c$)$ A penguin has the following information associated with it:

$z_{{?}_{?}}=(0\mathrm{.}450,0,0,0\mathrm{.}550{)}^{TT}$

What is $z_{{?}_{?}}?$ What cluster is this penguin assigned to$?$ Explain your answer.

Calculate the associated uncertainty with the cluster assignment.

$($d$)$ Briefly outline what happens in the two main steps of the EM algorithm

used to fit the MBC model. What is the main disadvantage of using the EM

algorithm and how can we try to mitigate against its impact?