procedure with $30$ replicntions is implemented for tuning. How many classifiens

need to be trained in total? Justify your nnswer.

$($d$)$ Consider two competing classifers, a logistic regression classifier and a random

farest classifier, both trained and evaluated on the same dnta splits $($training

and validation$).$ Is it ressonable to expect that the random forest classifier

will always outperform on average the logistic regression clnsifier? Justify

your nnswer.

$($e$)$ Four hundred labeled snmples are used to train two clnsifiers $M_1$ and $M_2.$ For

classifier $M_1,$ the dataset is divided into training and validation sets of $200$

samples each and the classifer is trained on the training set. The performance

of $M_1$ ou this validation set provides a $95\%$ nocuracy. For classifier $M_2,$ the

dataset is divided into a training set of $350$ samples and a validation set of

$50$ samples, and the classifier is trained on the training set. Tbe performance

of $M_2$ on the corresponding validation set provides an accurncy of $95\%.$ Is

it appeoprinte to consider classifier $M_2$ as having an equivalent predictive

performance relative to the predictive performance of classifier $M_1?$ Justify

your nnswer.Provide your answer and a coacise explanation for each of the following questivas.

$($a$)$ Using k$-$menns, a statisticinn wants to investignte the presenee of clusters in

a datnset with $2722$ ohservations.

A clustering of the data points with $K=4$ was found for this data, with a

total sum of squares of $457293$ and cluster$-$specific within sums of squares of

$6955,11329,10298,$ and $11411$ respectively.

Another clustering of the observations with $K=3$ wns found far the same

dataset, with a total sum of squares of $457293$ nnd cluster$-$specific within sums

of squares of $12123,7205,$ and $13027$ respectively.

Compare the two clustering solutions using an appropriate index. Which one

is preferred? Justify your answer.

$($b$)$ A clnssification tree algorithm is applied to a given data set with a target

binary variable $y$ and a numerical input variable $x.$ Consider the two following

splits Split A nnd Split B reported in the two tables below.

$($a$)$ Splat $B$

$($b$)$ Sple A

On the basis of the Gini messure, which split would be chosen by a classifies$-$

tion tree algorithm? Justify your nnswer.

$($c$)$ A SVM clnssifier with GRBF kernel, a classificntion tree classifer, and a ka$-$

gistic regression model are employed to deploy a system for malware detection

based on numerical features extracted from webpages. The SVM is tumed

cousidering a grid of hyperparameter values constructed using the get of cost

values $C=\{50,100,200,500\}$ and the set of $$ values $=\{0\mathrm{.}01,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}5\}.$

The classification tree is tuned using a set of complexity parameters $cp=$

$\{0\mathrm{.}1,0\mathrm{.}05,0\mathrm{.}01\}.$ No tuning is performed for legistic regression, which is im$-$

plemented rsing all the input varisbles available. A $10-$fold cross$-$valichation