In this problem, your task is to evaluate the performance of two Logistic$-$Regression models $($M$1$

and M$2)$ for a binary classification problem. The attributes in the test set you have chosen are

represented by X$.$ Table $1$ shows the output $($P$(+|$M$1)$ and P$(+|$M$2))$ of the models, which represent

the posterior probabilities for the positive class. As this is a binary classification problem, P$()=$

$1$ P$(+)$ and P$(|$ X$)=1$ P$(+|$ X$).$

Instance True Class P $(+|$ X$,$ M$1)$ P $(+|$ X$,$ M$2)$

$1+0\mathrm{.}780\mathrm{.}61$

$2+0\mathrm{.}620\mathrm{.}08$

$3-0\mathrm{.}440\mathrm{.}62$

$4-0\mathrm{.}550\mathrm{.}39$

$5+0\mathrm{.}610\mathrm{.}48$

$6+0\mathrm{.}470\mathrm{.}09$

$7-0\mathrm{.}070\mathrm{.}38$

$8-0\mathrm{.}140\mathrm{.}09$

$9+0\mathrm{.}480\mathrm{.}06$

$10-0\mathrm{.}320\mathrm{.}01$

Table $1$: Output of two models

$(1)$ For the two models M$1$ and M$2,$ suppose you choose the decision threshold to be $\backslash$tau $=0\mathrm{.}5.$ In

other words, any test instances whose output of the model is greater than $\backslash$tau will be classified as a

positive example. Write down the confusion matrix and calculate the precision, recall and accuracy

of the two models respectively.

$(2)$ Plot the ROC for both M$1$ and M$2,$ and calculate their AUC. Which model performs better

based on the AUC value? $($Hint: In this scenario, the ROC is a step line.$)$