$2.($a$)$ Four classification algorithms were fitted to a simulated dataset with a two$-$class classification problem: logistic regression, decision tree, random forest and a support vector machine with a radial basis kernel. The data with fits of four models are displayed on page $6.$ Which of the plots correspond to the classification algorithms? $[5$ marks$]($b$)$ Whatarethe restrictions on the types of variables in the four classification algorithms in $($a$)?[4$ marks$]($c$)$ Name three other methods that are direct competitors to the classifiers in $($a$).[3$ marks$]($d$)$ Suppose we take a dataset, divide it into equally$-$sized training and testing sets and then try out two different classification procedures. First we use logistic regression and get an error rate of $20\%$ on the training set and $30\%$ on the test set. Next we use a k$-$nearest neighbours classifier with k $=1$ and get an error rate of X$\%$ on the training set and Y $\%$ on the test set where $($X $+$Y$)/2=18.$ Based on these results, which method should we prefer to use for classification of new results? Explain. $[6$ marks$]($e$)$ A classifier gives the following result. In the table below, Group gives the true class, and Prob gives the estimated probability of Group A $($positive$)$ using the classifier. Group Prob $1$ A $0\mathrm{.}6672$ A $0\mathrm{.}4503$ A $0\mathrm{.}6414$ A $0\mathrm{.}4565$ A $0\mathrm{.}7816$ A $0\mathrm{.}9437$ B $0\mathrm{.}4238$ B $0\mathrm{.}7369$ B $0\mathrm{.}21510$ B $0\mathrm{.}021$ i$.$ What are the predicted classes? Use a threshold of $0\mathrm{.}5.$ What is the error rate? What is the false positive rate? The true positive rate? ii$.$ Describe how to construct the ROC curve for this problem. $[8$ marks$][7$ marks$]33.($a$)$ Explain how k$-$fold cross$-$validation is implemented