$(16$ marks$)$ In this question, you will predict Orange Juice sales with the OJ dataset. Each line in this dataset is a purchase of either a Minute Maid $($MM$)$ or Citrus Hill $($CH$)$ brand orange juice collected in five different stores over some period of time. We will now fit various tree$-$based classification algorithms to predict Purchase $($i$.$e$.,$ CH or MM$)$ from the remaining columns.

$($a$)$ Do some exploratory data analysis to get a feeling for the dataset. Answer the following questions:

Which brand tends to be more expensive?

Is one brand bought more often than the other one?

$($b$)$ Randomly split the dataset into a training and a test dataset with $80\%$ and $20\%$ relative size. In all following tasks, train on the training set. We will use the test set in the last step only.

$($c$)$ Using the tree package, fit a single classification tree and use $10-$fold cross$-$validation

to prune it to optimal size. Visualise the tree and record its misclassification error on the training set. Since trees allow relatively easy interpretation, give a one$-$or$-$two$-$sentence insight in what can be learned from the tree about the structure of the data.

$($d$)$ Using the randomForest package, use bagging of $500$ trees to predict Purchase.

Record the misclassification rate obtained.

$($e$)$ Same as the previous task, but use a random forest instead of bagging.

$($f$)$ Same as the previous task, but use boosting via the gbm package. You will need

to specify the option distribution $=$ "bernoulli" because we are considering a binary classification problem. You will also need to encode Purchase as a $0-1-$variable in order to be able to apply gbm$.$ You can do this by creating a new feature Purchase$\_01$ and removing the original feature Purchase afterwards.

$($g$)$ Compare all methods so far by predicting them on the test set, and computing the misclassification rate. Which method performs best?