Question $5.$ Since $2019,$ a novel coronavirus disease COVID$-19$ has subsequently more than $136,000,000$ COVID$-19$ cases globally. Assume you would like to develop a classification model to predict COVID$-19$ based on the symptom features. You have a dataset of $100$ patient records to consider in the training process. Each record describe a patient by their attributes of Is$-$Fever, IsTiredness, and Is$-$Cough. Each attribute has two values of $\text{'}T\text{'}($indicating True$)$ and $\text{'}F\text{'}($indicating False$).$ The class label tells us whether a patient gets COVID$-19$ infection, where $\text{'}-\text{'}$ represents False, and $\text{'}+\text{'}$ represents True. The following table summarizes the patient dataset and two class labels $\text{'}+\text{'},\text{'}-\text{'}.$

$\backslash$table$[[$Is$-$ Fever,$\backslash$table$[[$Is$-],[$Tiredness$]],\backslash$table$[[$Is$-],[$Cough$]],\backslash$table$[[$Number of Patients$],[$with Class $+]],\backslash$table$[[$Number of Patients$],[$with Class $-]]],[$T$,$T$,$T$,15,0],[$F$,$T$,$T$,0,10],[$T$,$F$,$T$,15,0],[$F$,$F$,$T$,0,0],[$T$,$T$,$F$,10,0],[$F$,$T$,$F$,0,25],[$T$,$F$,$F$,0,20],[$F$,$F$,$F$,5,]]$

$($a$)$ Construct a decision tree classifier based on the records of $100$ labeled patients. Calculate the gain in the Gini index when splitting on attributes Is$-$Fever and Is$-$Tiredness, respectively. According to the Gini index based gains, which one will you choose as the first attribute to split in the decision tree induction?

$(10$ marks$)$

$($b$)$ Write a completed decision tree using the CART algorithm, which uses the Gini index based impurity measurement.

$(25$ marks$)$

$($c$)$ Calculate the training instances are misclassified by the resulting decision tree.

$(5$ marks$)$

$($d$)$ Predict the class labels of three new patients X$,$ Y$,$ and Z below, according to your constructed decision tree classifier.

$\backslash$table$[[$Patient$,$Is$-$Fever,Is$-$Tiredness,Is$-$Cough,Is$-$COVID$-19],[x,F,F,F,?$