Question $4$: Decision trees

$[28]$

The data in Table $1$ shows attributes that might influence a family's decision on where to go for their vacation. The target variable is whether the family chose the destination $($Yes or No$).$

$\backslash$table$[[$Destination$,$Type,Distance,Budget,Kids$-$Friendly,Chosen?$],[d_1,$Beach,Short,Medium,Yes,Yes$],[d_2,$Mountain,Long,High,Yes,Yes$],[d_3,$City,Short,Low,No$,$No$],[d_4,$Beach,Long,Medium,Yes,No$],[d_5,$City,Short,Medium,Yes,Yes$],[d_6,$Mountain,Long,Low,No$,$No$],[d_7,$Beach,Short,High,Yes,Yes$],[d_8,$City,Short,High,Yes,Yes$],[d_9,$Mountain,Long,Medium,No$,$No$],[d_{10},$Beach,Short,Low,No$,$No$]]$

Table $1$: Vacation destination attributes and family's choices.

$($a$)$ Explain what Entropy and Information Gain represent in the context of the ID$3$ algorithm to construct decision trees. Explain what these values mean in practice.

$($b$)$ Calculate the base Entropy of this data set. Show every step of your calculation, including how you determine the ratios.

$($c$)$ Calculate the Information Gain for each of the attributes. Show all the ratios, steps and calculations.

$($d$)$ Based on your calculations, which attribute$($s$)$ will be used as the root node of the decision tree? Explain why.

$($e$)$ Discuss how ID$3$ proceeds once it has determined the root node. Use your root node as an example to start your discussion. Indicate how the dataset gets modified in the next steps of the algorithm.

$(6)$