$($b$)$ Decision tree is a straightforward algorithm and easy to understand in classification

problem. Entropy is a measure of purity and it can be used to calculate the expected

information for an attribute $($Fig$.3$b$).$

Entropy:

$E(S)=-{\displaystyle \underset{i=1}{\overset{c}{}}}{p}_{i}lo{g}_2{p}_i$

where $S$ is training set, $p_i$ is the probability of examples in class $i,$ and $c$ is the

number of class labels.

Expected information of an attribute:

$E(T,x)={\displaystyle \underset{\text{x}\text{i}\text{n}\text{x}}{\overset{?}{}}}{p}_{x}E(x)$

where $T$ is the training set, $x$ is the attribute, $x$ is a label in attribute $x,p_x$ is the

probability of examples in the training set, and $E(x)$ is the entropy of the label of

attribute $x.$

Fig. $3$b

Information gain can determine which attribute is better for being a splitting node by

subtracting the information before and after splitting, i$.$e$.$ info$\_$gain $(T,x)=E(T)-$

$E(T,x).$

Based on the dataset in Table $3,$ determine the information gain of each input

attribute and suggest which input attribute should be the root node of a decision tree.

Show your steps clearly.