Tree Entropy in R Programming

Entropy in R Programming is said to be a measure of the contaminant or ambiguity existing in the data. It is a deciding constituent while splitting the data through a decision tree. An unsplit sample has an entropy equal to zero while a sample with equally split parts has entropy equal to one. Two major factors that are considered while choosing an appropriate tree are- information gain (IG) and entropy.

Formula :

Entropy = - \sum p(x) \log p(x)

where, p(x) is the probability

For example, consider a school data set of a decision tree whose entropy needs to be calculated.



Library available Coaching joined Parent’s education Student’s performance
yes yes uneducated bad
yes no uneducated bad
no no educated good
no no uneducated bad

Hence, it is clearly seen that a student’s performance is affected by three factors – library available, coaching joined, and parent’s education. A decision tree can be constructed using the information of these three variables for the prediction of student’s performance and hence are called predictor variables. The variables with more information are considered a better splitter of the decision tree.

So to calculate the entropy of parent node – Student’s performance, the above entropy formula is used but probability needs to be calculated first.

There are four values in the Student’s performance column out of which two performances are good and two are bad.

 P_{good} = \frac{\text {good performance in sample}}{\text {total performance in sample}} = \frac{1}{4} = 0.25 \newline \newline  P_{bad} = \frac{\text {bad performance in sample}}{\text {total performance in sample}} = \frac{3}{4} = 0.75

Hence, total entropy of parent can be calculated as below

Entropy = - \sum p(x) \log p(x)  = - (\sum p_{good} \log_{2}{P_{good}}+ \sum p_{bad} \log_{2}{P_{bad}}) = - (0.25  \log_{2}{0.25} + 0.75  \log_{2}{0.75}) = 0.811

Information Gain using Entropy

Information gain is a parameter used to decide the best variable available for splitting the data at every node in the decision tree. So IG of every predictor variable can be calculated and the variable with the highest IG wins the race of deciding factor for splitting of root nodes.

Formula:



Information Gain(IG) = Entropyparent – (weighted average * Entropychildren)

Now to calculate IG of the predictor variable coaching joined, firstly split the parent node according to this variable.

Now there are two parts and their entropy is first calculated individually.

The entropy of the left part

There are two types of output available – good and bad. On the left part, there are three total outcomes with two being bad and one being good. Hence, Pgood and Pbad is calculated again as follows:

P_{good} = \frac {1}{3}= 0.334 \newline \newline P_{bad} = \frac {2}{3}= 0.667 \newline \newline Entropy_{left} = -(0.667 \log_2{0.667} + 0.334 \log_2{0.334}) = 0.9

The entropy of the right part

There is only one component in right, ie, bad performance. Hence, the probability becomes one. And the entropy becomes 0 because there is only one category to which output can belong to. 
 

Calculating the weighted average with Entropy of children

 \text{weighted average} \times Entropy_{children} = \frac{\text{no. of outcomes in left child node}}{\text{total outcomes in parent node}} \times {entropy_\text{left node}}  + \frac{\text{no. of outcomes in right child node}} {\text{total outcomes in parent node}} \times {entropy_{\text{right node}}}

There are 3 outcomes in left child node and 1 in the right node. While, Entropyleft node has been calculated as 0.9 and Entropyright node is 0.

Now keeping the values in the formula above we get a weighted average for this example:
 

 \text{weighted average} \times Entropy_{children} = \frac{3}{4} \times 0.9  + \frac{1} {4} \times 0 = 0.675

Calculating IG

Now putting the calculated weighted average in IG formula simply to obtain IG of ‘coaching joined’. 
 

IG(coaching joined) = Entropyparent - (weighted average * Entropychildren)
IG(coaching joined) = 0.811 - 0.675 = 0.136

Using the same steps and formula IG of other predictor variables is calculated, compared, and variable with the highest IG is therefore selected for splitting the data at every node.
 




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