Understanding shannon entropy of a data set

Question

I'm reading Machine Learning In Action and am going through the decision tree chapter. I understand that decision trees are built such that splitting the data set gives you a way to structure your branches and leafs. This gives you more likely information at the top of the tree and limits how many decisions you need to go through.

The book shows a function determining the shannon entropy of a data set:

def calcShannonEnt(dataSet):
    numEntries = len(dataSet)
    labelCounts = {}
    for featVec in dataSet: #the the number of unique elements and their occurance
        currentLabel = featVec[-1]
        if currentLabel not in labelCounts.keys(): labelCounts[currentLabel] = 0
        labelCounts[currentLabel] += 1
    shannonEnt = 0.0
    for key in labelCounts:
        prob = float(labelCounts[key])/numEntries
        shannonEnt -= prob * log(prob,2) #log base 2
    return shannonEnt

Where the input data set is an array of arrays where each array represents a potential classifiable feature:

dataSet = [[1, 1, 'yes'],
    [1, 1, 'yes'],
    [1, 0, 'no'],
    [0, 1, 'no'],
    [0, 1, 'no']]

What I don't get is why the shannon entropy function in this book is only ever looking at the last element in the feature array? It looks like its only calculating the entropy for "yes" or "no" items, and not the entropy of any of the other features?

It doesn't make sense to me because the entropy for this data set

dataSet = [[1, 1, 'yes'],
    [1, 'asdfasdf', 'yes'],
    [1900, 0, 'no'],
    [0, 1, 'no'],
    ['ddd', 1, 'no']]

Is the same as the entropy above, even though it has a lot more diverse data.

Shouldn't the other feature elements be counted as well in order to give the total entropy of the data set, or am I misunderstanding what the entropy calculation is supposed to do?

If anyone is curious, the full source (which is where this code came from) for the book is here under the Chapter03 folder.

Answer 1

The potential ambiguity here is that the dataset you are looking at contains both features and outcome variable, the outcome variable being in the last column. The problem you are trying to solve for is "Do feature 1 and feature 2 help me predict the Outcome"?

Another way to state this is, if I split my data according to feature 1, do I get better information on the Outcome?

In this case, without splitting, the Outcome variable is [ yes, yes, no, no, no ]. If I split on Feature 1, I get 2 groups: Feature 1 = 0 -> Outcome is [ no, no ] Feature 1 = 1 -> Ouctome is [ yes, yes, no ]

The idea here is to see if you are better off with that split. Initially, you had a certain information, described by the Shannon Entropy of [ yes, yes, no, no, no ]. After the split, you have two groups, with "better information" for the group where Feature 1 = 0: you know in that case that the Outcome is no, and that is measured by the Entropy of [ no, no ].

In other words, the approach is to figure out if out of the Features you have available, there is one which, if used, increased your information on what you care about, that is, the Outcome variable. The tree building will greedily pick the feature with the highest information gain at each step, and then see if it's worth splitting even further the resulting groups.