Gini coefficient vs Gini impurity - decision trees

The problem refers to decision trees building. According to Wikipedia 'Gini coefficient' should not be confused with 'Gini impurity'. However both measures can be used when building a decision tree - these can support our choices when splitting the set of items.

1) 'Gini impurity' - it is a standard decision-tree splitting metric (see in the link above);

2) 'Gini coefficient' - each splitting can be assessed based on the AUC criterion. For each splitting scenario we can build a ROC curve and compute AUC metric. According to Wikipedia AUC=(GiniCoeff+1)/2;

Question is: are both these measures equivalent? On the one hand, I am informed that Gini coefficient should not be confused with Gini impurity. On the other hand, both these measures can be used in doing the same thing - assessing the quality of a decision tree split.

No, despite their names they are not equivalent or even that similar.

• Gini impurity is a measure of misclassification, which applies in a multiclass classifier context.
• Gini coefficient applies to binary classification and requires a classifier that can in some way rank examples according to the likelihood of being in a positive class.

Both could be applied in some cases, but they are different measures for different things. Impurity is what is commonly used in decision trees.

I took an example of Data with two people A and B with wealth of unit 1 and unit 3 respectively. Gini Impurity as per Wikipedia = 1 - [ (1/4)^2 + (3/4)^2 ] = 3/8

Gini coefficient as per Wikipedia would be ratio of area between red and blue line to the total area under blue line in the following graph Area under red line is 1/2 + 1 + 3/2 = 3

Total area under blue line = 4

So Gini coefficient = 3/4

Clearly the two numbers are different. I will check more cases to see if they are proportional or there is an exact relationship and edit the answer.

Edit: I checked for other combinations as well, the ratio is not constant. Below is a list of few combinations I tried. • What an explanation!! – Outlier Aug 7 '17 at 9:32

I think they both represent the same concept.

In classification trees, the Gini Index is used to compute the impurity of a data partition. So Assume the data partition D consisiting of 4 classes each with equal probability. Then the Gini Index (Gini Impurity) will be: Gini(D) = 1 - (0.25^2 + 0.25^2 + 0.25^2 + 0.25^2)

In CART we perform binary splits. So The gini index will be computed as the weighted sum of the resulting partitions and we select the split with the smallest gini index.

So the use of Gini Impurity (Gini Index) is not limited to binary situations.

Another term for Gini Impurity is Gini Coefficient which is used normally as a measure of income distribution.

• Gini coefficient is not Gini impurity. See the links in the question – Sean Owen Sep 10 '14 at 19:15
• Wikipedia ist not always a reliable source of information :-) – Pasmod Turing Sep 11 '14 at 13:40
• Sure. Go look it up somewhere else: mathworld.wolfram.com/GiniCoefficient.html What makes you think Gini coefficient = Gini impurity? – Sean Owen Sep 11 '14 at 14:03
• Look it up: books.google.de/… – Pasmod Turing Sep 11 '14 at 15:10
• I think we are talking about decision trees. So we are in the field of machine learning! Please read the question more carefully – Pasmod Turing Sep 19 '14 at 12:38