Attractive Mathematical Properties Of The Roc Curve

ROC Curve

Most of us use the ROC curve to assess our binary classifiers everyday. Sometimes we take for granted its theoretical properties. In this post, I will take some time and analyze why the properties are what they are.

2 properties will be covered:

  1. The baseline is the diagonal line
  2. The area can be interpreted as how strong the classifier is

1. The baseline is the diagonal line

So as we know, the x-axis of the ROC curve is the False Positive Rate (FPR) and the y-axis is the True Positive Rate (TPR).

If we have a dataset with $\pi \in [0,1] $ positives, and $ 1 - \pi $ in the negatives, and we predict randomly with a positive rate of $ p $ and negative rate of $ 1 - p $, then we can calculate the TPR and FPR as a ratio.

Since TPR and FPR are both p, a random classifier (baseline) will have a ROC curve of slope 1 (the diagonal) and an AUC of 0.5.

2. The area can be interpreted as how strong the classifier is

Technically, the area is a bit different from I described.

Let’s go into why.

The integral means that for each negative example, count the number of positive examples with a higher score than this negative example.

If we have a perfect classifier, then all P positives will be scored higher than the negative example, so the integral will result in a maximum area of $ P * N$.

Note: Intuitively, the perfect classifier has {TPR = 1 and FPR = 0}, which is the upper left point

Combined together, the 2 terms give us a nice interpretation of the $A_{AUC} \in [0,1] $ as the classifier’s ability to discern positive and negative data.


Unfortunately, I did not go over other properties such as linear correlation with accuracy, pareto optimality and relationships with the calibration curve. That’s for another day.

The 2 main properties outlined in this post make the ROC curve a fairly good way to compare binary classifiers. These are great theoretical advantages that other popular metrics (such as the precision-recall or the calibration curves) don’t have.