• ## ROC Area-Under-the-Curve Explained

Posted by Konstantin 12.10.2011 16 Comments

The Receiver Operating Characteristic Area Under the Curve (ROC AUC) is a measure of classifier performance, which is widely used in machine learning. Unfortunately, the obscure way the concept is explained in most sources makes it fairly hard to grasp its intuitive meaning. The name "area under the curve" itself is ill-conceived and is utterly useless in helping the intution. The aim of this post is to aid those struggling with the concept, and also present a simple and intuitive interpretation of the ROC AUC metric as the "average positive rank" which I, so far, have not seen stated explicitly elsewhere.

To facilitate explanation, let us consider a hypothetical classification problem: classifying chocolate bars to "very good" or "just OK" (hereby we presume a chocolate bar may not be considered "bad" by definition of chocolate). Suppose we have a sample of particular chocolate bar types, for which we have carefully assigned the proper classification, e.g.:

Item Class
Mars 0
Milka 1
Kalev 0
Lindt 1
Ferrero-Rocher 1
Laima 0
Hershey's 0

where class label "1" denotes "very good" and "0" means "just OK". Suppose we have derived a classifier which, based on some features of a chocolate bar, such as its chocolate content, packaging, country of origin, etc, attempts to assign a "goodness score" which should resemble our notion of goodness. For concreteness' sake, let us say the classifier assigned the following scores:

Item True class Assigned score
Milka 1 8
Mars 0 5
Lindt 1 3
Ferrero-Rocher 1 3
Kalev 0 1
Laima 0 -3
Hershey's 0 -5

How do we assess the goodness of such a classifier? The "ROC approach" suggests we do the following. First, we shall plot a ROC curve. To do that, we order the chocolate bars according to their assigned score (this is already done in the table above) and use this ordering to write out the vector of the class labels (the "sorted prediction vector"):

Highest rated item's true class Lowest rated item's true class
1 0 1 1 0 0 0

Obviously, the best possible classifier would order the instances so that all the ones would be on the left of all the zeroes. The particular classifier we chose here looks good, but not perfect.

Next, we use the obtained vector to plot the ROC curve. To do this plot, forget all of the "True Positive Rate" and "False Positive Rate" nonsense you might have read about in most textbooks, and follow the following simple recipe:

1. Draw a square with sides of length 1. Denote its its lower left corner as the "origin", its horizontal axis as "the negatives" and its vertical axis as "the positives". Split the negatives axis in as many parts as there are examples of class "0" in the dataset. Split the positives axis in as many parts as there are examples of class "1" in your dataset. In our case, the square will look as follows:

2. Next, take the sorted prediction vector, and draw a path, starting from the origin. For each "1" in the sorted prediction vector, the path will move one step "up", and for each "0" in the vector, the path will move one step "right" (note that if you have multiple predictions with the same score, make all the corresponding "up" and "right" moves at once by drawing the corresponding diagonal). In our case, the path goes ("up", "right", "up", "up", "right", "right", "right") and hence looks as follows:

Voilà, this is your ROC curve. Obviously, the "perfect" classifier would make all the "up" steps first and the "right" steps last, hence the ideal ROC curve looks like a perfect corner. A random classifier would mix his "up" and "right" steps randomly, and the curve would most probably follow a diagonal-ish path. The one-number summary of the "goodness" of such a path is the "area under the curve" (coloured gray here), which will be 1.0 for the ideal case, and somewhere around 0.5 for the worst case. In our case the area is 10/12 ~ 0.83.

Here comes the important question — what does this number actually mean? One good answer is given by the following probabilistic interpretation:

The area under ROC curve specifies the probability that, when we draw one positive and one negative example at random, the decision function assigns a higher value to the positive than to the negative example.

There is, however, yet another, to my mind even more intuitive, interpretation. Define the rank of a positive example as the proportion of negative examples "to the right of it" in the sorted prediction vector. For example, if the sorted prediction vector is (1, 0, 0, 0, 0), then the rank of the sole positive item there is 1.0 (all negative examples are to the right). If the sorted prediction vector is (0, 0, 1, 0, 0), the rank is 0.5 (half of negative examples is to the right), and if the vector is (0, 0, 0, 0, 1), the rank is 0. In our chocolate bar example, the ranks of the positive examples are the following:

Highest rated item's true class Lowest rated item's true class
Sorted prediction vector 1 0 1 1 0 0 0
Rank 1.0 0.75 0.75

The "rank" thus denotes where would a classifier position a given positive item within the set of negative examples, whether closer to the "left" (rank 1) or closer to the "right" (rank 0).

Now the average rank of all the positive examples seems like a reasonable metric of classifier performance, doesn't it? In fact, is exactly the ROC AUC! Now the value of 0.80 ROC score can be interpreted intuitively as follows: "the average positive example has about 20% of negative examples scored higher than it". This, so far, seems to me personally to be the "simplest" interpretation of the ROC AUC. So write it down:

ROC AUC is the average positive rank.

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