ROC-AUC from scratch

Background

The ROC curve plots the true-positive rate against the false-positive rate as the decision threshold sweeps from high to low; the AUC (area under it) is a single, threshold-independent score for a binary classifier's ranking quality. AUC $= 1$ is perfect ranking, $0.5$ is random, and it equals the probability that a randomly chosen positive is scored above a randomly chosen negative.

Problem statement

Implement roc_auc(scores, labels) returning the area under the ROC curve via the equivalent rank/pairwise definition (the Mann-Whitney U statistic): over all positive-negative pairs, count how often the positive outscores the negative, with ties counting a half:

$$\text{AUC} = \frac{1}{|P|\,|N|}\sum_{i \in P}\sum_{j \in N}\Big[\mathbb{1}(s_i > s_j) + \tfrac{1}{2}\,\mathbb{1}(s_i = s_j)\Big]$$

Input

• scores — array-like of float: predicted scores/probabilities (higher = more positive).
• labels — array-like of {0, 1}: the true binary labels.

Output

Returns a float in $[0, 1]$.

Examples

Example 1

Input:  scores = [0.1, 0.4, 0.35, 0.8], labels = [0, 0, 1, 1]
Output: 0.75

Explanation: positives score $\{0.35, 0.8\}$ and negatives $\{0.1, 0.4\}$. Of the 4 positive-negative pairs, 3 have the positive ranked above the negative (only $0.35 < 0.4$ is out of order), so $\text{AUC} = 3/4 = 0.75$.

Constraints

• AUC is the fraction of positive-negative pairs ordered correctly, with ties counting $0.5$.
• Requires at least one positive and one negative; the result lies in $[0, 1]$.
• Equivalent to integrating the ROC curve, but the pairwise form avoids threshold bookkeeping.

Notes

• AUC depends only on the ordering of scores, not their calibration — any monotonic rescaling leaves it unchanged.
• A perfect classifier scores every positive above every negative (AUC 1); reversing the scores gives 0; random scores give 0.5 in expectation.