ROC AUC

Formula

\[ \mathrm{AUC} = P(s(x^+) > s(x^-)) + \tfrac{1}{2}P(s(x^+)=s(x^-)) \]

Plot

fns: x | sqrt(x)
colors: #111111 | #ff6b2c
labels: Random baseline | Example model
xmin: 0
xmax: 1
ymin: 0
ymax: 1.05
height: 280
title: AUC intuition from area under ROC (illustrative)

Parameters

• \(s(\cdot)\): score function
• \(x^+\): positive example
• \(x^-\): negative example

What it means

AUC is the area under the ROC curve (TPR vs FPR) as you sweep a score threshold.

What it's used for

• Measuring ranking quality of binary classifiers.
• Comparing models independent of a threshold.

Key properties

• Threshold-independent ranking metric
• Invariant to any strictly monotone transform of scores

Common gotchas

• AUC can look “good” under extreme class imbalance even when precision is poor
• If you care about top-of-list performance, consider PR AUC / precision@k

Example

If all positives are scored above all negatives, AUC = 1.0.

How to Compute (Pseudocode)

Input: ROC curve points (FPR, TPR) sorted by FPR
Output: ROC AUC

auc <- 0
for each consecutive pair of ROC points:
  auc <- auc + trapezoid_area_between_points
return auc

Complexity

• Time: \(O(L)\) once \(L\) ROC curve points are available
• Space: \(O(1)\) extra space (excluding stored curve points)
• Assumptions: If starting from raw scores, ROC construction typically dominates at \(O(n\log n)\)