ROC Curve¶

Formula¶

\[ \mathcal{C} = \{(\mathrm{FPR}(t),\,\mathrm{TPR}(t)) : t \in \mathbb{R}\} \]

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: ROC curve with random baseline (illustrative)

Parameters¶

\(\mathrm{TPR} = \mathrm{Recall}\)
\(\mathrm{FPR} = \mathrm{FP}/(\mathrm{FP}+\mathrm{TN})\)
\(t\): decision threshold

What it means¶

Tradeoff between true-positive rate and false-positive rate across thresholds.

What it's used for¶

Visualizing TPR vs FPR across thresholds.
Comparing classifiers independent of a threshold.

Key properties¶

AUC-ROC is threshold-invariant
Equivalent to ranking performance under mild assumptions

Common gotchas¶

Can look overly optimistic under extreme class imbalance.
Uses \(\mathrm{TN}\), unlike PR curves.

Example¶

Two thresholds might give points \((\mathrm{FPR},\mathrm{TPR})=(0.1,0.8)\) and \((0.3,0.9)\).

How to Compute (Pseudocode)¶

Input: scores p_hat[1..n], labels y[1..n]
Output: ROC curve points

sort examples by score descending
sweep a threshold from high to low through unique scores
at each threshold, update confusion-matrix counts incrementally
record the corresponding curve point (TPR/FPR for ROC or Precision/Recall for PR)
return all curve points

Complexity¶

Time: \(O(n\log n)\) due to sorting, plus a linear threshold sweep
Space: \(O(n)\) for sorted scores/labels and output curve points
Assumptions: Binary ranking scores; ties and interpolation conventions depend on the implementation