Confusion Matrix¶

Formula¶

\[ \begin{array}{c|cc} & \hat y=1 & \hat y=0 \\\hline y=1 & TP & FN \\ y=0 & FP & TN \end{array} \]

Plot¶

type: bars
xs: 0 | 1 | 2 | 3
ys: 42 | 8 | 6 | 44
xmin: -0.5
xmax: 3.5
ymin: 0
ymax: 50
height: 280
title: Example counts: TP, FP, FN, TN (bar view)

Parameters¶

TP, FP, TN, FN: classification outcome counts

What it means¶

Tabulates prediction outcomes against true labels for a chosen threshold.

What it's used for¶

Deriving threshold-dependent metrics like precision, recall, specificity, and accuracy.
Error analysis by type.

Key properties¶

Changes when the decision threshold changes.
Supports cost-sensitive evaluation by weighting error types.

Common gotchas¶

A single confusion matrix can hide threshold tradeoffs.
Class imbalance can make accuracy look misleading.

Example¶

If fraud is rare, inspect FP and FN directly rather than relying on accuracy alone.

How to Compute (Pseudocode)¶

Input: true labels y[1..n], predicted labels y_hat[1..n] (binary)
Output: TP, FP, FN, TN counts

initialize TP, FP, FN, TN <- 0
for i from 1 to n:
  update the appropriate count based on (y[i], y_hat[i])
return TP, FP, FN, TN

Complexity¶

Time: \(O(n)\) for \(n\) labeled predictions
Space: \(O(1)\) extra space for the four counts
Assumptions: Binary classification confusion matrix; multiclass confusion matrices use \(K\times K\) count tables