Matthews Correlation Coefficient

Formula

\[ \operatorname{MCC} = \frac{\mathrm{TP}\cdot\mathrm{TN}-\mathrm{FP}\cdot\mathrm{FN}}{\sqrt{(\mathrm{TP}+\mathrm{FP})(\mathrm{TP}+\mathrm{FN})(\mathrm{TN}+\mathrm{FP})(\mathrm{TN}+\mathrm{FN})}} \]

Parameters

• \(\mathrm{TP},\mathrm{TN},\mathrm{FP},\mathrm{FN}\): confusion matrix counts

What it means

Correlation between predicted and true labels for binary classification.

What it's used for

• Balanced metric for imbalanced binary classification.
• Single-number summary of confusion matrix.

Key properties

• Range \([-1,1]\); 1 is perfect, 0 is random, -1 is total disagreement
• Robust under class imbalance

Common gotchas

• Undefined when any denominator term is zero.
• For multiclass, use the generalized MCC formula.

Example

With \(\mathrm{TP}=8,\mathrm{FP}=2,\mathrm{FN}=1,\mathrm{TN}=9\), \(\mathrm{MCC}=70/\sqrt{9900}\approx0.70\).

How to Compute (Pseudocode)

Input: TP, FP, FN, TN
Output: MCC

den <- sqrt((TP+FP)(TP+FN)(TN+FP)(TN+FN))
if den == 0:
  return undefined (or use a library convention)
num <- TP*TN - FP*FN
return num / den

Complexity

• Time: \(O(1)\) once confusion-matrix counts are available
• Space: \(O(1)\)
• Assumptions: Binary MCC formula shown; multiclass MCC uses a generalized confusion-matrix computation

See also

• Confusion Matrix