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Fowlkes-Mallows Index

Formula

\[ \mathrm{FM}=\frac{\mathrm{TP}}{\sqrt{(\mathrm{TP}+\mathrm{FP})(\mathrm{TP}+\mathrm{FN})}} =\sqrt{\mathrm{Precision}\cdot \mathrm{Recall}} \]

Parameters

  • \(\mathrm{TP},\mathrm{FP},\mathrm{FN}\): pair-counts over point pairs for two partitions
  • Precision/Recall: computed on pairwise same-cluster decisions

What it means

Geometric mean of pairwise precision and recall for clustering agreement.

What it's used for

  • External clustering validation against reference labels.
  • Comparing clustering outputs using pairwise membership decisions.

Key properties

  • Range \([0,1]\), higher is better
  • Invariant to cluster label permutations

Common gotchas

  • Pairwise counts scale as \(O(n^2)\) conceptually (implementations optimize this).
  • Not adjusted for chance (unlike ARI).

Example

If pairwise precision is \(0.81\) and recall is \(0.64\), then \(\mathrm{FM}=\sqrt{0.81\cdot0.64}=0.72\).

How to Compute (Pseudocode)

Input: true labels and predicted labels (or sets/masks, depending on the metric)
Output: Fowlkes-Mallows score

build the contingency table / overlap counts needed by the metric
compute the metric numerator and denominator from those counts
apply any normalization/adjustment terms required by the definition
return the score

Complexity

  • Time: Typically \(O(n)\) to accumulate counts over \(n\) labeled examples once labels/sets are aligned (plus optional \(O(k^2)\) work on contingency tables for some metrics)
  • Space: Depends on the contingency-table size (from \(O(1)\) count accumulators to \(O(k_1 k_2)\) for label-table storage)
  • Assumptions: Exact complexity depends on binary-mask vs multiclass-label formulation and whether pair-count terms are computed from counts or explicit pairs

See also