Fowlkes-Mallows Index

Hook problem: focus on pairs placed together

Rand Index includes TN: pairs that are different in the reference labels and different in the clustering.

Sometimes you want a metric that focuses on the positive clustering decision: “these two items belong together.”

Fowlkes-Mallows Index uses only TP, FP, and FN.

First naive idea: use pair precision only

Pair precision asks:

That measures how trustworthy same-cluster pairs are. But it ignores same-label pairs that the clustering split apart.

Core invention: balance pair precision and pair recall

Pair recall asks:

FMI combines pair precision and pair recall with a geometric mean:

The equivalent count form is:

For the fixture:

Interactive preset lab

Why TN is absent

TN means two items were apart in both the reference labels and the clustering. That is real agreement for RI, but it does not tell us whether predicted same-cluster groups are useful.

FMI therefore focuses on the co-clustered pair decision.

Edge cases

If TP + FP = 0, there are no predicted same-cluster pairs, so pair precision is unavailable.

If TP + FN = 0, there are no reference same-label pairs, so pair recall is unavailable.

In either case, render FMI as not available rather than NaN.

Implementation sketch

function fowlkesMallows(tp: number, fp: number, fn: number) {
  const denominator = Math.sqrt((tp + fp) * (tp + fn));
  return denominator === 0 ? null : tp / denominator;
}

Complexity

Once pair counts are known, FMI is O(1). Building pair counts directly is O(n^2), or it can be derived from a contingency table.

Common confusions

• FMI is not classifier F1, though both balance two positive-side quantities.
• FMI does not adjust for chance the way ARI does.
• A clustering that merges everything can have high pair recall but weak pair precision.

Exercises

1. Why does FMI ignore TN?
2. What happens to FMI in the singleton over-split preset?
3. Which mistake hurts pair precision: FP or FN?

Graph connections : Fowlkes-Mallows Index