Calinski-Harabasz Index

Hook problem: summarize the whole geometry

Silhouette score looks point by point. Calinski-Harabasz Index asks for a coarser summary:

Are cluster centers far apart compared with how spread out their own points are?

It is still an internal clustering metric. It uses geometry and cluster assignments, not reference labels.

First naive idea: use only within-cluster spread

Small within-cluster spread is good, but it is not enough.

Three tiny clusters sitting almost on top of each other should not receive the same judgment as three tiny clusters separated across the plane.

Core invention: between scatter divided by within scatter

CH uses two sums:

• W_k: within-cluster scatter, the sum of squared distances from each point to its own cluster centroid.
• B_k: between-cluster scatter, the cluster-size-weighted squared distance from each cluster centroid to the global centroid.

Formal version

For n points and k clusters:

Plainly: the numerator rewards cluster centers that are far apart; the denominator penalizes points that wander far from their own centers.

Higher CH is better. The score is unavailable when k < 2, when n <= k, or when within-cluster scatter is zero.

Interactive preset lab

Implementation sketch

function calinskiHarabasz(between: number, within: number, n: number, k: number) {
  if (k < 2 || n <= k || within === 0) return null;
  return (between / (k - 1)) / (within / (n - k));
}

Complexity

After assignments are known, computing centroids and squared-distance sums is O(n) for fixed-dimensional points. If dimensions are variable, include the dimension factor.

Common confusions

• CH is higher-better, unlike Davies-Bouldin.
• CH is not an external score; it does not know the true labels.
• Squared distances make far-off points matter more than small local noise.

Exercises

1. Which part of the formula grows when cluster centroids move farther apart?
2. Which part grows when each cluster gets stretched?
3. Why is n <= k a degenerate case?

Graph connections : Calinski-Harabasz Index