Davies-Bouldin index

The problem, from first principles

Silhouette scores each point. Sometimes you want a single number for the whole clustering that asks whether clusters are compact and far apart. The Davies-Bouldin index does that: for each cluster, find the most similar other cluster (similar means spread-to-separation ratio is high), then average those worst-case ratios. Lower is better, and 0 is the floor.

Implement davies_bouldin_index(X, labels).

1. For each cluster i, compute its centroid and its scatter S_i, the mean distance from its points to the centroid.
2. For each pair i ≠ j, compute R_ij = (S_i + S_j) / d(centroid_i, centroid_j).
3. Return the mean over clusters i of max over j ≠ i of R_ij.

Constraints

• Fewer than two distinct labels raises ValueError.
• Use Euclidean distance.

Hints

Open one at a time. Each gives away a little more than the last.