What do the Davies-Bouldin and Calinski-Harabasz indices measure, and why can they rank two partitions oppositely?

### Davies-Bouldin For each cluster `i`, define `S_i` as the average distance from its members to its own centroid — a radius. For each pair `i, j`, define `M_ij` as the distance between the two centroids. The similarity of the pair is `R_ij = (S_i + S_j) / M_ij`: two fat clusters whose centres are close score high, two tight clusters far apart score low. The index then takes, for each cluster `i`, the **maximum** `R_ij` over all other clusters — its single most confusable neighbour — and averages those maxima over the clusters. **Lower is better**, and 0 is the unreachable ideal of zero-radius clusters infinitely far apart. The defining property is the maximum. Davies-Bouldin does not care that eleven of your twelve cluster pairs are beautifully separated; each cluster is judged only by its worst neighbour, so a single confusable pair drags the whole index. ### Calinski-Harabasz Also called the variance ratio criterion. Within-cluster dispersion is the total squared distance from points to their own centroid; between-cluster dispersion is the total squared distance from cluster centroids to the overall mean, weighted by cluster size. The index is `CH = (between_dispersion / (k - 1)) / (within_dispersion / (n - k))` for `n` points and `k` clusters. **Higher is better** and there is no upper bound. The degrees-of-freedom divisors give it the shape of a variance-ratio statistic rather than a raw quotient. Here the defining property is that everything is pooled. Every point and every cluster contributes to both sums, and because the distances are squared, points far from their centroid contribute disproportionately. Calinski-Harabasz is a statement about the overall configuration, not about any single pair. ### Why they can invert Consider two candidate partitions of the same thirteen-column table of beverage chemistry measurements — acidity, sugar, phenolics and so on — computed at the same cluster count by two different methods. - **Partition A:** every cluster is compact, but two of them sit almost on top of each other. Within-dispersion is low overall, so Calinski-Harabasz is high. But that one nearly-touching pair produces a large `R_ij` for both clusters involved, so Davies-Bouldin looks bad. - **Partition B:** no two clusters are anywhere near each other, but one cluster is a sprawling, diffuse mass. Every cluster's worst neighbour is comfortably far, so Davies-Bouldin looks good. Yet the diffuse cluster inflates the squared within-dispersion, so Calinski-Harabasz drops. A ranks better on one index and B on the other. Neither index malfunctioned. They answer different questions: *how bad is my worst confusion?* versus *how much of the total dispersion is between clusters rather than inside them?* A second, subtler source of divergence is the exponent. Davies-Bouldin is built from average distances, Calinski-Harabasz from squared ones, so a few far-flung members hurt Calinski-Harabasz much more than they hurt Davies-Bouldin. ### Practical handling Report both, together with what they disagree about, rather than quietly picking the one that supports the answer you already prefer — an index chosen after seeing the result is not evidence. Decide in advance which failure the application cannot tolerate: if two segments being confusable makes them unusable downstream, the worst-pair criterion is the relevant one; if you care about how much structure the partition captures overall, the variance ratio is. Both are invariant to multiplying every feature by one constant — numerator and denominator scale together in each — but neither is invariant to rescaling features differently, which changes the geometry, the partition and both scores. And both are computed from centroids and straight-line spreads, so they share the assumption that clusters are compact, roughly round and comparably spread. Where that assumption fails, the two indices tend to be wrong in the same direction, and their agreement is no comfort at all.