DBI - Davies-Bouldin Index

The Davies-Bouldin Index (DBI) is an internal evaluation metric for clustering algorithms [36]. It is defined as the average similarity measure of each cluster with its most similar cluster, where similarity is the ratio of within-cluster distances to between-cluster distances.

Intuitively, DBI evaluates how well the clustering has separated the data. It answers the question: “For each cluster, how distinct is it from its closest neighboring cluster?” Clusters that are far apart and highly compact will result in a lower DBI score. A smaller DBI indicates a better clustering partition.

\[\text{DBI} = \frac{1}{K} \sum_{k=1}^{K} \max_{j \neq k} \left( \frac{\Delta_k + \Delta_j}{\delta(c_k, c_j)} \right)\]

Where:

  • \(K\) is the total number of clusters.

  • \(c_k\) and \(c_j\) are the centroids of clusters \(k\) and \(j\) respectively.

  • \(\Delta_k\) is the intra-cluster dispersion (average distance of all elements in cluster \(k\) to its centroid \(c_k\)).

  • \(\delta(c_k, c_j)\) is the inter-cluster separation (Euclidean distance between centroids \(c_k\) and \(c_j\)).

Handling Edge Cases (Finite Values)

The Davies-Bouldin index requires comparing at least two distinct clusters to calculate the inter-cluster separation \(\delta(c_k, c_j)\). It is mathematically undefined when there is only one cluster (\(K = 1\)).

  • force_finite (bool): If True, the function will catch the undefined mathematical operation and return a safe, finite number instead of raising a ValueError. Default is True.

  • finite_value (float): The specific fallback value returned when force_finite=True and the clustering has only 1 cluster. Because a smaller score is better for DBI, the default fallback is a large penalty value (1e10).

Properties

  • Best possible score: 0.0 (Smaller value is better. A score of 0 implies perfectly compact clusters that are infinitely far apart).

  • Worst possible score: +inf (or the defined penalty finite_value).

  • Range: [0.0, +inf)

  • References: Scikit-Learn Davies-Bouldin

Example Usage

from permetrics.clustering import ClusteringMetric
import numpy as np

# ==============================================================================
# SCENARIO 1: Normal Clustering Evaluation
# ==============================================================================
print("--- 1. BASIC DAVIES-BOULDIN INDEX EXAMPLE ---")

X_data = np.array([[1, 2], [1, 4], [1, 0], [10, 2], [10, 4], [10, 0]])
y_pred_labels = np.array([0, 0, 0, 1, 1, 1])

cm = ClusteringMetric(X=X_data, y_pred=y_pred_labels)
dbi_score = cm.DBI()
print(f"Davies-Bouldin Index: {dbi_score}")

# ==============================================================================
# SCENARIO 2: Edge Case with 1 Cluster (Demonstrating force_finite)
# ==============================================================================
print("\n--- 2. EDGE CASE (1 CLUSTER) EXAMPLE ---")

# All data points are predicted to be in the same single cluster (label 0)
y_pred_single = np.array([0, 0, 0, 0, 0, 0])
cm_single = ClusteringMetric(X=X_data, y_pred=y_pred_single)

# Returns the penalty finite_value (1e10) instead of crashing
dbi_safe = cm_single.DBI(force_finite=True, finite_value=1e10)
print(f"DBI with 1 cluster (Safe Mode): {dbi_safe}")