Davies–Bouldin indicator

Mainly for the data that has even density and distribution

The Davies–Bouldin index (DBI) is an indicator that measures the quality of clustering achieved by a clustering algorithm. Specifically, it provides a score that indicates the average similarity between each cluster and its most similar cluster, with the ideal score being the smallest possible value.

Here’s a breakdown of the Davies–Bouldin index:

Within-cluster scatter: For each cluster, compute the average distance between each data point in the cluster and the centroid of the cluster. This provides a measure of the scatter or spread of the points within a cluster.

Of course! Here are the formulas for the Davies–Bouldin index written in LaTeX:

1. Within-cluster scatter for each cluster: S_i = \frac{1}{|C_i|} \sum_{x \in C_i} \left\\| x - c_i \right\\| Where:

   • $C_i$ represents the i-th cluster.
   • $c_i$ denotes the centroid of the i-th cluster.
   • $|C_i|$ indicates the number of data points in the i-th cluster.

2. Between-cluster separation for each pair of clusters: d(c_i, c_j) = \left\\| c_i - c_j \right\\|

3. Similarity measure between two clusters: $R_{ij}=\frac{S_i+S_j}{d(c_i,c_j)}$ This formula computes the similarity between clusters $i$ and $j$.

4. Davies–Bouldin Index: $DBI=\frac{1}{N}{\sum }_{i=1}^N{\max }_{j\ne i}\left(R_{ij}\right)$ Where $N$ represents the total number of clusters.

These formulas provide a comprehensive representation of the Davies–Bouldin index in mathematical terms.

In summary, the Davies–Bouldin index calculates the ratio of within-cluster scatter to between-cluster separation for each cluster, and then averages these ratios across all clusters. A lower DBI value indicates better clustering since it means that clusters are compact (small within-cluster scatter) and well-separated (large between-cluster separation).

When evaluating clustering results, it’s important to consider multiple evaluation metrics alongside the DBI, as each has its strengths and limitations.

python

import numpy as np
from sklearn.cluster import KMeans
from sklearn.metrics import pairwise_distances
 
def davies_bouldin_index(X, labels):
    n_cluster = len(np.bincount(labels))
    cluster_k = [X[labels == k] for k in range(n_cluster)]
    centroids = [np.mean(k, axis=0) for k in cluster_k]
 
    # Calculate the scatter (S_i) for each cluster
    S = [np.mean(pairwise_distances(cluster_k[i], [centroids[i]])) for i in range(n_cluster)]
 
    # Calculate the pairwise centroid distances (d(c_i, c_j))
    centroid_distances = pairwise_distances(centroids)
    np.fill_diagonal(centroid_distances, float('inf'))
 
    # Calculate the similarity (R_ij) between each pair of clusters
    R = np.zeros((n_cluster, n_cluster))
    for i in range(n_cluster):
        for j in range(n_cluster):
            if i != j:
                R[i, j] = (S[i] + S[j]) / centroid_distances[i, j]
 
    # Calculate the Davies–Bouldin index
    dbi = np.mean(np.max(R, axis=1))
 
    return dbi
 
# Sample dataset and clustering
X = np.array([[1, 2], [5, 8], [1.5, 1.8], [8, 8], [1, 0.6], [9, 11]])
kmeans = KMeans(n_clusters=2).fit(X)
labels = kmeans.labels_
 
# Calculate DBI
dbi_value = davies_bouldin_index(X, labels)
print(f"Davies–Bouldin index: {dbi_value}")