Summary  
This chapter covers implementing and interpreting clustering evaluation metrics—such as inertia, silhouette score, Davies–Bouldin index, and Calinski–Harabasz index—to assess the compactness and separation of clusters.  

General domain of usage  
Unsupervised learning / cluster analysis

Evaluating clustering algorithms presents unique challenges because, unlike supervised learning, there are no ground truth labels to compare predictions against. This makes it essential to use specialized clustering evaluation metrics that assess how well the algorithm has grouped the data. These metrics help you understand whether clusters are compact, well-separated, and meaningful for your application, even in the absence of labeled data.

Several key metrics are commonly used to evaluate clustering quality. Each provides a different perspective on the clustering structure:

**Inertia (Within-Cluster Sum of Squares):**
$$
\text{Inertia} = \sum_{i=1}^{n} \min_{\mu_j \in C} \| x_i - \mu_j \|^2
$$
where $$x_i$$ is a data point, $$\mu_j$$ is the centroid of cluster $$j$$, and $$C$$ is the set of all centroids.

**Silhouette Score:**
$$
s(i) = \frac{b(i) - a(i)}{\max\{a(i), b(i)\}}
$$
where $$a(i)$$ is the mean distance between $$x_i$$ and all other points in the same cluster, and $$b(i)$$ is the lowest mean distance between $$x_i$$ and all points in any other cluster.

**Davies–Bouldin Index:**
$$
\text{DB} = \frac{1}{k} \sum_{i=1}^{k} \max_{j \ne i} \left( \frac{s_i + s_j}{d_{ij}} \right)
$$
where $$s_i$$ is the average distance of all points in cluster $$i$$ to its centroid, and $$d_{ij}$$ is the distance between centroids $$i$$ and $$j$$.

**Calinski–Harabasz Index:**
$$
\text{CH} = \frac{\text{Tr}(B_k)}{\text{Tr}(W_k)} \cdot \frac{n - k}{k - 1}
$$
where $$Tr(B_k)$$ is the between-cluster dispersion, $$Tr(W_k)$$ is the within-cluster dispersion, $$n$$ is the number of samples, and $$k$$ is the number of clusters.

from sklearn.datasets import make_blobs
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score, davies_bouldin_score, calinski_harabasz_score

# Generate synthetic data
X, _ = make_blobs(n_samples=300, centers=4, cluster_std=0.60, random_state=0)

# Fit KMeans clustering
kmeans = KMeans(n_clusters=4, random_state=0)
labels = kmeans.fit_predict(X)

# Compute Inertia
inertia = kmeans.inertia_

# Compute Silhouette Score
silhouette = silhouette_score(X, labels)

# Compute Davies–Bouldin Index
db_index = davies_bouldin_score(X, labels)

# Compute Calinski–Harabasz Index
ch_index = calinski_harabasz_score(X, labels)

print(f"Inertia: {inertia:.2f}")
print(f"Silhouette Score: {silhouette:.2f}")
print(f"Davies–Bouldin Index: {db_index:.2f}")
print(f"Calinski–Harabasz Index: {ch_index:.2f}")

Interpreting clustering metrics helps you assess clustering quality and select the best solution for your data:

- **Inertia**: lower values mean more compact clusters. Use inertia to compare models with the same number of clusters or to find the "elbow point" where adding more clusters gives little improvement; do not rely on it alone, as it always decreases with more clusters.

- **Silhouette Score**: ranges from -1 to 1, with higher values indicating better-separated clusters. Use it to compare different clustering solutions or pick the optimal number of clusters, but note it may be biased if clusters differ greatly in size or density.

- **Davies-Bouldin Index**: lower values reflect better clustering with well-separated, compact clusters. It provides a quick summary but can be sensitive to varying cluster shapes and densities.

- **Calinski-Harabasz Index**: higher values show better separation between clusters. It works best when clusters are similar in size and shape, but may not perform well with irregular or overlapping clusters.

**Use multiple metrics together** and visualize your clusters to get a complete, reliable evaluation.

Why do clustering algorithms require specialized evaluation metrics instead of using metrics like accuracy or precision?

A comprehensive course for intermediate learners to master the key evaluation metrics used in machine learning, covering both supervised and unsupervised tasks. The course progresses from foundational concepts to practical Python implementations, interpretation of results, and a comparative overview of metric types and their trade-offs.

Explore the fundamental metrics used to evaluate classification models, including their definitions, formulas, Python implementations, and interpretation. Includes advanced metrics and model validation techniques relevant to classification.

Delve into the core metrics for evaluating regression models, including their mathematical foundations, Python implementations, and result interpretation. Includes advanced validation techniques relevant to regression.

Examine the key metrics for evaluating unsupervised learning models, including clustering, dimensionality reduction, and anomaly detection. Learn their mathematical foundations, Python implementations, and interpretation.

Clustering Metrics

Clustering Metrics