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Chapter 134 – Agglomerative vs Divisive Clustering

1. Two Directions on the Same Tree

Hierarchical clustering builds nested groups. Agglomerative works bottom-up; divisive works top-down. Both end in a hierarchy, but they travel opposite paths.

2. Agglomerative (Bottom-Up)

Each point starts as its own cluster. Closest pairs are merged repeatedly until one root remains. Linkage choices:

• Single: nearest pair between clusters
• Complete: farthest pair
• Average: mean of all cross pairs
• Ward: minimize post-merge variance

Distances are usually Euclidean or Manhattan.

3. Divisive (Top-Down)

All points start in one root cluster. Dissimilar subsets are split until leaves (or a depth stop) are reached. Captures global structure but is much more expensive to run.

4. Practical Trade-Offs

• Divisive: heavy compute; less common in day-to-day work
• Agglomerative: simpler, widely supported, often cleaner boundaries

5. Scikit-Learn Sketch

from sklearn.cluster import AgglomerativeClustering, BisectingKMeans
from sklearn.datasets import make_blobs
import matplotlib.pyplot as plt

X, _ = make_blobs(n_samples=200, centers=3, cluster_std=0.9, random_state=42)

agg = AgglomerativeClustering(n_clusters=3, linkage="ward")
agg_labels = agg.fit_predict(X)

div = BisectingKMeans(n_clusters=3, random_state=42)
div_labels = div.fit_predict(X)

fig, axes = plt.subplots(1, 2, figsize=(9, 4))
axes[0].scatter(X[:, 0], X[:, 1], c=agg_labels, cmap="viridis", s=30)
axes[0].set_title("Agglomerative")
axes[1].scatter(X[:, 0], X[:, 1], c=div_labels, cmap="viridis", s=30)
axes[1].set_title("Divisive")
plt.tight_layout(); plt.show()

On synthetic blobs, agglomerative often separates boundaries cleanly; divisive may misassign a few boundary points.

6. Comparison Matrix

Aspect	Agglomerative	Divisive
Direction	Bottom-up	Top-down
Start	One cluster per point	One root for all
Operation	Merge closest	Split by dissimilarity
Cost	Lower / practical	Very expensive
Scikit-Learn	AgglomerativeClustering	BisectingKMeans