Clustering

The idea#

You have ten thousand customer records and no segmentation. Clustering lays them on a table so that similar things sit together, and draws circles around them. Which circles are meaningful is then decided by a person.

What it is good for#

• Customer segments as a starting point for a discussion, not as a result.
• Bundling similar error reports before anyone reads them.
• Organising product ranges, grouping sites, surveying text corpora.

The methods compared#

import numpy as np
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score, adjusted_rand_score
from sklearn.preprocessing import StandardScaler

rng = np.random.default_rng(0)
X = np.vstack([rng.normal(m, s, (200, 2))
               for m, s in ([(0,0), .5], [(5,5), .8], [(0,6), .4])])
X = StandardScaler().fit_transform(X)

# Stability check: two runs on two seeds, then compare.
a = KMeans(3, n_init=10, random_state=1).fit_predict(X)
b = KMeans(3, n_init=10, random_state=2).fit_predict(X)
print("silhouette", round(silhouette_score(X, a), 3))
print("stability ", round(adjusted_rand_score(a, b), 3))   # close to 1 is good

The k-means objective#

This quantity falls monotonically with k and is therefore worthless as a selection criterion on its own. The elbow criterion looks for the point where the gain bends: a heuristic, not a test.

Why k-means thinks spherically#

Assignment minimises Euclidean distance to a centroid. Cluster boundaries are therefore perpendicular bisectors and the cells form a Voronoi diagram: convex polygons. Elongated, curved or nested groups cannot be captured in principle, regardless of k.

Cost#

At n = 10⁶, agglomerative clustering with a 4-terabyte distance matrix is out of the question. In practice you reduce dimensionality first, see Dimensionality reduction, and then cluster.

The legal point#

Clusters over personal data are new personal attributes. They fall under purpose limitation and are subject to access requests. Where a cluster feeds a decision about a person, it carries a duty to justify, and "the procedure grouped them that way" is not a justification.