K-means: choose features before choosing the number of clusters

K-means minimizes squared Euclidean distance to centroids. Features and units therefore define what “close” means before the algorithm ever considers the number of clusters. Searching k on a weak representation only optimizes a weak question.

Hold k fixed and change the representation

Our synthetic fixture has 360 customers and three behaviour features: visits over 90 days, average order and return rate. The generator contains three authored behaviour types, but those labels never enter clustering. We use them only afterward to audit whether a representation recovered the construction.

With the three behaviour features standardized and k=3, mean silhouette is 0.6196 and adjusted Rand index against generator labels is 1.0. Clustering the raw units yields silhouette 0.6081 but ARI only 0.6587 because average order dominates smaller-scale variables.

Adding a standardized random irrelevant_code feature lowers silhouette to 0.4306. ARI remains high at 0.9917 in this easy fixture, but the weaker separation shows that equal scaling does not make an irrelevant feature useful.

The KMeans documentation defines its inertia objective. The metric treats every included coordinate as part of customer similarity.

from unsupervised_cases import kmeans_case

r = kmeans_case()
assert r['rows'] == 360
assert r['features'] == ['visits_90d', 'average_order', 'return_rate']
assert r['scaled_behavior']['generator_ari'] == 1
assert r['raw_unscaled']['generator_ari'] < .7
assert r['scaled_with_irrelevant_code']['silhouette'] < r['scaled_behavior']['silhouette']
print(r)

Run this in the unsupervised lab. Generator ARI is unavailable in a real segmentation project unless an independent reference taxonomy exists.

Define similarity from the decision

Include features only when their meaning, availability and time window match the segmentation use case. Monetary, frequency and rate features may need transformations for skew and recency. Highly correlated variables can double-count one behaviour. Identifiers and arbitrary codes should not enter a Euclidean space.

Scaling expresses a weighting choice. Standardization gives each feature unit variance in the fitting population; it does not prove equal business importance. Robust scaling may help with extreme values, while domain weights require documented justification and sensitivity analysis.

Fit transformations on the intended training or reference period. When assigning later customers, reuse the saved scaler and centroids. Recomputing scaling each period changes the coordinate system and makes movement harder to interpret.

After the feature contract is stable, compare plausible k values with silhouette, stability, profiles and minimum useful segment size. A cluster should support a distinguishable action or analysis; a high internal score alone does not create value.

Exercise: remove each behaviour feature one at a time and repeat the cluster audit. Add a duplicated average-order column and show how it changes distance weighting. Write the feature contract you would require before any customer receives a segment-driven action.

NeuraPath's Data Science course teaches clustering as a representation decision. Feature meaning comes before elbow plots and cluster counts.

Continue learning

This article is part of the Clustering, reduction and recommendations sequence. Use the neighbouring tasks when you need the prerequisite or the next application.

• Continue with Silhouette scores: useful signal and common misinterpretations.
• Then apply it in DBSCAN: distinguish noise from a bad density setting.