Distance Metrics for Clustering · Euclidean, Manhattan, Cosine, and More

The distance metric is the most under-appreciated parameter in clustering. Changing the metric changes which points are considered “close” and thus fundamentally changes the clusters. DBSCAN defaults to Euclidean distance but supports custom metrics.

Key Distance Metrics

Euclidean ($L_2$): Straight-line distance. Default. Works well for continuous features on similar scales. Sensitive to scale differences — normalise/standardise first.

Manhattan ($L_1$): Sum of absolute differences. More robust to outliers than Euclidean. Useful for grid-like data (city blocks) and when features represent counts.

Cosine: 1 minus the cosine of the angle between vectors. Ignores magnitude — two documents with similar word proportions but different lengths are similar. Dominant in text clustering (TF-IDF vectors).

Mahalanobis: Euclidean distance scaled by the inverse covariance matrix. Accounts for feature correlations and different scales. Computationally expensive ($O(d^2)$ per distance calculation).

Haversine: Great-circle distance between two points on a sphere. For geographic coordinates (latitude/longitude). DBSCAN with Haversine distance clusters GPS coordinates correctly (unlike Euclidean, which treats lat/lon as Cartesian).

Choosing a Metric

  • Continuous, normalised: Euclidean
  • Count data, heavy tails: Manhattan
  • Text (TF-IDF, embeddings): Cosine
  • Correlated features: Mahalanobis
  • Geographic (lat/lon): Haversine
  • Mixed types: Gower distance (pre-compute)

Scikit-learn’s DBSCAN(metric='precomputed') accepts a pre-computed distance matrix — use this for custom or non-standard metrics. The matrix is $n \times n$ and must be symmetric.

Impact Example

Clustering the same dataset with Euclidean vs Manhattan vs Cosine can produce completely different results. For a dataset of customer transactions, Euclidean clusters by total spending; Manhattan clusters by distribution across categories; Cosine clusters by spending pattern (ignoring total). All are “correct” — they answer different questions.