Custom distance metrics

Custom Distance Metrics for Clustering

When to Use

Use this skill whenever you need a non-standard distance metric for DBSCAN, hierarchical clustering, or any sklearn/scipy algorithm that accepts a metric parameter. Typical cases: weighted axes, domain-specific similarity, or anisotropic distance.

DBSCAN with Custom Metrics

DBSCAN accepts a callable metric via the metric parameter. The callable receives two 1-D arrays (points) and must return a scalar distance.

from sklearn.cluster import DBSCAN
import numpy as np

def weighted_euclidean(a, b, w):
    """Weighted Euclidean: d = sqrt((w*dx)^2 + ((2-w)*dy)^2)"""
    dx = a[0] - b[0]
    dy = a[1] - b[1]
    return np.sqrt((w * dx) ** 2 + ((2 - w) * dy) ** 2)

# Use functools.partial or a closure to bind extra parameters
from functools import partial
metric_fn = partial(weighted_euclidean, w=1.2)

db = DBSCAN(eps=10, min_samples=5, metric=metric_fn)
labels = db.fit_predict(points)  # points: array of shape (n, 2)

Performance Considerations

• Custom callable metrics are slower than built-in metrics because sklearn cannot use optimized C routines.
• For large datasets, precompute the full distance matrix and pass metric='precomputed':

from scipy.spatial.distance import pdist, squareform

# Precompute pairwise distances
dists = pdist(points, metric=metric_fn)
dist_matrix = squareform(dists)

db = DBSCAN(eps=10, min_samples=5, metric='precomputed')
labels = db.fit_predict(dist_matrix)

• Precomputed matrices use O(n^2) memory but are significantly faster when DBSCAN is called repeatedly on the same data or when the custom metric is expensive.

Extracting Cluster Centroids

After clustering, compute centroids from the original coordinates (not the distance matrix):

centroids = []
for label in set(labels):
    if label == -1:
        continue  # skip noise
    mask = labels == label
    centroid = points[mask].mean(axis=0)
    centroids.append(centroid)