custom-dbscan-metric

Custom Distance Metric for DBSCAN

When using DBSCAN with a non-standard distance metric, you can either provide a callable to the metric parameter or precompute the distance matrix.

Mathematical Formulation

For the Mars cloud task, the distance is defined as: d(a, b) = sqrt((w * Δx)² + ((2 - w) * Δy)²)

Implementation using scipy.spatial.distance.cdist

Precomputing the distance matrix is often more efficient for grid searches if the metric is reused or if you want to use sklearn.cluster.DBSCAN.

import numpy as np
from scipy.spatial.distance import cdist
from sklearn.cluster import DBSCAN

def custom_metric(p1, p2, w):
    dx = p1[0] - p2[0]
    dy = p1[1] - p2[1]
    return np.sqrt((w * dx)**2 + ((2 - w) * dy)**2)

# Vectorized version for efficiency
def precompute_custom_distance(X, w):
    # X is (N, 2)
    # Using cdist with a custom lambda can be slow, 
    # better to use vectorized numpy if possible.
    X_weighted = X * np.array([w, 2 - w])
    # Note: the formula is sqrt((w*dx)^2 + ((2-w)*dy)^2)
    # which is equivalent to standard Euclidean distance on weighted coordinates
    return cdist(X_weighted, X_weighted, metric='euclidean')

# Using DBSCAN with precomputed metric
# dist_matrix = precompute_custom_distance(X, w)
# db = DBSCAN(eps=epsilon, min_samples=min_samples, metric='precomputed')
# labels = db.fit_predict(dist_matrix)

Considerations

• epsilon in DBSCAN will be compared against the distances produced by this custom metric.
• Ensure shape_weight (w) is applied correctly to the coordinates before distance calculation if using standard Euclidean as a shortcut.