HDBSCAN | Hierarchical density-based clustering

1. Overview #

Classic DBSCAN relies on a single eps radius; picking a value that suits every region is tricky when densities differ. HDBSCAN removes eps and instead sweeps all radius values, constructing a hierarchy of clusters. From that hierarchy it extracts the most persistent clusters—those that remain intact across a wide range of density levels.

Key parameters:

• min_cluster_size: minimum number of points that makes a cluster worth keeping.
• min_samples: how many neighbours are needed for a point to be core (defaults to min_cluster_size). Larger values make the algorithm more conservative.
• cluster_selection_method: "eom" (excess of mass) prefers stable clusters, whereas "leaf" keeps the finest-grained leaves.

2. Core distance and mutual reachability #

For each point (x) we compute the core distance

$$ d_{\mathrm{core}}(x) = \text{distance to the } \texttt{min_samples}\text{-th nearest neighbour}. $$

The mutual reachability distance between (x) and (y) is

$$ d_{\mathrm{mreach}}(x, y) = \max\left{ d_{\mathrm{core}}(x),; d_{\mathrm{core}}(y),; \lVert x - y \rVert \right}. $$

HDBSCAN builds a minimum spanning tree over these distances, condenses the hierarchy, and assigns each cluster a persistence score (integrated stability). Clusters with high persistence are reported; samples that never belong to a persistent component are labelled noise ((-1)).

3. Python example #

Use the hdbscan library (install via pip install hdbscan) to cluster a two-moons dataset:

import numpy as np
import matplotlib.pyplot as plt
import hdbscan
from sklearn.datasets import make_moons

X, _ = make_moons(n_samples=400, noise=0.08, random_state=42)

clusterer = hdbscan.HDBSCAN(
    min_cluster_size=20,
    min_samples=10,
    cluster_selection_method="eom",
)
labels = clusterer.fit_predict(X)

plt.figure(figsize=(6, 5))
plt.scatter(X[:, 0], X[:, 1], c=labels, cmap="tab10", s=20)
plt.title("HDBSCAN clustering result")
plt.xlabel("feature 1")
plt.ylabel("feature 2")
plt.grid(alpha=0.2)
plt.show()

print("Cluster persistence:", clusterer.cluster_persistence_)
print("Noise points:", np.sum(labels == -1))

cluster_persistence_ indicates which clusters are the most stable; if persistence is low, consider increasing min_cluster_size or reducing noise in the data.

4. Practical tips #

• Standardise features before clustering—distance-based methods are sensitive to scale.
• If you want more clusters, decrease min_cluster_size or switch cluster_selection_method="leaf".
• To remove tiny noisy fragments, leave min_cluster_size high and keep min_samples modest (e.g., 5–10).
• Combine with UMAP: fit UMAP to project high-dimensional data to 2–10 dimensions, then run HDBSCAN on the embedding for robust unsupervised pipelines.

5. References #