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Plot Optics

=================================== Demo of OPTICS clustering algorithm

… currentmodule:: sklearn

Finds core samples of high density and expands clusters from them. This example uses data that is generated so that the clusters have different densities.

The :class:~cluster.OPTICS is first used with its Xi cluster detection method, and then setting specific thresholds on the reachability, which corresponds to :class:~cluster.DBSCAN. We can see that the different clusters of OPTICS’s Xi method can be recovered with different choices of thresholds in DBSCAN.

Imports for OPTICS Density-Based Clustering

OPTICS (Ordering Points To Identify the Clustering Structure) produces an ordering of the dataset that encodes density-based clustering structure at all scales, visualized through a reachability plot. Unlike DBSCAN which requires a single eps threshold, OPTICS computes a reachability distance for each point (the minimum radius needed to connect it to a denser region) and orders points so that spatially close points are neighbors in the ordering. Cutting the reachability plot at different heights yields clusterings equivalent to DBSCAN at those eps values, while the Xi method automatically extracts clusters by detecting steep changes in the reachability profile.

Why OPTICS matters for varying-density data: The cluster_optics_dbscan function demonstrates that a single OPTICS fit can produce multiple DBSCAN-equivalent clusterings at different epsilon thresholds without refitting. At eps=0.5, only the tightest, densest clusters are recovered; at eps=2.0, looser clusters merge together. The reachability plot provides a powerful diagnostic: valleys correspond to dense clusters, and the depth of each valley indicates cluster density. Parameters min_samples controls the smoothness of the reachability plot (higher values produce smoother profiles), while xi and min_cluster_size control the sensitivity of the automatic cluster extraction from the reachability ordering.

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

import matplotlib.gridspec as gridspec
import matplotlib.pyplot as plt
import numpy as np

from sklearn.cluster import OPTICS, cluster_optics_dbscan

# Generate sample data

np.random.seed(0)
n_points_per_cluster = 250

C1 = [-5, -2] + 0.8 * np.random.randn(n_points_per_cluster, 2)
C2 = [4, -1] + 0.1 * np.random.randn(n_points_per_cluster, 2)
C3 = [1, -2] + 0.2 * np.random.randn(n_points_per_cluster, 2)
C4 = [-2, 3] + 0.3 * np.random.randn(n_points_per_cluster, 2)
C5 = [3, -2] + 1.6 * np.random.randn(n_points_per_cluster, 2)
C6 = [5, 6] + 2 * np.random.randn(n_points_per_cluster, 2)
X = np.vstack((C1, C2, C3, C4, C5, C6))

clust = OPTICS(min_samples=50, xi=0.05, min_cluster_size=0.05)

# Run the fit
clust.fit(X)

labels_050 = cluster_optics_dbscan(
    reachability=clust.reachability_,
    core_distances=clust.core_distances_,
    ordering=clust.ordering_,
    eps=0.5,
)
labels_200 = cluster_optics_dbscan(
    reachability=clust.reachability_,
    core_distances=clust.core_distances_,
    ordering=clust.ordering_,
    eps=2,
)

space = np.arange(len(X))
reachability = clust.reachability_[clust.ordering_]
labels = clust.labels_[clust.ordering_]

plt.figure(figsize=(10, 7))
G = gridspec.GridSpec(2, 3)
ax1 = plt.subplot(G[0, :])
ax2 = plt.subplot(G[1, 0])
ax3 = plt.subplot(G[1, 1])
ax4 = plt.subplot(G[1, 2])

# Reachability plot
colors = ["g.", "r.", "b.", "y.", "c."]
for klass, color in enumerate(colors):
    Xk = space[labels == klass]
    Rk = reachability[labels == klass]
    ax1.plot(Xk, Rk, color, alpha=0.3)
ax1.plot(space[labels == -1], reachability[labels == -1], "k.", alpha=0.3)
ax1.plot(space, np.full_like(space, 2.0, dtype=float), "k-", alpha=0.5)
ax1.plot(space, np.full_like(space, 0.5, dtype=float), "k-.", alpha=0.5)
ax1.set_ylabel("Reachability (epsilon distance)")
ax1.set_title("Reachability Plot")

# OPTICS
colors = ["g.", "r.", "b.", "y.", "c."]
for klass, color in enumerate(colors):
    Xk = X[clust.labels_ == klass]
    ax2.plot(Xk[:, 0], Xk[:, 1], color, alpha=0.3)
ax2.plot(X[clust.labels_ == -1, 0], X[clust.labels_ == -1, 1], "k+", alpha=0.1)
ax2.set_title("Automatic Clustering\nOPTICS")

# DBSCAN at 0.5
colors = ["g.", "r.", "b.", "c."]
for klass, color in enumerate(colors):
    Xk = X[labels_050 == klass]
    ax3.plot(Xk[:, 0], Xk[:, 1], color, alpha=0.3)
ax3.plot(X[labels_050 == -1, 0], X[labels_050 == -1, 1], "k+", alpha=0.1)
ax3.set_title("Clustering at 0.5 epsilon cut\nDBSCAN")

# DBSCAN at 2.
colors = ["g.", "m.", "y.", "c."]
for klass, color in enumerate(colors):
    Xk = X[labels_200 == klass]
    ax4.plot(Xk[:, 0], Xk[:, 1], color, alpha=0.3)
ax4.plot(X[labels_200 == -1, 0], X[labels_200 == -1, 1], "k+", alpha=0.1)
ax4.set_title("Clustering at 2.0 epsilon cut\nDBSCAN")

plt.tight_layout()
plt.show()