Hierarchical clustering: structured vs unstructured ward

Example builds a swiss roll dataset and runs hierarchical clustering on their position.

For more information, see Hierarchical clustering.

In a first step, the hierarchical clustering is performed without connectivity constraints on the structure and is solely based on distance, whereas in a second step the clustering is restricted to the k-Nearest Neighbors graph: it’s a hierarchical clustering with structure prior.

Some of the clusters learned without connectivity constraints do not respect the structure of the swiss roll and extend across different folds of the manifolds. On the opposite, when opposing connectivity constraints, the clusters form a nice parcellation of the swiss roll.

# Authors : Vincent Michel, 2010
#           Alexandre Gramfort, 2010
#           Gael Varoquaux, 2010
# License: BSD 3 clause

import time as time

# The following import is required
# for 3D projection to work with matplotlib < 3.2

import mpl_toolkits.mplot3d  # noqa: F401

import numpy as np

Generate data

We start by generating the Swiss Roll dataset.

from sklearn.datasets import make_swiss_roll

n_samples = 1500
noise = 0.05
X, _ = make_swiss_roll(n_samples, noise=noise)
# Make it thinner
X[:, 1] *= 0.5

Compute clustering

We perform AgglomerativeClustering which comes under Hierarchical Clustering without any connectivity constraints.

from sklearn.cluster import AgglomerativeClustering

print("Compute unstructured hierarchical clustering...")
st = time.time()
ward = AgglomerativeClustering(n_clusters=6, linkage="ward").fit(X)
elapsed_time = time.time() - st
label = ward.labels_
print(f"Elapsed time: {elapsed_time:.2f}s")
print(f"Number of points: {label.size}")

Compute unstructured hierarchical clustering...
Elapsed time: 0.03s
Number of points: 1500

Plot result

Plotting the unstructured hierarchical clusters.

import matplotlib.pyplot as plt

fig1 = plt.figure()
ax1 = fig1.add_subplot(111, projection="3d", elev=7, azim=-80)
ax1.set_position([0, 0, 0.95, 1])
for l in np.unique(label):
    ax1.scatter(
        X[label == l, 0],
        X[label == l, 1],
        X[label == l, 2],
        color=plt.cm.jet(float(l) / np.max(label + 1)),
        s=20,
        edgecolor="k",
    )
_ = fig1.suptitle(f"Without connectivity constraints (time {elapsed_time:.2f}s)")

We are defining k-Nearest Neighbors with 10 neighbors

from sklearn.neighbors import kneighbors_graph

connectivity = kneighbors_graph(X, n_neighbors=10, include_self=False)

Compute clustering

We perform AgglomerativeClustering again with connectivity constraints.

print("Compute structured hierarchical clustering...")
st = time.time()
ward = AgglomerativeClustering(
    n_clusters=6, connectivity=connectivity, linkage="ward"
).fit(X)
elapsed_time = time.time() - st
label = ward.labels_
print(f"Elapsed time: {elapsed_time:.2f}s")
print(f"Number of points: {label.size}")

Compute structured hierarchical clustering...
Elapsed time: 0.05s
Number of points: 1500

Plot result

Plotting the structured hierarchical clusters.

fig2 = plt.figure()
ax2 = fig2.add_subplot(121, projection="3d", elev=7, azim=-80)
ax2.set_position([0, 0, 0.95, 1])
for l in np.unique(label):
    ax2.scatter(
        X[label == l, 0],
        X[label == l, 1],
        X[label == l, 2],
        color=plt.cm.jet(float(l) / np.max(label + 1)),
        s=20,
        edgecolor="k",
    )
fig2.suptitle(f"With connectivity constraints (time {elapsed_time:.2f}s)")

plt.show()