Comparing different hierarchical linkage methods on toy datasets#

This example shows characteristics of different linkage methods for hierarchical clustering on datasets that are “interesting” but still in 2D.

The main observations to make are:

  • single linkage is fast, and can perform well on non-globular data, but it performs poorly in the presence of noise.

  • average and complete linkage perform well on cleanly separated globular clusters, but have mixed results otherwise.

  • Ward is the most effective method for noisy data.

While these examples give some intuition about the algorithms, this intuition might not apply to very high dimensional data.

import time
import warnings
from itertools import cycle, islice

import matplotlib.pyplot as plt
import numpy as np

from sklearn import cluster, datasets
from sklearn.preprocessing import StandardScaler

Generate datasets. We choose the size big enough to see the scalability of the algorithms, but not too big to avoid too long running times

n_samples = 1500
noisy_circles = datasets.make_circles(
    n_samples=n_samples, factor=0.5, noise=0.05, random_state=170
noisy_moons = datasets.make_moons(n_samples=n_samples, noise=0.05, random_state=170)
blobs = datasets.make_blobs(n_samples=n_samples, random_state=170)
rng = np.random.RandomState(170)
no_structure = rng.rand(n_samples, 2), None

# Anisotropicly distributed data
X, y = datasets.make_blobs(n_samples=n_samples, random_state=170)
transformation = [[0.6, -0.6], [-0.4, 0.8]]
X_aniso =, transformation)
aniso = (X_aniso, y)

# blobs with varied variances
varied = datasets.make_blobs(
    n_samples=n_samples, cluster_std=[1.0, 2.5, 0.5], random_state=170

Run the clustering and plot

# Set up cluster parameters
plt.figure(figsize=(9 * 1.3 + 2, 14.5))
    left=0.02, right=0.98, bottom=0.001, top=0.96, wspace=0.05, hspace=0.01

plot_num = 1

default_base = {"n_neighbors": 10, "n_clusters": 3}

datasets = [
    (noisy_circles, {"n_clusters": 2}),
    (noisy_moons, {"n_clusters": 2}),
    (varied, {"n_neighbors": 2}),
    (aniso, {"n_neighbors": 2}),
    (blobs, {}),
    (no_structure, {}),

for i_dataset, (dataset, algo_params) in enumerate(datasets):
    # update parameters with dataset-specific values
    params = default_base.copy()

    X, y = dataset

    # normalize dataset for easier parameter selection
    X = StandardScaler().fit_transform(X)

    # ============
    # Create cluster objects
    # ============
    ward = cluster.AgglomerativeClustering(
        n_clusters=params["n_clusters"], linkage="ward"
    complete = cluster.AgglomerativeClustering(
        n_clusters=params["n_clusters"], linkage="complete"
    average = cluster.AgglomerativeClustering(
        n_clusters=params["n_clusters"], linkage="average"
    single = cluster.AgglomerativeClustering(
        n_clusters=params["n_clusters"], linkage="single"

    clustering_algorithms = (
        ("Single Linkage", single),
        ("Average Linkage", average),
        ("Complete Linkage", complete),
        ("Ward Linkage", ward),

    for name, algorithm in clustering_algorithms:
        t0 = time.time()

        # catch warnings related to kneighbors_graph
        with warnings.catch_warnings():
                message="the number of connected components of the "
                + "connectivity matrix is [0-9]{1,2}"
                + " > 1. Completing it to avoid stopping the tree early.",

        t1 = time.time()
        if hasattr(algorithm, "labels_"):
            y_pred = algorithm.labels_.astype(int)
            y_pred = algorithm.predict(X)

        plt.subplot(len(datasets), len(clustering_algorithms), plot_num)
        if i_dataset == 0:
            plt.title(name, size=18)

        colors = np.array(
                    int(max(y_pred) + 1),
        plt.scatter(X[:, 0], X[:, 1], s=10, color=colors[y_pred])

        plt.xlim(-2.5, 2.5)
        plt.ylim(-2.5, 2.5)
            ("%.2fs" % (t1 - t0)).lstrip("0"),
        plot_num += 1
Single Linkage, Average Linkage, Complete Linkage, Ward Linkage

Total running time of the script: (0 minutes 2.356 seconds)

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Comparing different clustering algorithms on toy datasets

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A demo of structured Ward hierarchical clustering on an image of coins

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