Various Agglomerative Clustering on a 2D embedding of digits

An illustration of various linkage option for agglomerative clustering on a 2D embedding of the digits dataset.

The goal of this example is to show intuitively how the metrics behave, and not to find good clusters for the digits. This is why the example works on a 2D embedding.

What this example shows us is the behavior “rich getting richer” of agglomerative clustering that tends to create uneven cluster sizes.

This behavior is pronounced for the average linkage strategy, that ends up with a couple of clusters with few datapoints.

The case of single linkage is even more pathologic with a very large cluster covering most digits, an intermediate size (clean) cluster with most zero digits and all other clusters being drawn from noise points around the fringes.

The other linkage strategies lead to more evenly distributed clusters that are therefore likely to be less sensible to a random resampling of the dataset.

  • ward linkage
  • average linkage
  • complete linkage
  • single linkage


Computing embedding
ward :  0.05s
average :       0.05s
complete :      0.05s
single :        0.02s

# Authors: Gael Varoquaux
# License: BSD 3 clause (C) INRIA 2014

from time import time

import numpy as np
from matplotlib import pyplot as plt

from sklearn import manifold, datasets

digits = datasets.load_digits()
X, y =,
n_samples, n_features = X.shape


# ----------------------------------------------------------------------
# Visualize the clustering
def plot_clustering(X_red, labels, title=None):
    x_min, x_max = np.min(X_red, axis=0), np.max(X_red, axis=0)
    X_red = (X_red - x_min) / (x_max - x_min)

    plt.figure(figsize=(6, 4))
    for digit in digits.target_names:
            *X_red[y == digit].T,
  [y == digit] / 10),

    if title is not None:
        plt.title(title, size=17)
    plt.tight_layout(rect=[0, 0.03, 1, 0.95])

# ----------------------------------------------------------------------
# 2D embedding of the digits dataset
print("Computing embedding")
X_red = manifold.SpectralEmbedding(n_components=2).fit_transform(X)

from sklearn.cluster import AgglomerativeClustering

for linkage in ("ward", "average", "complete", "single"):
    clustering = AgglomerativeClustering(linkage=linkage, n_clusters=10)
    t0 = time()
    print("%s :\t%.2fs" % (linkage, time() - t0))

    plot_clustering(X_red, clustering.labels_, "%s linkage" % linkage)

Total running time of the script: ( 0 minutes 1.352 seconds)

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