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Various Agglomerative Clustering on a 2D embedding of digitsΒΆ

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 especially pronounced for the average linkage strategy, that ends up with a couple of singleton clusters.

  • ../../_images/plot_digits_linkage_0011.png
  • ../../_images/plot_digits_linkage_0021.png
  • ../../_images/plot_digits_linkage_0031.png

Script output:

Computing embedding
ward : 41.25s
average : 31.28s
complete : 26.34s

Python source code:

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

from time import time

import numpy as np
from scipy import ndimage
from matplotlib import pyplot as plt

from sklearn import manifold, datasets

digits = datasets.load_digits(n_class=10)
X =
y =
n_samples, n_features = X.shape


def nudge_images(X, y):
    # Having a larger dataset shows more clearly the behavior of the
    # methods, but we multiply the size of the dataset only by 2, as the
    # cost of the hierarchical clustering methods are strongly
    # super-linear in n_samples
    shift = lambda x: ndimage.shift(x.reshape((8, 8)),
                                  .3 * np.random.normal(size=2),
    X = np.concatenate([X, np.apply_along_axis(shift, 1, X)])
    Y = np.concatenate([y, y], axis=0)
    return X, Y

X, y = nudge_images(X, y)

# Visualize the clustering
def plot_clustering(X_red, X, 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 i in range(X_red.shape[0]):
        plt.text(X_red[i, 0], X_red[i, 1], str(y[i]),
       [i] / 10.),
                 fontdict={'weight': 'bold', 'size': 9})

    if title is not None:
        plt.title(title, size=17)

# 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'):
    clustering = AgglomerativeClustering(linkage=linkage, n_clusters=10)
    t0 = time()
    print("%s : %.2fs" % (linkage, time() - t0))

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

Total running time of the example: 117.36 seconds ( 1 minutes 57.36 seconds)