scikit-learn · glemaitre · Dec 5, 2022 · Nov 16, 2022 · Nov 17, 2022 · Nov 17, 2022
diff --git a/doc/modules/clustering.rst b/doc/modules/clustering.rst
@@ -170,7 +170,7 @@ It suffers from various drawbacks:
   k-means clustering can alleviate this problem and speed up the
   computations.
 
-.. image:: ../auto_examples/cluster/images/sphx_glr_plot_kmeans_assumptions_001.png
+.. image:: ../auto_examples/cluster/images/sphx_glr_plot_kmeans_assumptions_002.png
    :target: ../auto_examples/cluster/plot_kmeans_assumptions.html
    :align: center
    :scale: 50

diff --git a/examples/cluster/plot_kmeans_assumptions.py b/examples/cluster/plot_kmeans_assumptions.py
@@ -3,67 +3,176 @@
 Demonstration of k-means assumptions
 ====================================
 
-This example is meant to illustrate situations where k-means will produce
-unintuitive and possibly unexpected clusters. In the first three plots, the
-input data does not conform to some implicit assumption that k-means makes and
-undesirable clusters are produced as a result. In the last plot, k-means
-returns intuitive clusters despite unevenly sized blobs.
+This example is meant to illustrate situations where k-means produces
+unintuitive and possibly undesirable clusters.
 
 """
 
 # Author: Phil Roth <[email protected]>
+#         Arturo Amor <[email protected]>
 # License: BSD 3 clause
 
-import numpy as np
-import matplotlib.pyplot as plt
+# %%
+# Data generation
+# ---------------
+#
+# The function :func:`~sklearn.datasets.make_blobs` generates isotropic
+# (spherical) gaussian blobs. To obtain anisotropic (elliptical) gaussian blobs
+# one has to define a linear `transformation`.
 
-from sklearn.cluster import KMeans
+import numpy as np
 from sklearn.datasets import make_blobs
 
-plt.figure(figsize=(12, 12))
-
 n_samples = 1500
 random_state = 170
+transformation = [[0.60834549, -0.63667341], [-0.40887718, 0.85253229]]
+
 X, y = make_blobs(n_samples=n_samples, random_state=random_state)
+X_aniso = np.dot(X, transformation)  # Anisotropic blobs
+X_varied, y_varied = make_blobs(
+    n_samples=n_samples, cluster_std=[1.0, 2.5, 0.5], random_state=random_state
+)  # Unequal variance
+X_filtered = np.vstack(
+    (X[y == 0][:500], X[y == 1][:100], X[y == 2][:10])
+)  # Unevenly sized blobs
+y_filtered = [0] * 500 + [1] * 100 + [2] * 10
 
-# Incorrect number of clusters
-y_pred = KMeans(n_clusters=2, n_init="auto", random_state=random_state).fit_predict(X)
+# %%
+# We can visualize the resulting data:
 
-plt.subplot(221)
-plt.scatter(X[:, 0], X[:, 1], c=y_pred)
-plt.title("Incorrect Number of Blobs")
+import matplotlib.pyplot as plt
 
-# Anisotropicly distributed data
-transformation = [[0.60834549, -0.63667341], [-0.40887718, 0.85253229]]
-X_aniso = np.dot(X, transformation)
-y_pred = KMeans(n_clusters=3, n_init="auto", random_state=random_state).fit_predict(
-    X_aniso
-)
+fig, axs = plt.subplots(nrows=2, ncols=2, figsize=(12, 12))
 
-plt.subplot(222)
-plt.scatter(X_aniso[:, 0], X_aniso[:, 1], c=y_pred)
-plt.title("Anisotropicly Distributed Blobs")
+axs[0, 0].scatter(X[:, 0], X[:, 1], c=y)
+axs[0, 0].set_title("Mixture of Gaussian Blobs")
 
-# Different variance
-X_varied, y_varied = make_blobs(
-    n_samples=n_samples, cluster_std=[1.0, 2.5, 0.5], random_state=random_state
-)
-y_pred = KMeans(n_clusters=3, n_init="auto", random_state=random_state).fit_predict(
-    X_varied
-)
+axs[0, 1].scatter(X_aniso[:, 0], X_aniso[:, 1], c=y)
+axs[0, 1].set_title("Anisotropically Distributed Blobs")
+
+axs[1, 0].scatter(X_varied[:, 0], X_varied[:, 1], c=y_varied)
+axs[1, 0].set_title("Unequal Variance")
+
+axs[1, 1].scatter(X_filtered[:, 0], X_filtered[:, 1], c=y_filtered)
+axs[1, 1].set_title("Unevenly Sized Blobs")
+
+plt.suptitle("Ground truth clusters").set_y(0.95)
+plt.show()
+
+# %%
+# Fit models and plot results
+# ---------------------------
+#
+# The previously generated data is now used to show how
+# :class:`~sklearn.cluster.KMeans` behaves in the following scenarios:
+#
+# - Non-optimal number of clusters: in a real setting there is no uniquely
+#   defined **true** number of clusters. An appropriate number of clusters has
+#   to be decided from data-based criteria and knowledge of the intended goal.
+# - Anisotropically distributed blobs: k-means consists of minimizing sample's
+#   euclidean distances to the centroid of the cluster they are assigned to. As
+#   a consequence, k-means is more appropriate for clusters that are isotropic
+#   and normally distributed (i.e. spherical gaussians).
+# - Unequal variance: k-means is equivalent to taking the maximum likelihood
+#   estimator for a "mixture" of k gaussian distributions with the same
+#   variances but with possibly different means.
+# - Unevenly sized blobs: there is no theoretical result about k-means that
+#   states that it requires similar cluster sizes to perform well, yet
+#   minimizing euclidean distances does mean that the more sparse and
+#   high-dimensional the problem is, the higher is the need to run the algorithm
+#   with different centroid seeds to ensure a global minimal inertia.
 
-plt.subplot(223)
-plt.scatter(X_varied[:, 0], X_varied[:, 1], c=y_pred)
-plt.title("Unequal Variance")
+from sklearn.cluster import KMeans
+
+common_params = {
+    "n_init": "auto",
+    "random_state": random_state,
+}
+
+fig, axs = plt.subplots(nrows=2, ncols=2, figsize=(12, 12))
+
+y_pred = KMeans(n_clusters=2, **common_params).fit_predict(X)
+axs[0, 0].scatter(X[:, 0], X[:, 1], c=y_pred)
+axs[0, 0].set_title("Non-optimal Number of Clusters")
+
+y_pred = KMeans(n_clusters=3, **common_params).fit_predict(X_aniso)
+axs[0, 1].scatter(X_aniso[:, 0], X_aniso[:, 1], c=y_pred)
+axs[0, 1].set_title("Anisotropically Distributed Blobs")
+
+y_pred = KMeans(n_clusters=3, **common_params).fit_predict(X_varied)
+axs[1, 0].scatter(X_varied[:, 0], X_varied[:, 1], c=y_pred)
+axs[1, 0].set_title("Unequal Variance")
+
+y_pred = KMeans(n_clusters=3, **common_params).fit_predict(X_filtered)
+axs[1, 1].scatter(X_filtered[:, 0], X_filtered[:, 1], c=y_pred)
+axs[1, 1].set_title("Unevenly Sized Blobs")
+
+plt.suptitle("Unexpected KMeans clusters").set_y(0.95)
+plt.show()
+
+# %%
+# Possible solutions
+# ------------------
+#
+# For an example on how to find a correct number of blobs, see
+# :ref:`sphx_glr_auto_examples_cluster_plot_kmeans_silhouette_analysis.py`.
+# In this case it suffices to set `n_clusters=3`.
+
+y_pred = KMeans(n_clusters=3, **common_params).fit_predict(X)
+plt.scatter(X[:, 0], X[:, 1], c=y_pred)
+plt.title("Optimal Number of Clusters")
+plt.show()
+
+# %%
+# To deal with unevenly sized blobs one can increase the number of random
+# initializations. In this case we set `n_init=10` to avoid finding a
+# sub-optimal local minimum. For more details see :ref:`kmeans_sparse_high_dim`.
 
-# Unevenly sized blobs
-X_filtered = np.vstack((X[y == 0][:500], X[y == 1][:100], X[y == 2][:10]))
 y_pred = KMeans(n_clusters=3, n_init=10, random_state=random_state).fit_predict(
     X_filtered
 )
-
-plt.subplot(224)
 plt.scatter(X_filtered[:, 0], X_filtered[:, 1], c=y_pred)
-plt.title("Unevenly Sized Blobs")
+plt.title("Unevenly Sized Blobs \nwith several initializations")
+plt.show()
+
+# %%
+# As anisotropic and unequal variances are real limitations of the k-means
+# algorithm, here we propose instead the use of
+# :class:`~sklearn.mixture.GaussianMixture`, which also assumes gaussian
+# clusters but does not impose any constraints on their variances. Notice that
+# one still has to find the correct number of blobs (see
+# :ref:`sphx_glr_auto_examples_mixture_plot_gmm_selection.py`).
+#
+# For an example on how other clustering methods deal with anisotropic or
+# unequal variance blobs, see the example
+# :ref:`sphx_glr_auto_examples_cluster_plot_cluster_comparison.py`.
 
+from sklearn.mixture import GaussianMixture
+
+fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize=(12, 6))
+
+y_pred = GaussianMixture(n_components=3).fit_predict(X_aniso)
+ax1.scatter(X_aniso[:, 0], X_aniso[:, 1], c=y_pred)
+ax1.set_title("Anisotropically Distributed Blobs")
+
+y_pred = GaussianMixture(n_components=3).fit_predict(X_varied)
+ax2.scatter(X_varied[:, 0], X_varied[:, 1], c=y_pred)
+ax2.set_title("Unequal Variance")
+
+plt.suptitle("Gaussian mixture clusters").set_y(0.95)
 plt.show()
+
+# %%
+# Final remarks
+# -------------
+#
+# In high-dimensional spaces, Euclidean distances tend to become inflated
+# (not shown in this example). Running a dimensionality reduction algorithm
+# prior to k-means clustering can alleviate this problem and speed up the
+# computations (see the example
+# :ref:`sphx_glr_auto_examples_text_plot_document_clustering.py`).
+#
+# In the case where clusters are known to be isotropic, have similar variance
+# and are not too sparse, the k-means algorithm is quite effective and is one of
+# the fastest clustering algorithms available. This advantage is lost if one has
+# to restart it several times to avoid convergence to a local minimum.