# Install the necessary dependencies

import os
import sys
!{sys.executable} -m pip install --quiet pandas scikit-learn numpy matplotlib jupyterlab_myst ipython

17. Unsupervised learning#

First, let’s import a few common modules, ensure MatplotLib plots figures inline and prepare a function to save the figures. We also check that Python 3.5 or later is installed (although Python 2.x may work, it is deprecated so we strongly recommend you use Python 3 instead), as well as Scikit-Learn ≥0.20.

# Python ≥3.5 is required
import sys
assert sys.version_info >= (3, 5)

# Scikit-Learn ≥0.20 is required
import sklearn
assert sklearn.__version__ >= "0.20"

# Common imports
import numpy as np
import os

# to make this notebook's output stable across runs
np.random.seed(42)

# To plot pretty figures
%matplotlib inline
import matplotlib as mpl
import matplotlib.pyplot as plt

import warnings
warnings.filterwarnings("ignore", category=FutureWarning, module="sklearn")

mpl.rc('axes', labelsize=14)
mpl.rc('xtick', labelsize=12)
mpl.rc('ytick', labelsize=12)

# Where to save the figures
PROJECT_ROOT_DIR = "."
CHAPTER_ID = "unsupervised_learning"
IMAGES_PATH = os.path.join(PROJECT_ROOT_DIR, "images", CHAPTER_ID)
os.makedirs(IMAGES_PATH, exist_ok=True)

def save_fig(fig_id, tight_layout=True, fig_extension="png", resolution=300):
    path = os.path.join(IMAGES_PATH, fig_id + "." + fig_extension)
    print("Saving figure", fig_id)
    if tight_layout:
        plt.tight_layout()
    plt.savefig(path, format=fig_extension, dpi=resolution)

17.1. K-Means#

from IPython.display import HTML
display(HTML("""
<p style="text-align: center;">
<iframe src="https://static-1300131294.cos.ap-shanghai.myqcloud.com/html/k-means-demo/k-means.html" width="105%" height="800px;"
style="border:none;" scrolling="auto"></iframe>
A demo of linear-regression. <a
href="http://alekseynp.com/viz/k-means.html"> [source]</a>
</p>
"""))

A demo of linear-regression. [source]

Let’s start by generating some blobs:

from sklearn.datasets import make_blobs

blob_centers = np.array(
    [[ 0.2,  2.3],
     [-1.5 ,  2.3],
     [-2.8,  1.8],
     [-2.8,  2.8],
     [-2.8,  1.3]])
blob_std = np.array([0.4, 0.3, 0.1, 0.1, 0.1])

X, y = make_blobs(n_samples=2000, centers=blob_centers,
                  cluster_std=blob_std, random_state=7)

Now let’s plot them:

def plot_clusters(X, y=None):
    plt.scatter(X[:, 0], X[:, 1], c=y, s=1)
    plt.xlabel("$x_1$", fontsize=14)
    plt.ylabel("$x_2$", fontsize=14, rotation=0)

plt.figure(figsize=(8, 4))
plot_clusters(X)
save_fig("blobs_plot")
plt.show()

Saving figure blobs_plot

../_images/unsupervised-learning_13_1.png

Let’s train a K-Means clusterer on this dataset. It will try to find each blob’s center and assign each instance to the closest blob:

from sklearn.cluster import KMeans

k = 5
kmeans = KMeans(n_clusters=k, random_state=42)
y_pred = kmeans.fit_predict(X)

Each instance was assigned to one of the 5 clusters:

y_pred

array([4, 0, 1, ..., 2, 1, 0], dtype=int32)

y_pred is kmeans.labels_

True

And the following 5 centroids (i.e., cluster centers) were estimated:

kmeans.cluster_centers_

array([[-2.80389616,  1.80117999],
       [ 0.20876306,  2.25551336],
       [-2.79290307,  2.79641063],
       [-1.46679593,  2.28585348],
       [-2.80037642,  1.30082566]])

Note that the KMeans instance preserves the labels of the instances it was trained on. Somewhat confusingly, in this context, the label of an instance is the index of the cluster that instance gets assigned to:

kmeans.labels_

array([4, 0, 1, ..., 2, 1, 0], dtype=int32)

Of course, we can predict the labels of new instances:

X_new = np.array([[0, 2], [3, 2], [-3, 3], [-3, 2.5]])
kmeans.predict(X_new)

array([1, 1, 2, 2], dtype=int32)

17.1.1. Decision Boundaries#

Let’s plot the model’s decision boundaries. This gives us a Voronoi diagram:

def plot_data(X):
    plt.plot(X[:, 0], X[:, 1], 'k.', markersize=2)

def plot_centroids(centroids, weights=None, circle_color='w', cross_color='k'):
    if weights is not None:
        centroids = centroids[weights > weights.max() / 10]
    plt.scatter(centroids[:, 0], centroids[:, 1],
                marker='o', s=35, linewidths=8,
                color=circle_color, zorder=10, alpha=0.9)
    plt.scatter(centroids[:, 0], centroids[:, 1],
                marker='x', s=2, linewidths=12,
                color=cross_color, zorder=11, alpha=1)

def plot_decision_boundaries(clusterer, X, resolution=1000, show_centroids=True,
                             show_xlabels=True, show_ylabels=True):
    mins = X.min(axis=0) - 0.1
    maxs = X.max(axis=0) + 0.1
    xx, yy = np.meshgrid(np.linspace(mins[0], maxs[0], resolution),
                         np.linspace(mins[1], maxs[1], resolution))
    Z = clusterer.predict(np.c_[xx.ravel(), yy.ravel()])
    Z = Z.reshape(xx.shape)

    plt.contourf(Z, extent=(mins[0], maxs[0], mins[1], maxs[1]),
                cmap="Pastel2")
    plt.contour(Z, extent=(mins[0], maxs[0], mins[1], maxs[1]),
                linewidths=1, colors='k')
    plot_data(X)
    if show_centroids:
        plot_centroids(clusterer.cluster_centers_)

    if show_xlabels:
        plt.xlabel("$x_1$", fontsize=14)
    else:
        plt.tick_params(labelbottom=False)
    if show_ylabels:
        plt.ylabel("$x_2$", fontsize=14, rotation=0)
    else:
        plt.tick_params(labelleft=False)

plt.figure(figsize=(8, 4))
plot_decision_boundaries(kmeans, X)
save_fig("voronoi_plot")
plt.show()

Saving figure voronoi_plot

../_images/unsupervised-learning_29_1.png

Hard Clustering vs Soft Clustering

kmeans.transform(X_new)

array([[2.81093633, 0.32995317, 2.9042344 , 1.49439034, 2.88633901],
       [5.80730058, 2.80290755, 5.84739223, 4.4759332 , 5.84236351],
       [1.21475352, 3.29399768, 0.29040966, 1.69136631, 1.71086031],
       [0.72581411, 3.21806371, 0.36159148, 1.54808703, 1.21567622]])

You can verify that this is indeed the Euclidian distance between each instance and each centroid:

np.linalg.norm(np.tile(X_new, (1, k)).reshape(-1, k, 2) - kmeans.cluster_centers_, axis=2)

array([[2.81093633, 0.32995317, 2.9042344 , 1.49439034, 2.88633901],
       [5.80730058, 2.80290755, 5.84739223, 4.4759332 , 5.84236351],
       [1.21475352, 3.29399768, 0.29040966, 1.69136631, 1.71086031],
       [0.72581411, 3.21806371, 0.36159148, 1.54808703, 1.21567622]])

17.1.2. The K-Means Algorithm#

The KMeans class applies an optimized algorithm by default. To get the original K-Means algorithm (for educational purposes only), you must set init=“random”, n_init=1 and algorithm=“full”. These hyperparameters will be explained below.

Let’s run the K-Means algorithm for 1, 2 and 3 iterations, to see how the centroids move around:

kmeans_iter1 = KMeans(n_clusters=5, init="random", n_init=1,
                     algorithm="full", max_iter=1, random_state=0)
kmeans_iter2 = KMeans(n_clusters=5, init="random", n_init=1,
                     algorithm="full", max_iter=2, random_state=0)
kmeans_iter3 = KMeans(n_clusters=5, init="random", n_init=1,
                     algorithm="full", max_iter=3, random_state=0)
kmeans_iter1.fit(X)
kmeans_iter2.fit(X)
kmeans_iter3.fit(X)

KMeans(algorithm='full', init='random', max_iter=3, n_clusters=5, n_init=1,
       random_state=0)

In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.

And let’s plot this:

plt.figure(figsize=(10, 8))

plt.subplot(321)
plot_data(X)
plot_centroids(kmeans_iter1.cluster_centers_, circle_color='r', cross_color='w')
plt.ylabel("$x_2$", fontsize=14, rotation=0)
plt.tick_params(labelbottom=False)
plt.title("Update the centroids (initially randomly)", fontsize=14)

plt.subplot(322)
plot_decision_boundaries(kmeans_iter1, X, show_xlabels=False, show_ylabels=False)
plt.title("Label the instances", fontsize=14)

plt.subplot(323)
plot_decision_boundaries(kmeans_iter1, X, show_centroids=False, show_xlabels=False)
plot_centroids(kmeans_iter2.cluster_centers_)

plt.subplot(324)
plot_decision_boundaries(kmeans_iter2, X, show_xlabels=False, show_ylabels=False)

plt.subplot(325)
plot_decision_boundaries(kmeans_iter2, X, show_centroids=False)
plot_centroids(kmeans_iter3.cluster_centers_)

plt.subplot(326)
plot_decision_boundaries(kmeans_iter3, X, show_ylabels=False)

save_fig("kmeans_algorithm_plot")
plt.show()

Saving figure kmeans_algorithm_plot

../_images/unsupervised-learning_39_1.png

17.1.3. K-Means Variability#

In the original K-Means algorithm, the centroids are just initialized randomly, and the algorithm simply runs a single iteration to gradually improve the centroids, as we saw above.

However, one major problem with this approach is that if you run K-Means multiple times (or with different random seeds), it can converge to very different solutions, as you can see below:

def plot_clusterer_comparison(clusterer1, clusterer2, X, title1=None, title2=None):
    clusterer1.fit(X)
    clusterer2.fit(X)

    plt.figure(figsize=(10, 3.2))

    plt.subplot(121)
    plot_decision_boundaries(clusterer1, X)
    if title1:
        plt.title(title1, fontsize=14)

    plt.subplot(122)
    plot_decision_boundaries(clusterer2, X, show_ylabels=False)
    if title2:
        plt.title(title2, fontsize=14)

kmeans_rnd_init1 = KMeans(n_clusters=5, init="random", n_init=1,
                         algorithm="full", random_state=2)
kmeans_rnd_init2 = KMeans(n_clusters=5, init="random", n_init=1,
                         algorithm="full", random_state=5)

plot_clusterer_comparison(kmeans_rnd_init1, kmeans_rnd_init2, X,
                          "Solution 1", "Solution 2 (with a different random init)")

save_fig("kmeans_variability_plot")
plt.show()

Saving figure kmeans_variability_plot

../_images/unsupervised-learning_43_1.png

17.1.4. Inertia#

To select the best model, we will need a way to evaluate a K-Mean model’s performance. Unfortunately, clustering is an unsupervised task, so we do not have the targets. But at least we can measure the distance between each instance and its centroid. This is the idea behind the inertia metric:

kmeans.inertia_

211.5985372581684

As you can easily verify, inertia is the sum of the squared distances between each training instance and its closest centroid:

X_dist = kmeans.transform(X)
np.sum(X_dist[np.arange(len(X_dist)), kmeans.labels_]**2)

211.59853725816868

The score() method returns the negative inertia. Why negative? Well, it is because a predictor’s score() method must always respect the “greater is better” rule.

kmeans.score(X)

-211.59853725816836

17.1.5. Multiple Initializations#

So one approach to solve the variability issue is to simply run the K-Means algorithm multiple times with different random initializations, and select the solution that minimizes the inertia. For example, here are the inertias of the two “bad” models shown in the previous figure:

kmeans_rnd_init1.inertia_

219.43539442771396

kmeans_rnd_init2.inertia_

211.5985372581684

As you can see, they have a higher inertia than the first “good” model we trained, which means they are probably worse.

When you set the n_init hyperparameter, Scikit-Learn runs the original algorithm n_init times, and selects the solution that minimizes the inertia. By default, Scikit-Learn sets n_init=10.

kmeans_rnd_10_inits = KMeans(n_clusters=5, init="random", n_init=10,
                              algorithm="full", random_state=2)
kmeans_rnd_10_inits.fit(X)

KMeans(algorithm='full', init='random', n_clusters=5, n_init=10, random_state=2)

As you can see, we end up with the initial model, which is certainly the optimal K-Means solution (at least in terms of inertia, and assuming \(k=5\)).

plt.figure(figsize=(8, 4))
plot_decision_boundaries(kmeans_rnd_10_inits, X)
plt.show()

../_images/unsupervised-learning_59_0.png

17.1.6. Centroid initialization methods#

Instead of initializing the centroids entirely randomly, it is preferable to initialize them using the following algorithm, proposed in a 2006 paper by David Arthur and Sergei Vassilvitskii:

Take one centroid \(c_1\), chosen uniformly at random from the dataset.
Take a new center \(c_i\), choosing an instance \(\mathbf{x}_i\) with probability: \(D(\mathbf{x}_i)^2\) / \(\sum\limits_{j=1}^{m}{D(\mathbf{x}_j)}^2\) where \(D(\mathbf{x}_i)\) is the distance between the instance \(\mathbf{x}_i\) and the closest centroid that was already chosen. This probability distribution ensures that instances that are further away from already chosen centroids are much more likely be selected as centroids.
Repeat the previous step until all \(k\) centroids have been chosen.

The rest of the K-Means++ algorithm is just regular K-Means. With this initialization, the K-Means algorithm is much less likely to converge to a suboptimal solution, so it is possible to reduce n_init considerably. Most of the time, this largely compensates for the additional complexity of the initialization process.

To set the initialization to K-Means++, simply set init="k-means++" (this is actually the default):

KMeans()

KMeans()

good_init = np.array([[-3, 3], [-3, 2], [-3, 1], [-1, 2], [0, 2]])
kmeans = KMeans(n_clusters=5, init=good_init, n_init=1, random_state=42)
kmeans.fit(X)
kmeans.inertia_

211.5985372581684

17.1.7. Accelerated K-Means#

The K-Means algorithm can be significantly accelerated by avoiding many unnecessary distance calculations: this is achieved by exploiting the triangle inequality (given three points A, B and C, the distance AC is always such that AC ≤ AB + BC) and by keeping track of lower and upper bounds for distances between instances and centroids.

%timeit -n 50 KMeans(algorithm="elkan", random_state=42).fit(X)

The slowest run took 7.79 times longer than the fastest. This could mean that an intermediate result is being cached.
135 ms ± 134 ms per loop (mean ± std. dev. of 7 runs, 50 loops each)

%timeit -n 50 KMeans(algorithm="full", random_state=42).fit(X)

43 ms ± 14.6 ms per loop (mean ± std. dev. of 7 runs, 50 loops each)

There’s no big difference in this case, as the dataset is fairly small.

17.1.8. Mini-Batch K-Means#

Scikit-Learn also implements a variant of the K-Means algorithm that supports mini-batches.

from sklearn.cluster import MiniBatchKMeans

minibatch_kmeans = MiniBatchKMeans(n_clusters=5, random_state=42)
minibatch_kmeans.fit(X)

MiniBatchKMeans(n_clusters=5, random_state=42)

minibatch_kmeans.inertia_

211.65239850433215

17.2. PCA#

If the dataset does not fit in memory, the simplest option is to use the memmap class, just like we did for incremental PCA in the previous chapter. First let’s load MNIST:

Warning: since Scikit-Learn 0.24, fetch_openml() returns a Pandas DataFrame by default. To avoid this and keep the same code as in the book, we use as_frame=False.

import urllib.request
from sklearn.datasets import fetch_openml

mnist = fetch_openml('mnist_784', version=1, as_frame=False)
mnist.target = mnist.target.astype(np.int64)

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(
    mnist["data"], mnist["target"], random_state=42)

filename = "my_mnist.data"
X_mm = np.memmap(filename, dtype='float32', mode='write', shape=X_train.shape)
X_mm[:] = X_train

minibatch_kmeans = MiniBatchKMeans(n_clusters=10, batch_size=10, random_state=42)
minibatch_kmeans.fit(X_mm)

MiniBatchKMeans(batch_size=10, n_clusters=10, random_state=42)

If your data is so large that you cannot use memmap, things get more complicated. Let’s start by writing a function to load the next batch (in real life, you would load the data from disk):

def load_next_batch(batch_size):
    return X[np.random.choice(len(X), batch_size, replace=False)]

Now we can train the model by feeding it one batch at a time. We also need to implement multiple initializations and keep the model with the lowest inertia:

np.random.seed(42)

k = 5
n_init = 10
n_iterations = 100
batch_size = 100
init_size = 500  # more data for K-Means++ initialization
evaluate_on_last_n_iters = 10

best_kmeans = None

for init in range(n_init):
    minibatch_kmeans = MiniBatchKMeans(n_clusters=k, init_size=init_size)
    X_init = load_next_batch(init_size)
    minibatch_kmeans.partial_fit(X_init)

    minibatch_kmeans.sum_inertia_ = 0
    for iteration in range(n_iterations):
        X_batch = load_next_batch(batch_size)
        minibatch_kmeans.partial_fit(X_batch)
        if iteration >= n_iterations - evaluate_on_last_n_iters:
            minibatch_kmeans.sum_inertia_ += minibatch_kmeans.inertia_

    if (best_kmeans is None or
        minibatch_kmeans.sum_inertia_ < best_kmeans.sum_inertia_):
        best_kmeans = minibatch_kmeans

best_kmeans.score(X)

-211.62571878891146

Mini-batch K-Means is much faster than regular K-Means:

%timeit KMeans(n_clusters=5, random_state=42).fit(X)

23.9 ms ± 1.4 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)

%timeit MiniBatchKMeans(n_clusters=5, random_state=42).fit(X)

13.5 ms ± 7.84 ms per loop (mean ± std. dev. of 7 runs, 100 loops each)

That’s much faster! However, its performance is often lower (higher inertia), and it keeps degrading as k increases. Let’s plot the inertia ratio and the training time ratio between Mini-batch K-Means and regular K-Means:

from timeit import timeit

times = np.empty((100, 2))
inertias = np.empty((100, 2))
for k in range(1, 101):
    kmeans_ = KMeans(n_clusters=k, random_state=42)
    minibatch_kmeans = MiniBatchKMeans(n_clusters=k, random_state=42)
    print("\r{}/{}".format(k, 100), end="")
    times[k-1, 0] = timeit("kmeans_.fit(X)", number=10, globals=globals())
    times[k-1, 1]  = timeit("minibatch_kmeans.fit(X)", number=10, globals=globals())
    inertias[k-1, 0] = kmeans_.inertia_
    inertias[k-1, 1] = minibatch_kmeans.inertia_

100/100

plt.figure(figsize=(10,4))

plt.subplot(121)
plt.plot(range(1, 101), inertias[:, 0], "r--", label="K-Means")
plt.plot(range(1, 101), inertias[:, 1], "b.-", label="Mini-batch K-Means")
plt.xlabel("$k$", fontsize=16)
plt.title("Inertia", fontsize=14)
plt.legend(fontsize=14)
plt.axis([1, 100, 0, 100])

plt.subplot(122)
plt.plot(range(1, 101), times[:, 0], "r--", label="K-Means")
plt.plot(range(1, 101), times[:, 1], "b.-", label="Mini-batch K-Means")
plt.xlabel("$k$", fontsize=16)
plt.title("Training time (seconds)", fontsize=14)
plt.axis([1, 100, 0, 6])

save_fig("minibatch_kmeans_vs_kmeans")
plt.show()

Saving figure minibatch_kmeans_vs_kmeans

../_images/unsupervised-learning_95_1.png

17.2.1. Finding the optimal number of clusters#

What if the number of clusters was set to a lower or greater value than 5?

kmeans_k3 = KMeans(n_clusters=3, random_state=42)
kmeans_k8 = KMeans(n_clusters=8, random_state=42)

plot_clusterer_comparison(kmeans_k3, kmeans_k8, X, "$k=3$", "$k=8$")
save_fig("bad_n_clusters_plot")
plt.show()

Saving figure bad_n_clusters_plot

../_images/unsupervised-learning_98_1.png

Ouch, these two models don’t look great. What about their inertias?

kmeans_k3.inertia_

653.216719002155

kmeans_k8.inertia_

119.11983416102879

No, we cannot simply take the value of \(k\) that minimizes the inertia, since it keeps getting lower as we increase \(k\). Indeed, the more clusters there are, the closer each instance will be to its closest centroid, and therefore the lower the inertia will be. However, we can plot the inertia as a function of \(k\) and analyze the resulting curve:

kmeans_per_k = [KMeans(n_clusters=k, random_state=42).fit(X)
                for k in range(1, 10)]
inertias = [model.inertia_ for model in kmeans_per_k]

plt.figure(figsize=(8, 3.5))
plt.plot(range(1, 10), inertias, "bo-")
plt.xlabel("$k$", fontsize=14)
plt.ylabel("Inertia", fontsize=14)
plt.annotate('Elbow',
             xy=(4, inertias[3]),
             xytext=(0.55, 0.55),
             textcoords='figure fraction',
             fontsize=16,
             arrowprops=dict(facecolor='black', shrink=0.1)
            )
plt.axis([1, 8.5, 0, 1300])
save_fig("inertia_vs_k_plot")
plt.show()

Saving figure inertia_vs_k_plot

../_images/unsupervised-learning_104_1.png

As you can see, there is an elbow at \(k=4\), which means that less clusters than that would be bad, and more clusters would not help much and might cut clusters in half. So \(k=4\) is a pretty good choice. Of course in this example it is not perfect since it means that the two blobs in the lower left will be considered as just a single cluster, but it’s a pretty good clustering nonetheless.

plot_decision_boundaries(kmeans_per_k[4-1], X)
plt.show()

../_images/unsupervised-learning_106_0.png

Another approach is to look at the silhouette score, which is the mean silhouette coefficient over all the instances. An instance’s silhouette coefficient is equal to \((b - a)/\max(a, b)\) where \(a\) is the mean distance to the other instances in the same cluster (it is the mean intra-cluster distance), and \(b\) is the mean nearest-cluster distance, that is the mean distance to the instances of the next closest cluster (defined as the one that minimizes \(b\), excluding the instance’s own cluster). The silhouette coefficient can vary between -1 and +1: a coefficient close to +1 means that the instance is well inside its own cluster and far from other clusters, while a coefficient close to 0 means that it is close to a cluster boundary, and finally a coefficient close to -1 means that the instance may have been assigned to the wrong cluster.

Let’s plot the silhouette score as a function of \(k\):

from sklearn.metrics import silhouette_score

silhouette_score(X, kmeans.labels_)

0.655517642572828

silhouette_scores = [silhouette_score(X, model.labels_)
                     for model in kmeans_per_k[1:]]

plt.figure(figsize=(8, 3))
plt.plot(range(2, 10), silhouette_scores, "bo-")
plt.xlabel("$k$", fontsize=14)
plt.ylabel("Silhouette score", fontsize=14)
plt.axis([1.8, 8.5, 0.55, 0.7])
save_fig("silhouette_score_vs_k_plot")
plt.show()

Saving figure silhouette_score_vs_k_plot

../_images/unsupervised-learning_112_1.png

As you can see, this visualization is much richer than the previous one: in particular, although it confirms that \(k=4\) is a very good choice, but it also underlines the fact that \(k=5\) is quite good as well.

An even more informative visualization is given when you plot every instance’s silhouette coefficient, sorted by the cluster they are assigned to and by the value of the coefficient. This is called a silhouette diagram:

from sklearn.metrics import silhouette_samples
from matplotlib.ticker import FixedLocator, FixedFormatter

plt.figure(figsize=(11, 9))

for k in (3, 4, 5, 6):
    plt.subplot(2, 2, k - 2)

    y_pred = kmeans_per_k[k - 1].labels_
    silhouette_coefficients = silhouette_samples(X, y_pred)

    padding = len(X) // 30
    pos = padding
    ticks = []
    for i in range(k):
        coeffs = silhouette_coefficients[y_pred == i]
        coeffs.sort()

        color = mpl.cm.Spectral(i / k)
        plt.fill_betweenx(np.arange(pos, pos + len(coeffs)), 0, coeffs,
                          facecolor=color, edgecolor=color, alpha=0.7)
        ticks.append(pos + len(coeffs) // 2)
        pos += len(coeffs) + padding

    plt.gca().yaxis.set_major_locator(FixedLocator(ticks))
    plt.gca().yaxis.set_major_formatter(FixedFormatter(range(k)))
    if k in (3, 5):
        plt.ylabel("Cluster")

    if k in (5, 6):
        plt.gca().set_xticks([-0.1, 0, 0.2, 0.4, 0.6, 0.8, 1])
        plt.xlabel("Silhouette Coefficient")
    else:
        plt.tick_params(labelbottom=False)

    plt.axvline(x=silhouette_scores[k - 2], color="red", linestyle="--")
    plt.title("$k={}$".format(k), fontsize=16)

save_fig("silhouette_analysis_plot")
plt.show()

Saving figure silhouette_analysis_plot

../_images/unsupervised-learning_115_1.png

As you can see, \(k=5\) looks like the best option here, as all clusters are roughly the same size, and they all cross the dashed line, which represents the mean silhouette score.

17.2.2. Limits of K-Means#

X1, y1 = make_blobs(n_samples=1000, centers=((4, -4), (0, 0)), random_state=42)
X1 = X1.dot(np.array([[0.374, 0.95], [0.732, 0.598]]))
X2, y2 = make_blobs(n_samples=250, centers=1, random_state=42)
X2 = X2 + [6, -8]
X = np.r_[X1, X2]
y = np.r_[y1, y2]

plot_clusters(X)

../_images/unsupervised-learning_119_0.png

kmeans_good = KMeans(n_clusters=3, init=np.array([[-1.5, 2.5], [0.5, 0], [4, 0]]), n_init=1, random_state=42)
kmeans_bad = KMeans(n_clusters=3, random_state=42)
kmeans_good.fit(X)
kmeans_bad.fit(X)

KMeans(n_clusters=3, random_state=42)

plt.figure(figsize=(10, 3.2))

plt.subplot(121)
plot_decision_boundaries(kmeans_good, X)
plt.title("Inertia = {:.1f}".format(kmeans_good.inertia_), fontsize=14)

plt.subplot(122)
plot_decision_boundaries(kmeans_bad, X, show_ylabels=False)
plt.title("Inertia = {:.1f}".format(kmeans_bad.inertia_), fontsize=14)

save_fig("bad_kmeans_plot")
plt.show()

Saving figure bad_kmeans_plot

../_images/unsupervised-learning_121_1.png

17.2.3. Using Clustering for Image Segmentation#

# Download the ladybug image
images_path = os.path.join(PROJECT_ROOT_DIR, "images", "unsupervised_learning")
os.makedirs(images_path, exist_ok=True)
DOWNLOAD_ROOT = "https://raw.githubusercontent.com/ageron/handson-ml2/master/"
filename = "ladybug.png"
print("Downloading", filename)
url = DOWNLOAD_ROOT + "images/unsupervised_learning/" + filename
urllib.request.urlretrieve(url, os.path.join(images_path, filename))

Downloading ladybug.png

('./images/unsupervised_learning/ladybug.png',
 <http.client.HTTPMessage at 0x7e1538110d60>)

from matplotlib.image import imread
image = imread(os.path.join(images_path, filename))
image.shape

(533, 800, 3)

X = image.reshape(-1, 3)
kmeans = KMeans(n_clusters=8, random_state=42).fit(X)
segmented_img = kmeans.cluster_centers_[kmeans.labels_]
segmented_img = segmented_img.reshape(image.shape)

segmented_imgs = []
n_colors = (10, 8, 6, 4, 2)
for n_clusters in n_colors:
    kmeans = KMeans(n_clusters=n_clusters, random_state=42).fit(X)
    segmented_img = kmeans.cluster_centers_[kmeans.labels_]
    segmented_imgs.append(segmented_img.reshape(image.shape))

plt.figure(figsize=(10,5))
plt.subplots_adjust(wspace=0.05, hspace=0.1)

plt.subplot(231)
plt.imshow(image)
plt.title("Original image")
plt.axis('off')

for idx, n_clusters in enumerate(n_colors):
    plt.subplot(232 + idx)
    plt.imshow(segmented_imgs[idx])
    plt.title("{} colors".format(n_clusters))
    plt.axis('off')

save_fig('image_segmentation_diagram', tight_layout=False)
plt.show()

Saving figure image_segmentation_diagram

../_images/unsupervised-learning_127_1.png

17.2.4. Using Clustering for Preprocessing#

Let’s tackle the digits dataset which is a simple MNIST-like dataset containing 1,797 grayscale 8×8 images representing digits 0 to 9.

from sklearn.datasets import load_digits

X_digits, y_digits = load_digits(return_X_y=True)

Let’s split it into a training set and a test set:

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X_digits, y_digits, random_state=42)

17.4. DBSCAN#

from IPython.display import HTML
display(HTML("""
<p style="text-align: center;">
<iframe src="https://static-1300131294.cos.ap-shanghai.myqcloud.com/html/dbscan-demo/dbscan.html" width="105%" height="800px;"
style="border:none;" scrolling="auto"></iframe>
A demo of linear-regression. <a
href="https://www.naftaliharris.com/blog/visualizing-dbscan-clustering/"> [source]</a>
</p>
"""))