Chapter 8 – Dimensionality Reduction
This notebook contains all the sample code and solutions to the exercises in chapter 8.
Setup¶
First, let’s make sure this notebook works well in both python 2 and 3, import a few common modules, ensure MatplotLib plots figures inline and prepare a function to save the figures:
In [3]:
# To support both python 2 and python 3
from __future__ import division, print_function, unicode_literals
# 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
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”
def save_fig(fig_id, tight_layout=True):
path = os.path.join(PROJECT_ROOT_DIR, “images”, CHAPTER_ID, fig_id + “.png”)
print(“Saving figure”, fig_id)
if tight_layout:
plt.tight_layout()
plt.savefig(path, format=’png’, dpi=300)
# Ignore useless warnings (see SciPy issue #5998)
import warnings
warnings.filterwarnings(action=”ignore”, message=”^internal gelsd”)
Lecture 5 – Clustering¶
Introduction – Classification vs Clustering¶
In [10]:
from sklearn.datasets import load_iris
In [11]:
data = load_iris()
X = data.data
y = data.target
data.target_names
Out[11]:
array([‘setosa’, ‘versicolor’, ‘virginica’], dtype=’
plt.scatter(centroids[:, 0], centroids[:, 1],
marker=’o’, s=30, linewidths=8,
color=circle_color, zorder=10, alpha=0.9)
plt.scatter(centroids[:, 0], centroids[:, 1],
marker=’x’, s=50, linewidths=50,
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)
In [31]:
plt.figure(figsize=(8, 4))
plot_decision_boundaries(kmeans, X)
save_fig(“voronoi_diagram”)
plt.show()
Saving figure voronoi_diagram

Not bad! Some of the instances near the edges were probably assigned to the wrong cluster, but overall it looks pretty good.
EXERCISE 1: Vizualising Errors We can see that some points are not assigned to the correct clusters. Write some code to lable the data points that are not assigned by k-means to the cluster that generated them.
Hard Clustering vs Soft Clustering¶
Rather than arbitrarily choosing the closest cluster for each instance, which is called hard clustering, it might be better measure the distance of each instance to all 5 centroids. This is what the transform() method does:
In [32]:
kmeans.transform(X_new)
Out[32]:
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:
In [33]:
np.linalg.norm(np.tile(X_new, (1, k)).reshape(-1, k, 2) – kmeans.cluster_centers_, axis=2)
Out[33]:
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]])
EXERCISE 2: Vizualising membership Use the output of the `transform` method to mix different colour contributions from the different cluster centres so we can see how closely they fall to the Voronoi boundaries.
K-Means Algorithm¶
The K-Means algorithm is one of the fastest clustering algorithms, but also one of the simplest:
•First initialize $k$ centroids randomly: $k$ distinct instances are chosen randomly from the dataset and the centroids are placed at their locations.
•Repeat until convergence (i.e., until the centroids stop moving):
▪Assign each instance to the closest centroid.
▪Update the centroids to be the mean of the instances that are assigned to them.
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=1and 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:
In [34]:
kmeans_iter1 = KMeans(n_clusters=5, init=”random”, n_init=1,
algorithm=”full”, max_iter=1, random_state=1)
kmeans_iter2 = KMeans(n_clusters=5, init=”random”, n_init=1,
algorithm=”full”, max_iter=2, random_state=1)
kmeans_iter3 = KMeans(n_clusters=5, init=”random”, n_init=1,
algorithm=”full”, max_iter=3, random_state=1)
kmeans_iter1.fit(X)
kmeans_iter2.fit(X)
kmeans_iter3.fit(X)
Out[34]:
KMeans(algorithm=’full’, copy_x=True, init=’random’, max_iter=3, n_clusters=5,
n_init=1, n_jobs=None, precompute_distances=’auto’, random_state=1,
tol=0.0001, verbose=0)
And let’s plot this:
In [35]:
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_diagram”)
plt.show()
Saving figure kmeans_algorithm_diagram

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:
In [36]:
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)
In [37]:
kmeans_rnd_init1 = KMeans(n_clusters=5, init=”random”, n_init=1,
algorithm=”full”, random_state=11)
kmeans_rnd_init2 = KMeans(n_clusters=5, init=”random”, n_init=1,
algorithm=”full”, random_state=19)
plot_clusterer_comparison(kmeans_rnd_init1, kmeans_rnd_init2, X,
“Solution 1”, “Solution 2 (with a different random init)”)
save_fig(“kmeans_variability_diagram”)
plt.show()
Saving figure kmeans_variability_diagram

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:
In [38]:
kmeans.inertia_
Out[38]:
211.5985372581684
As you can easily verify, inertia is the sum of the squared distances between each training instance and its closest centroid:
In [39]:
X_dist = kmeans.transform(X)
np.sum(X_dist[np.arange(len(X_dist)), kmeans.labels_]**2)
Out[39]:
211.59853725816856
The score() method returns the negative inertia. Why negative? Well, it is because a predictor’s score() method must always respect the “great is better” rule.
In [40]:
kmeans.score(X)
Out[40]:
-211.59853725816856
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:
In [41]:
kmeans_rnd_init1.inertia_
Out[41]:
223.29108572819035
In [42]:
kmeans_rnd_init2.inertia_
Out[42]:
237.46249169442845
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.
In [43]:
kmeans_rnd_10_inits = KMeans(n_clusters=5, init=”random”, n_init=10,
algorithm=”full”, random_state=11)
kmeans_rnd_10_inits.fit(X)
Out[43]:
KMeans(algorithm=’full’, copy_x=True, init=’random’, max_iter=300, n_clusters=5,
n_init=10, n_jobs=None, precompute_distances=’auto’, random_state=11,
tol=0.0001, verbose=0)
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 $k=5$).
In [44]:
plt.figure(figsize=(8, 4))
plot_decision_boundaries(kmeans_rnd_10_inits, X)
plt.show()

K-Means++¶
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: $D(mathbf{x}_i)^2$ / $sumlimits_{j=1}^{m}{D(mathbf{x}_j)}^2$ 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):
In [45]:
KMeans()
Out[45]:
KMeans(algorithm=’auto’, copy_x=True, init=’k-means++’, max_iter=300,
n_clusters=8, n_init=10, n_jobs=None, precompute_distances=’auto’,
random_state=None, tol=0.0001, verbose=0)
In [46]:
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_
Out[46]:
211.5985372581684
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 (see this 2003 paper by Charles Elkan for more details).
To use Elkan’s variant of K-Means, just set algorithm=”elkan”. Note that it does not support sparse data, so by default, Scikit-Learn uses “elkan” for dense data, and “full” (the regular K-Means algorithm) for sparse data.
In [47]:
%timeit -n 50 KMeans(algorithm=”elkan”).fit(X)
95.7 ms ± 12.4 ms per loop (mean ± std. dev. of 7 runs, 50 loops each)
In [48]:
%timeit -n 50 KMeans(algorithm=”full”).fit(X)
97 ms ± 10 ms per loop (mean ± std. dev. of 7 runs, 50 loops each)
Mini-Batch K-Means¶
Scikit-Learn also implements a variant of the K-Means algorithm that supports mini-batches (see this paper):
In [49]:
from sklearn.cluster import MiniBatchKMeans
In [50]:
minibatch_kmeans = MiniBatchKMeans(n_clusters=5, random_state=42)
minibatch_kmeans.fit(X)
Out[50]:
MiniBatchKMeans(batch_size=100, compute_labels=True, init=’k-means++’,
init_size=None, max_iter=100, max_no_improvement=10,
n_clusters=5, n_init=3, random_state=42,
reassignment_ratio=0.01, tol=0.0, verbose=0)
In [51]:
minibatch_kmeans.inertia_
Out[51]:
211.93186531476775
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 [53]:
filename = “my_mnist.data”
m, n = 50000, 28*28
X_mm = np.memmap(filename, dtype=”float32″, mode=”readonly”, shape=(m, n))
In [54]:
minibatch_kmeans = MiniBatchKMeans(n_clusters=10, batch_size=10, random_state=42)
minibatch_kmeans.fit(X_mm)
Out[54]:
MiniBatchKMeans(batch_size=10, compute_labels=True, init=’k-means++’,
init_size=None, max_iter=100, max_no_improvement=10,
n_clusters=10, n_init=3, random_state=42,
reassignment_ratio=0.01, tol=0.0, verbose=0)
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):
In [55]:
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:
In [56]:
np.random.seed(42)
In [57]:
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_kmeansIn [58]:best_kmeans.score(X)Out[58]:-211.70999744411483Mini-batch K-Means is much faster than regular K-Means:In [59]:%timeit KMeans(n_clusters=5).fit(X)72.1 ms ± 13.8 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)In [60]:%timeit MiniBatchKMeans(n_clusters=5).fit(X)24.6 ms ± 2.91 ms per loop (mean ± std. dev. of 7 runs, 10 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:In [61]:from timeit import timeitIn [62]: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_100/100In [63]: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.ylabel(“Inertia”, fontsize=14)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.ylabel(“Training time (seconds)”, fontsize=14)plt.title(“Training time (seconds)”, fontsize=14)plt.axis([1, 100, 0, 6])#plt.legend(fontsize=14)save_fig(“minibatch_kmeans_vs_kmeans”)plt.show()Saving figure minibatch_kmeans_vs_kmeans Finding the optimal number of clusters¶What if the number of clusters was set to a lower or greater value than 5?In [97]:kmeans_k2 = KMeans(n_clusters=2, random_state=42)kmeans_k4 = KMeans(n_clusters=4, random_state=42)kmeans_k8 = KMeans(n_clusters=8, random_state=42)plot_clusterer_comparison(kmeans_k2, kmeans_k4, X, “$k=2$”, “$k=4$”)save_fig(“bad_n_clusters_diagram”)plt.show()Saving figure bad_n_clusters_diagram Ouch, these two models don’t look great. What about their inertias?In [65]:kmeans_k3.inertia_Out[65]:653.2167190021553In [66]:kmeans_k8.inertia_Out[66]:119.11983416102879No, 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:In [67]: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]In [68]: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_diagram”)plt.show()Saving figure inertia_vs_k_diagram 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.In [69]:plot_decision_boundaries(kmeans_per_k[4-1], X)plt.show() 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 $k$:In [70]:from sklearn.metrics import silhouette_scoreIn [71]:silhouette_score(X, kmeans.labels_)Out[71]:0.34507909442492657In [72]:silhouette_scores = [silhouette_score(X, model.labels_) for model in kmeans_per_k[1:]]In [73]: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_diagram”)plt.show()Saving figure silhouette_score_vs_k_diagram 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:In [74]:from sklearn.metrics import silhouette_samplesfrom matplotlib.ticker import FixedLocator, FixedFormatterplt.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) // 30pos = paddingticks = []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) + paddingplt.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_diagram”)plt.show()Saving figure silhouette_analysis_diagram Limits of K-Means¶In [75]: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]In [76]:plot_clusters(X) In [77]: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)Out[77]:KMeans(algorithm=’auto’, copy_x=True, init=’k-means++’, max_iter=300, n_clusters=3, n_init=10, n_jobs=None, precompute_distances=’auto’, random_state=42, tol=0.0001, verbose=0)In [78]: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_diagram”)plt.show()Saving figure bad_kmeans_diagram
TIP: Scaling It is important to scale the input features before you run K-Means, or the clusters may be very stretched and K-Means will perform poorly. Scaling the features does not guarantee that all the clusters will be nice and spherical, but it generally improves things. EXERCISE 3: Scaling Generate some new toy data and try K-means with and without different scaling functions. Find some examples where scaling improves things, and where it does not really make any difference. Comment on what you notice. Using clustering for image segmentation¶In [100]:from matplotlib.image import imreadimage = imread(os.path.join(“images”,”unsupervised_learning”,”ladybug.png”))image.shapeOut[100]:(533, 800, 3)In [102]: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)In [103]: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))In [104]: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 EXERCISE 4: Colour space – showing clusters in 3 dimensions. In the figure above we can see the different representations as we use fewer and fewer colours to describe the image – the cluster centres. Plot the orginal pixel data in 3D and show the different cluster centres as we reduce the number of colours. Clearly show what is happening (eg. highlight the pixels) when the insect disapears when reducing from 6 to 4 clusters. 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.In [105]:from sklearn.datasets import load_digitsIn [106]:X_digits, y_digits = load_digits(return_X_y=True)Let’s split it into a training set and a test set:In [107]:from sklearn.model_selection import train_test_splitIn [108]:X_train, X_test, y_train, y_test = train_test_split(X_digits, y_digits, random_state=42)Now let’s fit a Logistic Regression model and evaluate it on the test set:In [109]:from sklearn.linear_model import LogisticRegressionIn [110]:log_reg = LogisticRegression(multi_class=”ovr”, solver=”liblinear”, random_state=42)log_reg.fit(X_train, y_train)Out[110]:LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True, intercept_scaling=1, l1_ratio=None, max_iter=100, multi_class=’ovr’, n_jobs=None, penalty=’l2′, random_state=42, solver=’liblinear’, tol=0.0001, verbose=0, warm_start=False)In [111]:log_reg.score(X_test, y_test)Out[111]:0.9666666666666667Okay, that’s our baseline: 96.7% accuracy. Let’s see if we can do better by using K-Means as a preprocessing step. We will create a pipeline that will first cluster the training set into 50 clusters and replace the images with their distances to the 50 clusters, then apply a logistic regression model:In [112]:from sklearn.pipeline import PipelineIn [113]:pipeline = Pipeline([(“kmeans”, KMeans(n_clusters=50, random_state=42)),(“log_reg”, LogisticRegression(multi_class=”ovr”, solver=”liblinear”, random_state=42)),])pipeline.fit(X_train, y_train)Out[113]:Pipeline(memory=None, steps=[(‘kmeans’, KMeans(algorithm=’auto’, copy_x=True, init=’k-means++’,max_iter=300, n_clusters=50, n_init=10, n_jobs=None,precompute_distances=’auto’, random_state=42,tol=0.0001, verbose=0)),(‘log_reg’, LogisticRegression(C=1.0, class_weight=None, dual=False,fit_intercept=True, intercept_scaling=1,l1_ratio=None, max_iter=100,multi_class=’ovr’, n_jobs=None,penalty=’l2′, random_state=42,solver=’liblinear’, tol=0.0001, verbose=0,warm_start=False))], verbose=False)In [114]:pipeline.score(X_test, y_test)Out[114]:0.9822222222222222In [115]:1 – (1 – 0.9822222) / (1 – 0.9666666)Out[115]:0.4666670666658673How about that? We almost divided the error rate by a factor of 2! But we chose the number of clusters $k$ completely arbitrarily, we can surely do better. Since K-Means is just a preprocessing step in a classification pipeline, finding a good value for $k$ is much simpler than earlier: there’s no need to perform silhouette analysis or minimize the inertia, the best value of $k$ is simply the one that results in the best classification performance.In [116]:from sklearn.model_selection import GridSearchCVIn [117]:param_grid = dict(kmeans__n_clusters=range(2, 100))grid_clf = GridSearchCV(pipeline, param_grid, cv=3, verbose=2)grid_clf.fit(X_train, y_train)Fitting 3 folds for each of 98 candidates, totalling 294 fits[CV] kmeans__n_clusters=2 ……………………………………..[CV] ……………………….. kmeans__n_clusters=2, total= 0.1s[CV] kmeans__n_clusters=2 ……………………………………..[CV] ……………………….. kmeans__n_clusters=2, total= 0.1s[CV] kmeans__n_clusters=2 ……………………………………..[Parallel(n_jobs=1)]: Using backend SequentialBackend with 1 concurrent workers.[Parallel(n_jobs=1)]: Done 1 out of 1 | elapsed:0.1s remaining:0.0s[CV] ……………………….. kmeans__n_clusters=2, total= 0.1s[CV] kmeans__n_clusters=3 ……………………………………..[CV] ……………………….. kmeans__n_clusters=3, total= 0.1s[CV] kmeans__n_clusters=3 ……………………………………..[CV] ……………………….. kmeans__n_clusters=3, total= 0.1s[CV] kmeans__n_clusters=3 ……………………………………..[CV] ……………………….. kmeans__n_clusters=3, total= 0.1s[CV] kmeans__n_clusters=4 ……………………………………..[CV] ……………………….. kmeans__n_clusters=4, total= 0.1s[CV] kmeans__n_clusters=4 ……………………………………..[CV] ……………………….. kmeans__n_clusters=4, total= 0.1s[CV] kmeans__n_clusters=4 ……………………………………..[CV] ……………………….. kmeans__n_clusters=4, total= 0.2s[CV] kmeans__n_clusters=5 ……………………………………..[CV] ……………………….. kmeans__n_clusters=5, total= 0.2s[CV] kmeans__n_clusters=5 ……………………………………..[CV] ……………………….. kmeans__n_clusters=5, total= 0.2s[CV] kmeans__n_clusters=5 ……………………………………..[CV] ……………………….. kmeans__n_clusters=5, total= 0.2s[CV] kmeans__n_clusters=6 ……………………………………..[CV] ……………………….. kmeans__n_clusters=6, total= 0.2s[CV] kmeans__n_clusters=6 ……………………………………..[CV] ……………………….. kmeans__n_clusters=6, total= 0.3s[CV] kmeans__n_clusters=6 ……………………………………..[CV] ……………………….. kmeans__n_clusters=6, total= 0.3s[CV] kmeans__n_clusters=7 ……………………………………..[CV] ……………………….. kmeans__n_clusters=7, total= 0.2s[CV] kmeans__n_clusters=7 ……………………………………..[CV] ……………………….. kmeans__n_clusters=7, total= 0.2s[CV] kmeans__n_clusters=7 ……………………………………..[CV] ……………………….. kmeans__n_clusters=7, total= 0.2s[CV] kmeans__n_clusters=8 ……………………………………..[CV] ……………………….. kmeans__n_clusters=8, total= 0.2s[CV] kmeans__n_clusters=8 ……………………………………..[CV] ……………………….. kmeans__n_clusters=8, total= 0.2s[CV] kmeans__n_clusters=8 ……………………………………..[CV] ……………………….. kmeans__n_clusters=8, total= 0.1s[CV] kmeans__n_clusters=9 ……………………………………..[CV] ……………………….. kmeans__n_clusters=9, total= 0.2s[CV] kmeans__n_clusters=9 ……………………………………..[CV] ……………………….. kmeans__n_clusters=9, total= 0.2s[CV] kmeans__n_clusters=9 ……………………………………..[CV] ……………………….. kmeans__n_clusters=9, total= 0.2s[CV] kmeans__n_clusters=10 …………………………………….[CV] ………………………. kmeans__n_clusters=10, total= 0.2s[CV] kmeans__n_clusters=10 …………………………………….[CV] ………………………. kmeans__n_clusters=10, total= 0.2s[CV] kmeans__n_clusters=10 …………………………………….[CV] ………………………. kmeans__n_clusters=10, total= 0.3s[CV] kmeans__n_clusters=11 …………………………………….[CV] ………………………. kmeans__n_clusters=11, total= 0.2s[CV] kmeans__n_clusters=11 …………………………………….[CV] ………………………. kmeans__n_clusters=11, total= 0.4s[CV] kmeans__n_clusters=11 …………………………………….[CV] ………………………. kmeans__n_clusters=11, total= 0.6s[CV] kmeans__n_clusters=12 …………………………………….[CV] ………………………. kmeans__n_clusters=12, total= 0.4s[CV] kmeans__n_clusters=12 …………………………………….[CV] ………………………. kmeans__n_clusters=12, total= 0.2s[CV] kmeans__n_clusters=12 …………………………………….[CV] ………………………. kmeans__n_clusters=12, total= 0.2s[CV] kmeans__n_clusters=13 …………………………………….[CV] ………………………. kmeans__n_clusters=13, total= 0.3s[CV] kmeans__n_clusters=13 …………………………………….[CV] ………………………. kmeans__n_clusters=13, total= 0.3s[CV] kmeans__n_clusters=13 …………………………………….[CV] ………………………. kmeans__n_clusters=13, total= 0.2s[CV] kmeans__n_clusters=14 …………………………………….[CV] ………………………. kmeans__n_clusters=14, total= 0.3s[CV] kmeans__n_clusters=14 …………………………………….[CV] ………………………. kmeans__n_clusters=14, total= 0.2s[CV] kmeans__n_clusters=14 …………………………………….[CV] ………………………. kmeans__n_clusters=14, total= 0.2s[CV] kmeans__n_clusters=15 …………………………………….[CV] ………………………. kmeans__n_clusters=15, total= 0.2s[CV] kmeans__n_clusters=15 …………………………………….[CV] ………………………. kmeans__n_clusters=15, total= 0.2s[CV] kmeans__n_clusters=15 …………………………………….[CV] ………………………. kmeans__n_clusters=15, total= 0.2s[CV] kmeans__n_clusters=16 …………………………………….[CV] ………………………. kmeans__n_clusters=16, total= 0.2s[CV] kmeans__n_clusters=16 …………………………………….[CV] ………………………. kmeans__n_clusters=16, total= 0.2s[CV] kmeans__n_clusters=16 …………………………………….[CV] ………………………. kmeans__n_clusters=16, total= 0.2s[CV] kmeans__n_clusters=17 …………………………………….[CV] ………………………. kmeans__n_clusters=17, total= 0.3s[CV] kmeans__n_clusters=17 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…………………………………….[CV] ………………………. kmeans__n_clusters=35, total= 0.4s[CV] kmeans__n_clusters=35 …………………………………….[CV] ………………………. kmeans__n_clusters=35, total= 0.4s[CV] kmeans__n_clusters=36 …………………………………….[CV] ………………………. kmeans__n_clusters=36, total= 0.4s[CV] kmeans__n_clusters=36 …………………………………….[CV] ………………………. kmeans__n_clusters=36, total= 0.4s[CV] kmeans__n_clusters=36 …………………………………….[CV] ………………………. kmeans__n_clusters=36, total= 0.4s[CV] kmeans__n_clusters=37 …………………………………….[CV] ………………………. kmeans__n_clusters=37, total= 0.4s[CV] kmeans__n_clusters=37 …………………………………….[CV] ………………………. kmeans__n_clusters=37, total= 0.4s[CV] kmeans__n_clusters=37 …………………………………….[CV] ………………………. kmeans__n_clusters=37, total= 0.4s[CV] kmeans__n_clusters=38 …………………………………….[CV] ………………………. kmeans__n_clusters=38, total= 0.4s[CV] kmeans__n_clusters=38 …………………………………….[CV] ………………………. kmeans__n_clusters=38, total= 0.4s[CV] kmeans__n_clusters=38 …………………………………….[CV] ………………………. kmeans__n_clusters=38, total= 0.4s[CV] kmeans__n_clusters=39 …………………………………….[CV] ………………………. kmeans__n_clusters=39, total= 0.4s[CV] kmeans__n_clusters=39 …………………………………….[CV] ………………………. kmeans__n_clusters=39, total= 0.4s[CV] kmeans__n_clusters=39 …………………………………….[CV] ………………………. kmeans__n_clusters=39, total= 0.4s[CV] kmeans__n_clusters=40 …………………………………….[CV] ………………………. kmeans__n_clusters=40, total= 0.4s[CV] kmeans__n_clusters=40 …………………………………….[CV] ………………………. kmeans__n_clusters=40, total= 0.4s[CV] kmeans__n_clusters=40 …………………………………….[CV] ………………………. kmeans__n_clusters=40, total= 0.4s[CV] kmeans__n_clusters=41 …………………………………….[CV] ………………………. kmeans__n_clusters=41, total= 0.4s[CV] kmeans__n_clusters=41 …………………………………….[CV] ………………………. kmeans__n_clusters=41, total= 0.4s[CV] kmeans__n_clusters=41 …………………………………….[CV] ………………………. kmeans__n_clusters=41, total= 0.4s[CV] kmeans__n_clusters=42 …………………………………….[CV] ………………………. kmeans__n_clusters=42, total= 0.4s[CV] kmeans__n_clusters=42 …………………………………….[CV] ………………………. kmeans__n_clusters=42, total= 0.4s[CV] kmeans__n_clusters=42 …………………………………….[CV] ………………………. kmeans__n_clusters=42, total= 0.4s[CV] kmeans__n_clusters=43 …………………………………….[CV] ………………………. kmeans__n_clusters=43, total= 0.4s[CV] kmeans__n_clusters=43 …………………………………….[CV] ………………………. kmeans__n_clusters=43, total= 0.4s[CV] kmeans__n_clusters=43 …………………………………….[CV] ………………………. kmeans__n_clusters=43, total= 0.4s[CV] kmeans__n_clusters=44 …………………………………….[CV] ………………………. kmeans__n_clusters=44, total= 0.4s[CV] kmeans__n_clusters=44 …………………………………….[CV] ………………………. kmeans__n_clusters=44, total= 0.4s[CV] kmeans__n_clusters=44 …………………………………….[CV] ………………………. kmeans__n_clusters=44, total= 0.4s[CV] kmeans__n_clusters=45 …………………………………….[CV] ………………………. kmeans__n_clusters=45, total= 0.4s[CV] kmeans__n_clusters=45 …………………………………….[CV] ………………………. kmeans__n_clusters=45, total= 0.5s[CV] kmeans__n_clusters=45 …………………………………….[CV] ………………………. kmeans__n_clusters=45, total= 0.5s[CV] kmeans__n_clusters=46 …………………………………….[CV] ………………………. kmeans__n_clusters=46, total= 0.5s[CV] kmeans__n_clusters=46 …………………………………….[CV] ………………………. kmeans__n_clusters=46, total= 0.6s[CV] kmeans__n_clusters=46 …………………………………….[CV] ………………………. kmeans__n_clusters=46, total= 0.5s[CV] kmeans__n_clusters=47 …………………………………….[CV] ………………………. kmeans__n_clusters=47, total= 0.5s[CV] kmeans__n_clusters=47 …………………………………….[CV] ………………………. kmeans__n_clusters=47, total= 0.6s[CV] kmeans__n_clusters=47 …………………………………….[CV] ………………………. kmeans__n_clusters=47, total= 0.5s[CV] kmeans__n_clusters=48 …………………………………….[CV] ………………………. kmeans__n_clusters=48, total= 0.5s[CV] kmeans__n_clusters=48 …………………………………….[CV] ………………………. kmeans__n_clusters=48, total= 0.6s[CV] kmeans__n_clusters=48 …………………………………….[CV] ………………………. kmeans__n_clusters=48, total= 0.5s[CV] kmeans__n_clusters=49 …………………………………….[CV] ………………………. kmeans__n_clusters=49, total= 0.6s[CV] kmeans__n_clusters=49 …………………………………….[CV] ………………………. kmeans__n_clusters=49, total= 0.6s[CV] kmeans__n_clusters=49 …………………………………….[CV] ………………………. kmeans__n_clusters=49, total= 0.5s[CV] kmeans__n_clusters=50 …………………………………….[CV] ………………………. kmeans__n_clusters=50, total= 0.5s[CV] kmeans__n_clusters=50 …………………………………….[CV] ………………………. kmeans__n_clusters=50, total= 0.5s[CV] kmeans__n_clusters=50 …………………………………….[CV] ………………………. kmeans__n_clusters=50, total= 0.5s[CV] kmeans__n_clusters=51 …………………………………….[CV] ………………………. kmeans__n_clusters=51, total= 0.5s[CV] kmeans__n_clusters=51 …………………………………….[CV] ………………………. kmeans__n_clusters=51, total= 0.5s[CV] kmeans__n_clusters=51 …………………………………….[CV] ………………………. kmeans__n_clusters=51, total= 0.5s[CV] kmeans__n_clusters=52 …………………………………….[CV] ………………………. kmeans__n_clusters=52, total= 0.5s[CV] kmeans__n_clusters=52 …………………………………….[CV] ………………………. kmeans__n_clusters=52, total= 0.5s[CV] kmeans__n_clusters=52 …………………………………….[CV] ………………………. kmeans__n_clusters=52, total= 0.5s[CV] kmeans__n_clusters=53 …………………………………….[CV] ………………………. kmeans__n_clusters=53, total= 0.5s[CV] kmeans__n_clusters=53 …………………………………….[CV] ………………………. kmeans__n_clusters=53, total= 0.5s[CV] kmeans__n_clusters=53 …………………………………….[CV] ………………………. kmeans__n_clusters=53, total= 0.5s[CV] kmeans__n_clusters=54 …………………………………….[CV] ………………………. kmeans__n_clusters=54, total= 0.5s[CV] kmeans__n_clusters=54 …………………………………….[CV] ………………………. kmeans__n_clusters=54, total= 0.5s[CV] kmeans__n_clusters=54 …………………………………….[CV] ………………………. kmeans__n_clusters=54, total= 0.5s[CV] kmeans__n_clusters=55 …………………………………….[CV] ………………………. kmeans__n_clusters=55, total= 0.5s[CV] kmeans__n_clusters=55 …………………………………….[CV] ………………………. kmeans__n_clusters=55, total= 0.6s[CV] kmeans__n_clusters=55 …………………………………….[CV] ………………………. kmeans__n_clusters=55, total= 0.6s[CV] kmeans__n_clusters=56 …………………………………….[CV] ………………………. kmeans__n_clusters=56, total= 0.5s[CV] kmeans__n_clusters=56 …………………………………….[CV] ………………………. kmeans__n_clusters=56, total= 0.5s[CV] kmeans__n_clusters=56 …………………………………….[CV] ………………………. kmeans__n_clusters=56, total= 0.5s[CV] kmeans__n_clusters=57 …………………………………….[CV] ………………………. kmeans__n_clusters=57, total= 0.5s[CV] kmeans__n_clusters=57 …………………………………….[CV] ………………………. kmeans__n_clusters=57, total= 0.5s[CV] kmeans__n_clusters=57 …………………………………….[CV] ………………………. kmeans__n_clusters=57, total= 0.5s[CV] kmeans__n_clusters=58 …………………………………….[CV] ………………………. kmeans__n_clusters=58, total= 0.5s[CV] kmeans__n_clusters=58 …………………………………….[CV] ………………………. kmeans__n_clusters=58, total= 0.5s[CV] kmeans__n_clusters=58 …………………………………….[CV] ………………………. kmeans__n_clusters=58, total= 0.5s[CV] kmeans__n_clusters=59 …………………………………….[CV] ………………………. kmeans__n_clusters=59, total= 0.5s[CV] kmeans__n_clusters=59 …………………………………….[CV] ………………………. kmeans__n_clusters=59, total= 0.5s[CV] kmeans__n_clusters=59 …………………………………….[CV] ………………………. kmeans__n_clusters=59, total= 0.6s[CV] kmeans__n_clusters=60 …………………………………….[CV] ………………………. kmeans__n_clusters=60, total= 0.6s[CV] kmeans__n_clusters=60 …………………………………….[CV] ………………………. kmeans__n_clusters=60, total= 0.6s[CV] kmeans__n_clusters=60 …………………………………….[CV] ………………………. kmeans__n_clusters=60, total= 0.6s[CV] kmeans__n_clusters=61 …………………………………….[CV] ………………………. kmeans__n_clusters=61, total= 0.6s[CV] kmeans__n_clusters=61 …………………………………….[CV] ………………………. kmeans__n_clusters=61, total= 0.6s[CV] kmeans__n_clusters=61 …………………………………….[CV] ………………………. kmeans__n_clusters=61, total= 0.7s[CV] kmeans__n_clusters=62 …………………………………….[CV] ………………………. kmeans__n_clusters=62, total= 0.5s[CV] kmeans__n_clusters=62 …………………………………….[CV] ………………………. kmeans__n_clusters=62, total= 0.6s[CV] kmeans__n_clusters=62 …………………………………….[CV] ………………………. kmeans__n_clusters=62, total= 0.6s[CV] kmeans__n_clusters=63 …………………………………….[CV] ………………………. kmeans__n_clusters=63, total= 0.7s[CV] kmeans__n_clusters=63 …………………………………….[CV] ………………………. kmeans__n_clusters=63, total= 0.6s[CV] kmeans__n_clusters=63 …………………………………….[CV] ………………………. kmeans__n_clusters=63, total= 1.2s[CV] kmeans__n_clusters=64 …………………………………….[CV] ………………………. kmeans__n_clusters=64, total= 1.0s[CV] kmeans__n_clusters=64 …………………………………….[CV] ………………………. kmeans__n_clusters=64, total= 1.8s[CV] kmeans__n_clusters=64 …………………………………….[CV] ………………………. kmeans__n_clusters=64, total= 0.8s[CV] kmeans__n_clusters=65 …………………………………….[CV] ………………………. kmeans__n_clusters=65, total= 0.8s[CV] kmeans__n_clusters=65 …………………………………….[CV] ………………………. kmeans__n_clusters=65, total= 1.0s[CV] kmeans__n_clusters=65 …………………………………….[CV] ………………………. kmeans__n_clusters=65, total= 1.0s[CV] kmeans__n_clusters=66 …………………………………….[CV] ………………………. kmeans__n_clusters=66, total= 0.7s[CV] kmeans__n_clusters=66 …………………………………….[CV] ………………………. kmeans__n_clusters=66, total= 0.7s[CV] kmeans__n_clusters=66 …………………………………….[CV] ………………………. kmeans__n_clusters=66, total= 0.7s[CV] kmeans__n_clusters=67 …………………………………….[CV] ………………………. kmeans__n_clusters=67, total= 0.6s[CV] kmeans__n_clusters=67 …………………………………….[CV] ………………………. kmeans__n_clusters=67, total= 0.8s[CV] kmeans__n_clusters=67 …………………………………….[CV] ………………………. kmeans__n_clusters=67, total= 0.9s[CV] kmeans__n_clusters=68 …………………………………….[CV] ………………………. kmeans__n_clusters=68, total= 1.1s[CV] kmeans__n_clusters=68 …………………………………….[CV] ………………………. kmeans__n_clusters=68, total= 0.9s[CV] kmeans__n_clusters=68 …………………………………….[CV] ………………………. kmeans__n_clusters=68, total= 0.7s[CV] kmeans__n_clusters=69 …………………………………….[CV] ………………………. kmeans__n_clusters=69, total= 0.6s[CV] kmeans__n_clusters=69 …………………………………….[CV] ………………………. kmeans__n_clusters=69, total= 0.6s[CV] kmeans__n_clusters=69 …………………………………….[CV] ………………………. kmeans__n_clusters=69, total= 1.0s[CV] kmeans__n_clusters=70 …………………………………….[CV] ………………………. kmeans__n_clusters=70, total= 0.7s[CV] kmeans__n_clusters=70 …………………………………….[CV] ………………………. kmeans__n_clusters=70, total= 0.7s[CV] kmeans__n_clusters=70 …………………………………….[CV] ………………………. kmeans__n_clusters=70, total= 0.7s[CV] kmeans__n_clusters=71 …………………………………….[CV] ………………………. kmeans__n_clusters=71, total= 0.7s[CV] kmeans__n_clusters=71 …………………………………….[CV] ………………………. kmeans__n_clusters=71, total= 0.7s[CV] kmeans__n_clusters=71 …………………………………….[CV] ………………………. kmeans__n_clusters=71, total= 0.6s[CV] kmeans__n_clusters=72 …………………………………….[CV] ………………………. kmeans__n_clusters=72, total= 0.6s[CV] kmeans__n_clusters=72 …………………………………….[CV] ………………………. kmeans__n_clusters=72, total= 0.7s[CV] kmeans__n_clusters=72 …………………………………….[CV] ………………………. kmeans__n_clusters=72, total= 0.8s[CV] kmeans__n_clusters=73 …………………………………….[CV] ………………………. kmeans__n_clusters=73, total= 0.8s[CV] kmeans__n_clusters=73 …………………………………….[CV] ………………………. kmeans__n_clusters=73, total= 0.7s[CV] kmeans__n_clusters=73 …………………………………….[CV] ………………………. kmeans__n_clusters=73, total= 0.9s[CV] kmeans__n_clusters=74 …………………………………….[CV] ………………………. kmeans__n_clusters=74, total= 0.8s[CV] kmeans__n_clusters=74 …………………………………….[CV] ………………………. kmeans__n_clusters=74, total= 0.8s[CV] kmeans__n_clusters=74 …………………………………….[CV] ………………………. kmeans__n_clusters=74, total= 0.8s[CV] kmeans__n_clusters=75 …………………………………….[CV] ………………………. kmeans__n_clusters=75, total= 0.8s[CV] kmeans__n_clusters=75 …………………………………….[CV] ………………………. kmeans__n_clusters=75, total= 1.1s[CV] kmeans__n_clusters=75 …………………………………….[CV] ………………………. kmeans__n_clusters=75, total= 0.8s[CV] kmeans__n_clusters=76 …………………………………….[CV] ………………………. kmeans__n_clusters=76, total= 1.0s[CV] kmeans__n_clusters=76 …………………………………….[CV] ………………………. kmeans__n_clusters=76, total= 0.9s[CV] kmeans__n_clusters=76 …………………………………….[CV] ………………………. kmeans__n_clusters=76, total= 0.9s[CV] kmeans__n_clusters=77 …………………………………….[CV] ………………………. kmeans__n_clusters=77, total= 0.9s[CV] kmeans__n_clusters=77 …………………………………….[CV] ………………………. kmeans__n_clusters=77, total= 1.0s[CV] kmeans__n_clusters=77 …………………………………….[CV] ………………………. kmeans__n_clusters=77, total= 0.8s[CV] kmeans__n_clusters=78 …………………………………….[CV] ………………………. kmeans__n_clusters=78, total= 0.9s[CV] kmeans__n_clusters=78 …………………………………….[CV] ………………………. kmeans__n_clusters=78, total= 0.9s[CV] kmeans__n_clusters=78 …………………………………….[CV] ………………………. kmeans__n_clusters=78, total= 0.8s[CV] kmeans__n_clusters=79 …………………………………….[CV] ………………………. kmeans__n_clusters=79, total= 0.7s[CV] kmeans__n_clusters=79 …………………………………….[CV] ………………………. kmeans__n_clusters=79, total= 0.7s[CV] kmeans__n_clusters=79 …………………………………….[CV] ………………………. kmeans__n_clusters=79, total= 0.7s[CV] kmeans__n_clusters=80 …………………………………….[CV] ………………………. kmeans__n_clusters=80, total= 0.7s[CV] kmeans__n_clusters=80 …………………………………….[CV] ………………………. kmeans__n_clusters=80, total= 0.8s[CV] kmeans__n_clusters=80 …………………………………….[CV] ………………………. kmeans__n_clusters=80, total= 0.7s[CV] kmeans__n_clusters=81 …………………………………….[CV] ………………………. kmeans__n_clusters=81, total= 0.7s[CV] kmeans__n_clusters=81 …………………………………….[CV] ………………………. kmeans__n_clusters=81, total= 0.7s[CV] kmeans__n_clusters=81 …………………………………….[CV] ………………………. kmeans__n_clusters=81, total= 0.7s[CV] kmeans__n_clusters=82 …………………………………….[CV] ………………………. kmeans__n_clusters=82, total= 0.8s[CV] kmeans__n_clusters=82 …………………………………….[CV] ………………………. kmeans__n_clusters=82, total= 0.9s[CV] kmeans__n_clusters=82 …………………………………….[CV] ………………………. kmeans__n_clusters=82, total= 0.8s[CV] kmeans__n_clusters=83 …………………………………….[CV] ………………………. kmeans__n_clusters=83, total= 0.7s[CV] kmeans__n_clusters=83 …………………………………….[CV] ………………………. kmeans__n_clusters=83, total= 0.7s[CV] kmeans__n_clusters=83 …………………………………….[CV] ………………………. kmeans__n_clusters=83, total= 0.8s[CV] kmeans__n_clusters=84 …………………………………….[CV] ………………………. kmeans__n_clusters=84, total= 0.7s[CV] kmeans__n_clusters=84 …………………………………….[CV] ………………………. kmeans__n_clusters=84, total= 0.7s[CV] kmeans__n_clusters=84 …………………………………….[CV] ………………………. kmeans__n_clusters=84, total= 0.7s[CV] kmeans__n_clusters=85 …………………………………….[CV] ………………………. kmeans__n_clusters=85, total= 0.7s[CV] kmeans__n_clusters=85 …………………………………….[CV] ………………………. kmeans__n_clusters=85, total= 0.8s[CV] kmeans__n_clusters=85 …………………………………….[CV] ………………………. kmeans__n_clusters=85, total= 0.8s[CV] kmeans__n_clusters=86 …………………………………….[CV] ………………………. kmeans__n_clusters=86, total= 0.7s[CV] kmeans__n_clusters=86 …………………………………….[CV] ………………………. kmeans__n_clusters=86, total= 0.7s[CV] kmeans__n_clusters=86 …………………………………….[CV] ………………………. kmeans__n_clusters=86, total= 0.8s[CV] kmeans__n_clusters=87 …………………………………….[CV] ………………………. kmeans__n_clusters=87, total= 0.8s[CV] kmeans__n_clusters=87 …………………………………….[CV] ………………………. kmeans__n_clusters=87, total= 0.8s[CV] kmeans__n_clusters=87 …………………………………….[CV] ………………………. kmeans__n_clusters=87, total= 0.8s[CV] kmeans__n_clusters=88 …………………………………….[CV] ………………………. kmeans__n_clusters=88, total= 0.7s[CV] kmeans__n_clusters=88 …………………………………….[CV] ………………………. kmeans__n_clusters=88, total= 0.7s[CV] kmeans__n_clusters=88 …………………………………….[CV] ………………………. kmeans__n_clusters=88, total= 0.9s[CV] kmeans__n_clusters=89 …………………………………….[CV] ………………………. kmeans__n_clusters=89, total= 0.8s[CV] kmeans__n_clusters=89 …………………………………….[CV] ………………………. kmeans__n_clusters=89, total= 0.8s[CV] kmeans__n_clusters=89 …………………………………….[CV] ………………………. kmeans__n_clusters=89, total= 0.8s[CV] kmeans__n_clusters=90 …………………………………….[CV] ………………………. kmeans__n_clusters=90, total= 0.9s[CV] kmeans__n_clusters=90 …………………………………….[CV] ………………………. kmeans__n_clusters=90, total= 0.8s[CV] kmeans__n_clusters=90 …………………………………….[CV] ………………………. kmeans__n_clusters=90, total= 0.8s[CV] kmeans__n_clusters=91 …………………………………….[CV] ………………………. kmeans__n_clusters=91, total= 0.8s[CV] kmeans__n_clusters=91 …………………………………….[CV] ………………………. kmeans__n_clusters=91, total= 0.8s[CV] kmeans__n_clusters=91 …………………………………….[CV] ………………………. kmeans__n_clusters=91, total= 0.8s[CV] kmeans__n_clusters=92 …………………………………….[CV] ………………………. kmeans__n_clusters=92, total= 0.8s[CV] kmeans__n_clusters=92 …………………………………….[CV] ………………………. kmeans__n_clusters=92, total= 0.9s[CV] kmeans__n_clusters=92 …………………………………….[CV] ………………………. kmeans__n_clusters=92, total= 1.0s[CV] kmeans__n_clusters=93 …………………………………….[CV] ………………………. kmeans__n_clusters=93, total= 0.9s[CV] kmeans__n_clusters=93 …………………………………….[CV] ………………………. kmeans__n_clusters=93, total= 0.9s[CV] kmeans__n_clusters=93 …………………………………….[CV] ………………………. kmeans__n_clusters=93, total= 0.8s[CV] kmeans__n_clusters=94 …………………………………….[CV] ………………………. kmeans__n_clusters=94, total= 0.9s[CV] kmeans__n_clusters=94 …………………………………….[CV] ………………………. kmeans__n_clusters=94, total= 1.1s[CV] kmeans__n_clusters=94 …………………………………….[CV] ………………………. kmeans__n_clusters=94, total= 1.2s[CV] kmeans__n_clusters=95 …………………………………….[CV] ………………………. kmeans__n_clusters=95, total= 0.8s[CV] kmeans__n_clusters=95 …………………………………….[CV] ………………………. kmeans__n_clusters=95, total= 0.9s[CV] kmeans__n_clusters=95 …………………………………….[CV] ………………………. kmeans__n_clusters=95, total= 0.8s[CV] kmeans__n_clusters=96 …………………………………….[CV] ………………………. kmeans__n_clusters=96, total= 0.9s[CV] kmeans__n_clusters=96 …………………………………….[CV] ………………………. kmeans__n_clusters=96, total= 0.9s[CV] kmeans__n_clusters=96 …………………………………….[CV] ………………………. kmeans__n_clusters=96, total= 0.9s[CV] kmeans__n_clusters=97 …………………………………….[CV] ………………………. kmeans__n_clusters=97, total= 0.8s[CV] kmeans__n_clusters=97 …………………………………….[CV] ………………………. kmeans__n_clusters=97, total= 0.9s[CV] kmeans__n_clusters=97 …………………………………….[CV] ………………………. kmeans__n_clusters=97, total= 0.9s[CV] kmeans__n_clusters=98 …………………………………….[CV] ………………………. kmeans__n_clusters=98, total= 0.9s[CV] kmeans__n_clusters=98 …………………………………….[CV] ………………………. kmeans__n_clusters=98, total= 0.9s[CV] kmeans__n_clusters=98 …………………………………….[CV] ………………………. kmeans__n_clusters=98, total= 0.9s[CV] kmeans__n_clusters=99 …………………………………….[CV] ………………………. kmeans__n_clusters=99, total= 1.0s[CV] kmeans__n_clusters=99 …………………………………….[CV] ………………………. kmeans__n_clusters=99, total= 1.0s[CV] kmeans__n_clusters=99 …………………………………….[CV] ………………………. kmeans__n_clusters=99, total= 1.0s[Parallel(n_jobs=1)]: Done 294 out of 294 | elapsed:2.7min finishedOut[117]:GridSearchCV(cv=3, error_score=’raise-deprecating’, estimator=Pipeline(memory=None,steps=[(‘kmeans’,KMeans(algorithm=’auto’, copy_x=True, init=’k-means++’, max_iter=300, n_clusters=50, n_init=10, n_jobs=None, precompute_distances=’auto’, random_state=42, tol=0.0001, verbose=0)), (‘log_reg’,LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True, intercept_scaling=1, l1_ratio=None, max_iter=100, multi_class=’ovr’, n_jobs=None, penalty=’l2′, random_state=42, solver=’liblinear’, tol=0.0001, verbose=0, warm_start=False))],verbose=False), iid=’warn’, n_jobs=None, param_grid={‘kmeans__n_clusters’: range(2, 100)}, pre_dispatch=’2*n_jobs’, refit=True, return_train_score=False, scoring=None, verbose=2)In [118]:grid_clf.best_params_Out[118]:{‘kmeans__n_clusters’: 90}In [119]:grid_clf.score(X_test, y_test)Out[119]:0.9844444444444445The performance is slightly improved when $k=90$, so 90 it is.EXERCISE 5: Explain this phenomena Create some vizualizations that give you insight into what is happening here. Explain why classification performnace go up as `k` increases. Show why `k=10`is unlikley to give good results. Clustering for Semi-supervised Learning¶Another use case for clustering is in semi-supervised learning, when we have plenty of unlabeled instances and very few labeled instances.Let’s look at the performance of a logistic regression model when we only have 50 labeled instances:In [120]:n_labeled = 50In [121]:log_reg = LogisticRegression(multi_class=”ovr”, solver=”liblinear”, random_state=42)log_reg.fit(X_train[:n_labeled], y_train[:n_labeled])log_reg.score(X_test, y_test)Out[121]:0.8266666666666667It’s much less than earlier of course. Let’s see how we can do better. First, let’s cluster the training set into 50 clusters, then for each cluster let’s find the image closest to the centroid. We will call these images the representative images:In [122]:k = 50In [123]:kmeans = KMeans(n_clusters=k, random_state=42)X_digits_dist = kmeans.fit_transform(X_train)representative_digit_idx = np.argmin(X_digits_dist, axis=0)X_representative_digits = X_train[representative_digit_idx]Now let’s plot these representative images and label them manually:In [124]:plt.figure(figsize=(8, 2))for index, X_representative_digit in enumerate(X_representative_digits):plt.subplot(k // 10, 10, index + 1)plt.imshow(X_representative_digit.reshape(8, 8), cmap=”binary”, interpolation=”bilinear”)plt.axis(‘off’)save_fig(“representative_images_diagram”, tight_layout=False)plt.show()Saving figure representative_images_diagram In [125]:y_representative_digits = np.array([4, 8, 0, 6, 8, 3, 7, 7, 9, 2,5, 5, 8, 5, 2, 1, 2, 9, 6, 1,1, 6, 9, 0, 8, 3, 0, 7, 4, 1,6, 5, 2, 4, 1, 8, 6, 3, 9, 2,4, 2, 9, 4, 7, 6, 2, 3, 1, 1])Now we have a dataset with just 50 labeled instances, but instead of being completely random instances, each of them is a representative image of its cluster. Let’s see if the performance is any better:In [126]:log_reg = LogisticRegression(multi_class=”ovr”, solver=”liblinear”, random_state=42)log_reg.fit(X_representative_digits, y_representative_digits)log_reg.score(X_test, y_test)Out[126]:0.9244444444444444Wow! We jumped from 82.7% accuracy to 92.4%, although we are still only training the model on 50 instances. Since it’s often costly and painful to label instances, especially when it has to be done manually by experts, it’s a good idea to make them label representative instances rather than just random instances.But perhaps we can go one step further: what if we propagated the labels to all the other instances in the same cluster?In [127]:y_train_propagated = np.empty(len(X_train), dtype=np.int32)for i in range(k):y_train_propagated[kmeans.labels_==i] = y_representative_digits[i]In [128]:log_reg = LogisticRegression(multi_class=”ovr”, solver=”liblinear”, random_state=42)log_reg.fit(X_train, y_train_propagated)Out[128]:LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True, intercept_scaling=1, l1_ratio=None, max_iter=100, multi_class=’ovr’, n_jobs=None, penalty=’l2′, random_state=42, solver=’liblinear’, tol=0.0001, verbose=0, warm_start=False)In [129]:log_reg.score(X_test, y_test)Out[129]:0.9288888888888889We got a tiny little accuracy boost. Better than nothing, but we should probably have propagated the labels only to the instances closest to the centroid, because by propagating to the full cluster, we have certainly included some outliers. Let’s only propagate the labels to the 20th percentile closest to the centroid:In [130]:percentile_closest = 20X_cluster_dist = X_digits_dist[np.arange(len(X_train)), kmeans.labels_]for i in range(k):in_cluster = (kmeans.labels_ == i)cluster_dist = X_cluster_dist[in_cluster]cutoff_distance = np.percentile(cluster_dist, percentile_closest)above_cutoff = (X_cluster_dist > cutoff_distance)
X_cluster_dist[in_cluster & above_cutoff] = -1
In [131]:
partially_propagated = (X_cluster_dist != -1)
X_train_partially_propagated = X_train[partially_propagated]
y_train_partially_propagated = y_train_propagated[partially_propagated]
In [132]:
log_reg = LogisticRegression(multi_class=”ovr”, solver=”liblinear”, random_state=42)
log_reg.fit(X_train_partially_propagated, y_train_partially_propagated)
Out[132]:
LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,
intercept_scaling=1, l1_ratio=None, max_iter=100,
multi_class=’ovr’, n_jobs=None, penalty=’l2′,
random_state=42, solver=’liblinear’, tol=0.0001, verbose=0,
warm_start=False)
In [133]:
log_reg.score(X_test, y_test)
Out[133]:
0.9422222222222222
Nice! With just 50 labeled instances (just 5 examples per class on average!), we got 94.2% performance, which is pretty close to the performance of logistic regression on the fully labeled digits dataset (which was 96.7%).
This is because the propagated labels are actually pretty good: their accuracy is very close to 99%:
In [134]:
np.mean(y_train_partially_propagated == y_train[partially_propagated])
Out[134]:
0.9896907216494846
You could now do a few iterations of active learning:
1.Manually label the instances that the classifier is least sure about, if possible by picking them in distinct clusters.
2.Train a new model with these additional labels.
DBSCAN¶
In [135]:
from sklearn.datasets import make_moons
In [136]:
X, y = make_moons(n_samples=1000, noise=0.05, random_state=42)
In [137]:
from sklearn.cluster import DBSCAN
In [138]:
dbscan = DBSCAN(eps=0.05, min_samples=5)
dbscan.fit(X)
Out[138]:
DBSCAN(algorithm=’auto’, eps=0.05, leaf_size=30, metric=’euclidean’,
metric_params=None, min_samples=5, n_jobs=None, p=None)
In [139]:
dbscan.labels_[:10]
Out[139]:
array([ 0,2, -1, -1,1,0,0,0,2,5])
In [140]:
len(dbscan.core_sample_indices_)
Out[140]:
808
In [141]:
dbscan.core_sample_indices_[:10]
Out[141]:
array([ 0,4,5,6,7,8, 10, 11, 12, 13])
In [142]:
dbscan.components_[:3]
Out[142]:
array([[-0.02137124,0.40618608],
[-0.84192557,0.53058695],
[ 0.58930337, -0.32137599]])
In [143]:
np.unique(dbscan.labels_)
Out[143]:
array([-1,0,1,2,3,4,5,6])
In [144]:
dbscan2 = DBSCAN(eps=0.2)
dbscan2.fit(X)
Out[144]:
DBSCAN(algorithm=’auto’, eps=0.2, leaf_size=30, metric=’euclidean’,
metric_params=None, min_samples=5, n_jobs=None, p=None)
In [145]:
def plot_dbscan(dbscan, X, size, show_xlabels=True, show_ylabels=True):
core_mask = np.zeros_like(dbscan.labels_, dtype=bool)
core_mask[dbscan.core_sample_indices_] = True
anomalies_mask = dbscan.labels_ == -1
non_core_mask = ~(core_mask | anomalies_mask)
cores = dbscan.components_
anomalies = X[anomalies_mask]
non_cores = X[non_core_mask]
plt.scatter(cores[:, 0], cores[:, 1],
c=dbscan.labels_[core_mask], marker=’o’, s=size, cmap=”Paired”)
plt.scatter(cores[:, 0], cores[:, 1], marker=’*’, s=20, c=dbscan.labels_[core_mask])
plt.scatter(anomalies[:, 0], anomalies[:, 1],
c=”r”, marker=”x”, s=100)
plt.scatter(non_cores[:, 0], non_cores[:, 1], c=dbscan.labels_[non_core_mask], marker=”.”)
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.title(“eps={:.2f}, min_samples={}”.format(dbscan.eps, dbscan.min_samples), fontsize=14)
In [146]:
plt.figure(figsize=(9, 3.2))
plt.subplot(121)
plot_dbscan(dbscan, X, size=100)
plt.subplot(122)
plot_dbscan(dbscan2, X, size=600, show_ylabels=False)
save_fig(“dbscan_diagram”)
plt.show()
Saving figure dbscan_diagram

In [147]:
dbscan = dbscan2
In [148]:
from sklearn.neighbors import KNeighborsClassifier
In [149]:
knn = KNeighborsClassifier(n_neighbors=50)
knn.fit(dbscan.components_, dbscan.labels_[dbscan.core_sample_indices_])
Out[149]:
KNeighborsClassifier(algorithm=’auto’, leaf_size=30, metric=’minkowski’,
metric_params=None, n_jobs=None, n_neighbors=50, p=2,
weights=’uniform’)
In [150]:
X_new = np.array([[-0.5, 0], [0, 0.5], [1, -0.1], [2, 1]])
knn.predict(X_new)
Out[150]:
array([1, 0, 1, 0])
In [151]:
knn.predict_proba(X_new)
Out[151]:
array([[0.18, 0.82],
[1., 0.],
[0.12, 0.88],
[1., 0.]])
In [152]:
plt.figure(figsize=(6, 3))
plot_decision_boundaries(knn, X, show_centroids=False)
plt.scatter(X_new[:, 0], X_new[:, 1], c=”b”, marker=”+”, s=200, zorder=10)
save_fig(“cluster_classification_diagram”)
plt.show()
Saving figure cluster_classification_diagram

In [153]:
y_dist, y_pred_idx = knn.kneighbors(X_new, n_neighbors=1)
y_pred = dbscan.labels_[dbscan.core_sample_indices_][y_pred_idx]
y_pred[y_dist > 0.2] = -1
y_pred.ravel()
Out[153]:
array([-1,0,1, -1])
Other Clustering Algorithms¶
Spectral Clustering¶
In [154]:
from sklearn.cluster import SpectralClustering
In [155]:
sc1 = SpectralClustering(n_clusters=2, gamma=100, random_state=42)
sc1.fit(X)
Out[155]:
SpectralClustering(affinity=’rbf’, assign_labels=’kmeans’, coef0=1, degree=3,
eigen_solver=None, eigen_tol=0.0, gamma=100,
kernel_params=None, n_clusters=2, n_init=10, n_jobs=None,
n_neighbors=10, random_state=42)
In [156]:
sc2 = SpectralClustering(n_clusters=2, gamma=1, random_state=42)
sc2.fit(X)
Out[156]:
SpectralClustering(affinity=’rbf’, assign_labels=’kmeans’, coef0=1, degree=3,
eigen_solver=None, eigen_tol=0.0, gamma=1,
kernel_params=None, n_clusters=2, n_init=10, n_jobs=None,
n_neighbors=10, random_state=42)
In [157]:
np.percentile(sc1.affinity_matrix_, 95)
Out[157]:
0.04251990648936265
In [158]:
def plot_spectral_clustering(sc, X, size, alpha, show_xlabels=True, show_ylabels=True):
plt.scatter(X[:, 0], X[:, 1], marker=’o’, s=size, c=’gray’, cmap=”Paired”, alpha=alpha)
plt.scatter(X[:, 0], X[:, 1], marker=’o’, s=30, c=’w’)
plt.scatter(X[:, 0], X[:, 1], marker=’.’, s=10, c=sc.labels_, cmap=”Paired”)
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.title(“RBF gamma={}”.format(sc.gamma), fontsize=14)
In [159]:
plt.figure(figsize=(9, 3.2))
plt.subplot(121)
plot_spectral_clustering(sc1, X, size=500, alpha=0.1)
plt.subplot(122)
plot_spectral_clustering(sc2, X, size=4000, alpha=0.01, show_ylabels=False)
plt.show()

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