Clustering: K-means, agglomerative with dendrograms, and DBSCAN
There are three types of clustering algorithms:
* Prototype based clustering: k-means which clusters into spherical shapes based on a specified number of cluster centroids. Downside is that it assumes this spherical structure and you need to input the number of centroids.
* Hierarchical clustering: Agglomerative clustering does not require specifying the number of clusters up front and the result can be visuzlized with a dendrogram.
* Density based clustering: DBSCAN groups points based on local densities and is capable of handling outliers and identifying non-globular shapes.
* Density based clustering: Not included here.
k-means vs. Agglomerative vs DBSCAN: Case of half moons
import matplotlib.pyplot as plt
from sklearn.datasets import make_moons
X, y = make_moons(n_samples=200, noise=0.05, random_state=0)
plt.scatter(X[:, 0], X[:, 1])
plt.tight_layout()
#plt.savefig('images/11_14.png', dpi=300)
plt.show()
f, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(16, 3))
# K-means
from sklearn.cluster import KMeans
km = KMeans(n_clusters=2, random_state=0)
y_km = km.fit_predict(X)
ax1.scatter(X[y_km == 0, 0], X[y_km == 0, 1],
edgecolor='black',
c='lightblue', marker='o', s=40, label='cluster 1')
ax1.scatter(X[y_km == 1, 0], X[y_km == 1, 1],
edgecolor='black',
c='red', marker='s', s=40, label='cluster 2')
ax1.set_title('K-means clustering')
# Agglomerative
from sklearn.cluster import AgglomerativeClustering
ac = AgglomerativeClustering(n_clusters=2,
affinity='euclidean',
linkage='complete')
y_ac = ac.fit_predict(X)
ax2.scatter(X[y_ac == 0, 0], X[y_ac == 0, 1], c='lightblue',
edgecolor='black',
marker='o', s=40, label='Cluster 1')
ax2.scatter(X[y_ac == 1, 0], X[y_ac == 1, 1], c='red',
edgecolor='black',
marker='s', s=40, label='Cluster 2')
ax2.set_title('Agglomerative clustering')
# DBSCAN
from sklearn.cluster import DBSCAN
db = DBSCAN(eps=0.2, min_samples=5, metric='euclidean')
y_db = db.fit_predict(X)
ax3.scatter(X[y_db == 0, 0], X[y_db == 0, 1], c='lightblue',
edgecolor='black',
marker='o', s=40, label='Cluster 1')
ax3.scatter(X[y_db == 1, 0], X[y_db == 1, 1], c='red',
edgecolor='black',
marker='s', s=40, label='Cluster 2')
ax3.set_title('DBSCAN clustering')
plt.legend()
plt.tight_layout()
#plt.savefig('images/11_15.png', dpi=300)
plt.show()
K-means with elbow plot and silhouette
# Make and plot data
import matplotlib.pyplot as plt
from sklearn.datasets import make_blobs
X, y = make_blobs(n_samples=150,
n_features=2,
centers=3,
cluster_std=0.5,
shuffle=True,
random_state=0)
plt.scatter(X[:, 0], X[:, 1],
c='white', marker='o', edgecolor='black', s=50)
plt.grid()
plt.tight_layout()
#plt.savefig('images/11_01.png', dpi=300)
plt.show()
# K-means clustering
from sklearn.cluster import KMeans
km = KMeans(n_clusters=3,
init='random',
n_init=10,
max_iter=300,
tol=1e-04,
random_state=0)
y_km = km.fit_predict(X)
# Plot the clusters with their centroids
plt.scatter(X[y_km == 0, 0],
X[y_km == 0, 1],
s=50, c='lightgreen',
marker='s', edgecolor='black',
label='Cluster 1')
plt.scatter(X[y_km == 1, 0],
X[y_km == 1, 1],
s=50, c='orange',
marker='o', edgecolor='black',
label='Cluster 2')
plt.scatter(X[y_km == 2, 0],
X[y_km == 2, 1],
s=50, c='lightblue',
marker='v', edgecolor='black',
label='Cluster 3')
plt.scatter(km.cluster_centers_[:, 0],
km.cluster_centers_[:, 1],
s=250, marker='*',
c='red', edgecolor='black',
label='Centroids')
plt.legend(scatterpoints=1)
plt.grid()
plt.tight_layout()
#plt.savefig('images/11_02.png', dpi=300)
plt.show()
Distortion: Within a cluster sum of squared error (SSE)
$SSE = \sum_{i=1}^n\sum_{j-1}^k w^{(i,j)} ||x^{(i)} - \mu^{(j)}||_2^2$
where i is the sample index, and j is the cluster index. $\mu$ is the centroid for cluster j, and $w^{(i,j)}=1$ if the sample $x^i$ is in cluster j, 0 otherwise.
# Find the elbow
distortions = []
for i in range(1, 11):
km = KMeans(n_clusters=i,
init='k-means++',
n_init=10,
max_iter=300,
random_state=0)
km.fit(X)
distortions.append(km.inertia_)
plt.plot(range(1, 11), distortions, marker='o')
plt.xlabel('Number of clusters')
plt.ylabel('Distortion')
plt.tight_layout()
#plt.savefig('images/11_03.png', dpi=300)
plt.show()
Quantifying the quality of clustering via silhouette plots
import numpy as np
from matplotlib import cm
from sklearn.metrics import silhouette_samples
km = KMeans(n_clusters=3,
init='k-means++',
n_init=10,
max_iter=300,
tol=1e-04,
random_state=0)
y_km = km.fit_predict(X)
cluster_labels = np.unique(y_km)
n_clusters = cluster_labels.shape[0]
silhouette_vals = silhouette_samples(X, y_km, metric='euclidean')
y_ax_lower, y_ax_upper = 0, 0
yticks = []
for i, c in enumerate(cluster_labels):
c_silhouette_vals = silhouette_vals[y_km == c]
c_silhouette_vals.sort()
y_ax_upper += len(c_silhouette_vals)
color = cm.jet(float(i) / n_clusters)
plt.barh(range(y_ax_lower, y_ax_upper), c_silhouette_vals, height=1.0,
edgecolor='none', color=color)
yticks.append((y_ax_lower + y_ax_upper) / 2.)
y_ax_lower += len(c_silhouette_vals)
silhouette_avg = np.mean(silhouette_vals)
plt.axvline(silhouette_avg, color="red", linestyle="--")
plt.yticks(yticks, cluster_labels + 1)
plt.ylabel('Cluster')
plt.xlabel('Silhouette coefficient')
plt.tight_layout()
#plt.savefig('images/11_04.png', dpi=300)
plt.show()
Quantifying the quality of clustering via silhouette plots for bad clusters
km = KMeans(n_clusters=2,
init='k-means++',
n_init=10,
max_iter=300,
tol=1e-04,
random_state=0)
y_km = km.fit_predict(X)
plt.scatter(X[y_km == 0, 0],
X[y_km == 0, 1],
s=50,
c='lightgreen',
edgecolor='black',
marker='s',
label='Cluster 1')
plt.scatter(X[y_km == 1, 0],
X[y_km == 1, 1],
s=50,
c='orange',
edgecolor='black',
marker='o',
label='Cluster 2')
plt.scatter(km.cluster_centers_[:, 0], km.cluster_centers_[:, 1],
s=250, marker='*', c='red', label='Centroids')
plt.legend()
plt.grid()
plt.tight_layout()
#plt.savefig('images/11_05.png', dpi=300)
plt.show()
cluster_labels = np.unique(y_km)
n_clusters = cluster_labels.shape[0]
silhouette_vals = silhouette_samples(X, y_km, metric='euclidean')
y_ax_lower, y_ax_upper = 0, 0
yticks = []
for i, c in enumerate(cluster_labels):
c_silhouette_vals = silhouette_vals[y_km == c]
c_silhouette_vals.sort()
y_ax_upper += len(c_silhouette_vals)
color = cm.jet(float(i) / n_clusters)
plt.barh(range(y_ax_lower, y_ax_upper), c_silhouette_vals, height=1.0,
edgecolor='none', color=color)
yticks.append((y_ax_lower + y_ax_upper) / 2.)
y_ax_lower += len(c_silhouette_vals)
silhouette_avg = np.mean(silhouette_vals)
plt.axvline(silhouette_avg, color="red", linestyle="--")
plt.yticks(yticks, cluster_labels + 1)
plt.ylabel('Cluster')
plt.xlabel('Silhouette coefficient')
plt.tight_layout()
#plt.savefig('images/11_06.png', dpi=300)
plt.show()
Understanding hierarchical clustering on a distance matrix
# Start with random numbers
import pandas as pd
import numpy as np
np.random.seed(123)
variables = ['X', 'Y', 'Z']
labels = ['ID_0', 'ID_1', 'ID_2', 'ID_3', 'ID_4']
X = np.random.random_sample([5, 3])*10
df = pd.DataFrame(X, columns=variables, index=labels)
df
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# Calcuate distance matrix
from scipy.spatial.distance import pdist, squareform
row_dist = pd.DataFrame(squareform(pdist(df, metric='euclidean')),
columns=labels,
index=labels)
row_dist
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# Calculate Linkage matrix
from scipy.cluster.hierarchy import linkage
row_clusters = linkage(pdist(df, metric='euclidean'), method='complete')
pd.DataFrame(row_clusters,
columns=['row label 1', 'row label 2',
'distance', 'no. of items in clust.'],
index=['cluster %d' % (i + 1)
for i in range(row_clusters.shape[0])])
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from scipy.cluster.hierarchy import dendrogram
# make dendrogram black (part 1/2)
# from scipy.cluster.hierarchy import set_link_color_palette
# set_link_color_palette(['black'])
row_dendr = dendrogram(row_clusters,
labels=labels,
# make dendrogram black (part 2/2)
# color_threshold=np.inf
)
plt.tight_layout()
plt.ylabel('Euclidean distance')
#plt.savefig('images/11_11.png', dpi=300,
# bbox_inches='tight')
plt.show()
Even fancier with heatmaps in the dendrogram
# plot row dendrogram
fig = plt.figure(figsize=(8, 8), facecolor='white')
axd = fig.add_axes([0.09, 0.1, 0.2, 0.6])
# note: for matplotlib < v1.5.1, please use orientation='right'
row_dendr = dendrogram(row_clusters, orientation='left')
# reorder data with respect to clustering
df_rowclust = df.iloc[row_dendr['leaves'][::-1]]
axd.set_xticks([])
axd.set_yticks([])
# remove axes spines from dendrogram
for i in axd.spines.values():
i.set_visible(False)
# plot heatmap
axm = fig.add_axes([0.23, 0.1, 0.6, 0.6]) # x-pos, y-pos, width, height
cax = axm.matshow(df_rowclust, interpolation='nearest', cmap='hot_r')
fig.colorbar(cax)
axm.set_xticklabels([''] + list(df_rowclust.columns))
axm.set_yticklabels([''] + list(df_rowclust.index))
#plt.savefig('images/11_12.png', dpi=300)
plt.show()
/var/folders/wn/k3vlc_7s4mjcszy_1l54z5qm0000gn/T/ipykernel_13006/1264724331.py:22: UserWarning: FixedFormatter should only be used together with FixedLocator
axm.set_xticklabels([''] + list(df_rowclust.columns))
/var/folders/wn/k3vlc_7s4mjcszy_1l54z5qm0000gn/T/ipykernel_13006/1264724331.py:23: UserWarning: FixedFormatter should only be used together with FixedLocator
axm.set_yticklabels([''] + list(df_rowclust.index))
