如何决定K-Means聚类个数——silhouette analysis

简介

在K-Means聚类时,我们常常会纠结,K该取多大呢?今天无意当中查看Sklearn时,发现了silhouette analysis,翻译过来,就是轮廓分析,具体来说,就是通过结果簇之间的分隔距离来辅助决定K的取值。偷个懒,直接参照手册,学习下。

数据集分布

下图是500个样本含有2个feature的数据分布情况:
scatter

轮廓系数

接下来看下,n_clusters 分别为 2,3,4,5,6时,平均的轮廓分值(结果簇之间平均的分隔距离)如下,这个轮廓分值是介于[-1,1]之间的度量指标。每次聚类后,每个样本都会得到一个轮廓系数,当它为1时,说明这个点与周围簇距离较远,结果非常好,当它为0,说明这个点可能处在两个簇的边界上,当值为负时,暗含该点可能被误分了。
从平均轮廓分值结果来看,K取3,5,6是不明智的。对于2,4的选择还是有点纠结的,因为它们值相差并不大。

For n_clusters = 2 The average silhouette_score is : 0.704978749608
For n_clusters = 3 The average silhouette_score is : 0.588200401213
For n_clusters = 4 The average silhouette_score is : 0.650518663273
For n_clusters = 5 The average silhouette_score is : 0.563764690262
For n_clusters = 6 The average silhouette_score is : 0.450466629437

轮廓宽度

下面是簇为2,3,4,5,6相对应的轮廓图,左图可以看出,当n_clusters = 2时,第0簇的宽度远宽于第1簇,但n_clusters = 4时,所聚的簇宽度相差不大,这样分析下来,自然而然会选择K=4,作为最终聚类个数。
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本例参考官方手册,详情戳这

附脚本

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from sklearn.datasets import make_blobs
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_samples, silhouette_score
import matplotlib.pyplot as plt
import matplotlib.cm as cm
import numpy as np
# Generating the sample data from make_blobs
# This particular setting has one distinct cluster and 3 clusters placed close
# together.
X, y = make_blobs(n_samples=500,
n_features=2,
centers=4,
cluster_std=1,
center_box=(-10.0, 10.0),
shuffle=True,
random_state=1) # For reproducibility
range_n_clusters = [2, 3, 4, 5, 6]
for n_clusters in range_n_clusters:
# Create a subplot with 1 row and 2 columns
fig, (ax1, ax2) = plt.subplots(1, 2)
fig.set_size_inches(18, 7)
# The 1st subplot is the silhouette plot
# The silhouette coefficient can range from -1, 1 but in this example all
# lie within [-0.1, 1]
ax1.set_xlim([-0.1, 1])
# The (n_clusters+1)*10 is for inserting blank space between silhouette
# plots of individual clusters, to demarcate them clearly.
ax1.set_ylim([0, len(X) + (n_clusters + 1) * 10])
# Initialize the clusterer with n_clusters value and a random generator
# seed of 10 for reproducibility.
clusterer = KMeans(n_clusters=n_clusters, random_state=10)
cluster_labels = clusterer.fit_predict(X)
# The silhouette_score gives the average value for all the samples.
# This gives a perspective into the density and separation of the formed
# clusters
silhouette_avg = silhouette_score(X, cluster_labels)
print("For n_clusters =", n_clusters,
"The average silhouette_score is :", silhouette_avg)
# Compute the silhouette scores for each sample
sample_silhouette_values = silhouette_samples(X, cluster_labels)
y_lower = 10
for i in range(n_clusters):
# Aggregate the silhouette scores for samples belonging to
# cluster i, and sort them
ith_cluster_silhouette_values = \
sample_silhouette_values[cluster_labels == i]
ith_cluster_silhouette_values.sort()
size_cluster_i = ith_cluster_silhouette_values.shape[0]
y_upper = y_lower + size_cluster_i
color = cm.spectral(float(i) / n_clusters)
ax1.fill_betweenx(np.arange(y_lower, y_upper),
0, ith_cluster_silhouette_values,
facecolor=color, edgecolor=color, alpha=0.7)
# Label the silhouette plots with their cluster numbers at the middle
ax1.text(-0.05, y_lower + 0.5 * size_cluster_i, str(i))
# Compute the new y_lower for next plot
y_lower = y_upper + 10 # 10 for the 0 samples
ax1.set_title("The silhouette plot for the various clusters.")
ax1.set_xlabel("The silhouette coefficient values")
ax1.set_ylabel("Cluster label")
# The vertical line for average silhouette score of all the values
ax1.axvline(x=silhouette_avg, color="red", linestyle="--")
ax1.set_yticks([]) # Clear the yaxis labels / ticks
ax1.set_xticks([-0.1, 0, 0.2, 0.4, 0.6, 0.8, 1])
# 2nd Plot showing the actual clusters formed
colors = cm.spectral(cluster_labels.astype(float) / n_clusters)
ax2.scatter(X[:, 0], X[:, 1], marker='.', s=30, lw=0, alpha=0.7,
c=colors)
# Labeling the clusters
centers = clusterer.cluster_centers_
# Draw white circles at cluster centers
ax2.scatter(centers[:, 0], centers[:, 1],
marker='o', c="white", alpha=1, s=200)
for i, c in enumerate(centers):
ax2.scatter(c[0], c[1], marker='$%d$' % i, alpha=1, s=50)
ax2.set_title("The visualization of the clustered data.")
ax2.set_xlabel("Feature space for the 1st feature")
ax2.set_ylabel("Feature space for the 2nd feature")
plt.suptitle(("Silhouette analysis for KMeans clustering on sample data "
"with n_clusters = %d" % n_clusters),
fontsize=14, fontweight='bold')
plt.show()
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