机器学习 | 聚类Clustering 算法

物以类聚人以群分。

什么是聚类呢？

1、核心思想和原理

聚类的目的

同簇高相似度

不同簇高相异度

2、K-means和分层聚类

2.1、基于划分的聚类方式

将对象划分为互斥的簇

每个对象仅属于一个簇

簇间相似性低，簇内相似性高

K-均值分类

根据样本点与簇质心距离判定

以样本间距离衡量簇内相似度

回顾一下

K均值聚类算法步骤：

选择k个初始质心，初始质心的选择是随机的，每一个质心是一个类
计算样本到各个质心欧式距离，归入最近的簇
计算新簇的质心，重复2 3，直到质心不再发生变化或者达到最大迭代次数

2.2、层次聚类

按照层次把数据划分到不同层的簇，形成树状结构，可以揭示数据间的分层结构

在树形结构上不同层次划分可以得到不同粒度的聚类

过程分为自底向上的聚合聚类和自顶向下的分裂聚类

自底向上的聚合聚类

将每个样本看做一个簇，初始状态下簇的数目 = 样本的数目

簇间距离最小的相似簇合并

下图纵轴不是合并的次序，而是合并的距离

簇间距离（簇间相似度）的度量

第一种即使已经离得很近，可能也老死不能合并。

第二种可能出现链条式的效果。

第三种相对合适。

自顶向下的分裂聚类

所有样本看成一个簇

逐渐分裂成更小的簇

目前大多数聚类算法使用的都是自底向上的聚合聚类方法

3、聚类算法代码实现

import numpy as np
import matplotlib.pyplot as plt

from sklearn.datasets import make_blobs
X, y = make_blobs(n_samples=250, centers=5, n_features=2, random_state=0)
plt.scatter(X[:,0], X[:,1], c=y)
plt.show()

plt.scatter(X[:,0], X[:,1])
plt.show()

KMeans 聚类法

from sklearn.cluster import KMeans
kmeans = KMeans(n_clusters=5, random_state=0).fit(X)

此处设定簇的个数为5

kmeans.labels_

注意，聚类不是分类，0-4只是相当于五个小组，治愈每个组是什么类型并不知道。

array([4, 4, 0, 1, 2, 0, 1, 0, 3, 2, 4, 0, 3, 2, 1, 2, 2, 4, 2, 0, 0, 0,4, 1, 1, 1, 0, 3, 4, 1, 0, 0, 2, 3, 4, 2, 2, 4, 2, 2, 4, 3, 1, 0,0, 3, 3, 2, 0, 1, 0, 1, 1, 1, 3, 2, 3, 4, 0, 0, 0, 3, 2, 3, 3, 0,4, 3, 4, 0, 0, 2, 4, 2, 1, 0, 2, 1, 1, 4, 1, 4, 0, 3, 2, 0, 3, 4,2, 3, 0, 3, 1, 0, 4, 1, 3, 2, 0, 2, 4, 3, 2, 0, 3, 3, 0, 4, 3, 0,0, 3, 3, 1, 3, 1, 0, 2, 3, 4, 4, 0, 2, 4, 3, 3, 4, 2, 2, 3, 4, 4,0, 2, 2, 4, 0, 1, 3, 3, 2, 0, 2, 1, 2, 3, 2, 0, 4, 0, 1, 0, 4, 3,1, 3, 3, 1, 3, 2, 2, 1, 1, 0, 1, 4, 0, 1, 2, 3, 3, 4, 2, 2, 0, 4,4, 1, 4, 2, 1, 3, 1, 1, 0, 4, 3, 1, 2, 1, 1, 2, 3, 4, 4, 2, 1, 2,1, 3, 4, 2, 1, 1, 1, 4, 3, 2, 0, 4, 3, 3, 2, 0, 3, 4, 4, 0, 3, 4,3, 2, 0, 1, 3, 3, 0, 0, 2, 2, 0, 2, 2, 1, 0, 1, 1, 4, 4, 1, 2, 1,4, 1, 4, 3, 4, 3, 1, 1], dtype=int32)

plt.scatter(X[:,0], X[:,1], c=kmeans.labels_)

<matplotlib.collections.PathCollection at 0x7f3285fb57c0>

center = kmeans.cluster_centers_
center

array([[ 0.93226669,  4.273606  ],[ 9.27996402, -2.3764533 ],[ 2.05849588,  0.9767519 ],[-1.39550161,  7.57857088],[-1.85199006,  2.98013351]])

plt.scatter(X[:,0], X[:,1], c=kmeans.labels_)
center = kmeans.cluster_centers_
plt.scatter(center[:,0],center[:,1], marker='x', c = 'red')
plt.show()

for n_clusters in [2, 3, 4, 5, 6, 7]:clusterer = KMeans(n_clusters=n_clusters, random_state=0).fit(X)z = clusterer.labels_center = clusterer.cluster_centers_plt.scatter(X[:,0], X[:,1], c=z)plt.scatter(center[:,0], center[:,1],marker = 'x', c='red')plt.title("k: {0}".format(n_clusters))plt.show()

层次聚类法没有聚类中心

from sklearn.cluster import AgglomerativeClustering
agg = AgglomerativeClustering(linkage='ward', n_clusters=5).fit(X)

plt.scatter(X[:,0], X[:,1], c=agg.labels_)
plt.show()

当我们对簇的个数没有预期时，不要一个一个试，可以传入距离的阈值。

agg = AgglomerativeClustering(distance_threshold=10, n_clusters=None).fit(X)
plt.scatter(X[:,0], X[:,1], c=agg.labels_)
plt.show()

from scipy.cluster.hierarchy import linkage, dendrogram
def show_dendrogram(model):counts = np.zeros(model.children_.shape[0])n_samples = len(model.labels_)for i, merge in enumerate(model.children_):current_count = 0for child_idx in merge:if child_idx < n_samples:current_count += 1  # leaf nodeelse:current_count += counts[child_idx - n_samples]counts[i] = current_countlinkage_matrix = np.column_stack([model.children_, model.distances_, counts]).astype(float)dendrogram(linkage_matrix)

show_dendrogram(agg)

import time
import warningsfrom sklearn import cluster, datasets
from sklearn.preprocessing import StandardScaler
from itertools import cycle, islice
n_samples = 1500
noisy_circles = datasets.make_circles(n_samples=n_samples, factor=0.5, noise=0.05)
noisy_moons = datasets.make_moons(n_samples=n_samples, noise=0.05)
blobs = datasets.make_blobs(n_samples=n_samples, random_state=8)
no_structure = np.random.rand(n_samples, 2), None# Anisotropicly distributed data
random_state = 170
X, y = datasets.make_blobs(n_samples=n_samples, random_state=random_state)
transformation = [[0.6, -0.6], [-0.4, 0.8]]
X_aniso = np.dot(X, transformation)
aniso = (X_aniso, y)# blobs with varied variances
varied = datasets.make_blobs(n_samples=n_samples, cluster_std=[1.0, 2.5, 0.5], random_state=random_state
)# Set up cluster parameters
plt.figure(figsize=(9 * 1.3 + 2, 14.5))
plt.subplots_adjust(left=0.02, right=0.98, bottom=0.001, top=0.96, wspace=0.05, hspace=0.01
)plot_num = 1default_base = {"n_neighbors": 10, "n_clusters": 3}datasets = [(noisy_circles, {"n_clusters": 2}),(noisy_moons, {"n_clusters": 2}),(varied, {"n_neighbors": 2}),(aniso, {"n_neighbors": 2}),(blobs, {}),(no_structure, {}),
]for i_dataset, (dataset, algo_params) in enumerate(datasets):# update parameters with dataset-specific valuesparams = default_base.copy()params.update(algo_params)X, y = dataset# normalize dataset for easier parameter selectionX = StandardScaler().fit_transform(X)# ============# Create cluster objects# ============ward = cluster.AgglomerativeClustering(n_clusters=params["n_clusters"], linkage="ward")complete = cluster.AgglomerativeClustering(n_clusters=params["n_clusters"], linkage="complete")average = cluster.AgglomerativeClustering(n_clusters=params["n_clusters"], linkage="average")single = cluster.AgglomerativeClustering(n_clusters=params["n_clusters"], linkage="single")clustering_algorithms = (("Single Linkage", single),("Average Linkage", average),("Complete Linkage", complete),("Ward Linkage", ward),)for name, algorithm in clustering_algorithms:t0 = time.time()# catch warnings related to kneighbors_graphwith warnings.catch_warnings():warnings.filterwarnings("ignore",message="the number of connected components of the "+ "connectivity matrix is [0-9]{1,2}"+ " > 1. Completing it to avoid stopping the tree early.",category=UserWarning,)algorithm.fit(X)t1 = time.time()if hasattr(algorithm, "labels_"):y_pred = algorithm.labels_.astype(int)else:y_pred = algorithm.predict(X)plt.subplot(len(datasets), len(clustering_algorithms), plot_num)if i_dataset == 0:plt.title(name, size=18)colors = np.array(list(islice(cycle(["#377eb8","#ff7f00","#4daf4a","#f781bf","#a65628","#984ea3","#999999","#e41a1c","#dede00",]),int(max(y_pred) + 1),)))plt.scatter(X[:, 0], X[:, 1], s=10, color=colors[y_pred])plt.xlim(-2.5, 2.5)plt.ylim(-2.5, 2.5)plt.xticks(())plt.yticks(())plt.text(0.99,0.01,("%.2fs" % (t1 - t0)).lstrip("0"),transform=plt.gca().transAxes,size=15,horizontalalignment="right",)plot_num += 1plt.show()

4、聚类评估代码实现

聚类效果评估方法

已知标签评价

调整兰德指数Adjusted Rand lndex

调整互信息分Adjusted mutual info score

基于预测簇向量与真实簇向量的互信息分数

V-Measure

同质性和完整性的调和平均值

————取值在-1到1，越接近1越好

未知标签评价

轮廓系数

通过计算样本与所在簇中其他样本的相似度

CHI (Calinski-Harabaz lndex/Variance Ratio Criterion)

群间离散度和群内离散度的比例

代码实现

import numpy as np
import matplotlib.pyplot as plt

from sklearn.datasets import make_blobs
X, y = make_blobs(n_samples=250, n_features=2, centers=5, random_state=0)

from sklearn.cluster import KMeans
kmeans = KMeans(n_clusters=5, random_state=0).fit(X)
z = kmeans.labels_

center = kmeans.cluster_centers_
plt.scatter(X[:,0], X[:,1], c=z)
plt.scatter(center[:,0], center[:,1], marker = 'x', c='red')
plt.show()

已知标签

from sklearn.metrics import adjusted_rand_score
adjusted_rand_score(y,z)

0.8676297613641788

from sklearn.metrics import adjusted_mutual_info_score
adjusted_mutual_info_score(y,z)

0.8579576361507845

from sklearn.metrics import v_measure_score
v_measure_score(y,z)

0.8608558483955058

ari_curve = []
ami_curve = []
vm_curve = []
clus = [2, 3, 4, 5, 6, 7]
for n_clusters in clus:clusterer = KMeans(n_clusters=n_clusters, random_state=0).fit(X)z = clusterer.labels_ari_curve.append(adjusted_rand_score(y,z))ami_curve.append(adjusted_mutual_info_score(y,z))vm_curve.append(v_measure_score(y,z))plt.plot(clus, ari_curve, label='ari')
plt.plot(clus, ami_curve, label='ami')
plt.plot(clus, vm_curve, label='vm')
plt.legend()
plt.show()

未知标签

from sklearn.metrics import silhouette_score
kmeans = KMeans(n_clusters=5, random_state=0).fit(X)
cluster_labels = kmeans.labels_
si = silhouette_score(X, cluster_labels)
si

0.5526930597314647

from sklearn.metrics import silhouette_samples
import matplotlib.cm as cmdef show_silhouette_plot(model, X):for n_clusters in [2, 3, 4, 5, 6, 7]:fig, (pic1, pic2) = plt.subplots(1, 2)fig.set_size_inches(15, 5)model.n_clusters = n_clustersclusterer = model.fit(X)cluster_labels = clusterer.labels_centers = clusterer.cluster_centers_silhouette_avg = silhouette_score(X, cluster_labels)sample_silhouette_values = silhouette_samples(X, cluster_labels)y_lower = 1for i in range(n_clusters):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_icolor = cm.nipy_spectral(float(i) / n_clusters)pic1.fill_betweenx(np.arange(y_lower, y_upper), ith_cluster_silhouette_values, facecolor = color)pic1.text(-0.02, y_lower + 0.5 * size_cluster_i, str(i))y_lower = y_upper + 1pic1.axvline(x = silhouette_avg, color = 'red', linestyle = "--")pic1.set_title("score: {0}".format(silhouette_avg))colors = cm.nipy_spectral(cluster_labels.astype(float) / n_clusters)pic2.scatter(X[:,0], X[:,1], marker = 'o', c = colors)pic2.scatter(centers[:, 0], centers[:, 1], marker = 'x', c = 'red', alpha = 1, s = 200)pic2.set_title("k: {0}".format(n_clusters))

show_silhouette_plot(kmeans, X)

from sklearn.metrics import calinski_harabasz_score
calinski_harabasz_score(X, y)

850.6346471314978

chi_curve = []clus = [2, 3, 4, 5, 6, 7]
for n_clusters in clus:clusterer = KMeans(n_clusters=n_clusters, random_state=0).fit(X)z = clusterer.labels_chi_curve.append(calinski_harabasz_score(X,z))plt.plot(clus, chi_curve, label='chi')
plt.legend()
plt.show()