聚类是一种无监督的学习，将相似的对象放到同一簇中，有点像是全自动分类，簇内的对象越相似，簇间的对象差别越大，则聚类效果越好。

1、k均值聚类算法

k均值聚类将数据分为k个簇，每个簇通过其质心，即簇中所有点的中心来描述。首先随机确定k个初始点作为质心，然后将数据集分配到距离最近的簇中。然后将每个簇的质心更新为所有数据集的平均值。然后再进行第二次划分数据集，直到聚类结果不再变化为止。

伪代码为

随机创建k个簇质心

当任意一个点的簇分配发生改变时：

对数据集中的每个数据点：

对每个质心：

计算数据集到质心的距离

将数据集分配到最近距离质心对应的簇

对每一个簇，计算簇中所有点的均值并将均值作为质心

python实现

import numpy as np

import matplotlib.pyplot as plt

def loadDataSet(fileName):

dataMat = []

with open(fileName) as f:

for line in f.readlines():

line = line.strip().split('\t')

dataMat.append(line)

dataMat = np.array(dataMat).astype(np.float32)

return dataMat

def distEclud(vecA,vecB):

return np.sqrt(np.sum(np.power((vecA-vecB),2)))

def randCent(dataSet,k):

m = np.shape(dataSet)[1]

center = np.mat(np.ones((k,m)))

for i in range(m):

centmin = min(dataSet[:,i])

centmax = max(dataSet[:,i])

center[:,i] = centmin + (centmax - centmin) * np.random.rand(k,1)

return center

def kMeans(dataSet,k,distMeans = distEclud,createCent = randCent):

m = np.shape(dataSet)[0]

clusterAssment = np.mat(np.zeros((m,2)))

centroids = createCent(dataSet,k)

clusterChanged = True

while clusterChanged:

clusterChanged = False

for i in range(m):

minDist = np.inf

minIndex = -1

for j in range(k):

distJI = distMeans(dataSet[i,:],centroids[j,:])

if distJI < minDist:

minDist = distJI

minIndex = j

if clusterAssment[i,0] != minIndex:

clusterChanged = True

clusterAssment[i,:] = minIndex,minDist**2

for cent in range(k):

ptsInClust = dataSet[np.nonzero(clusterAssment[:,0].A == cent)[0]]

centroids[cent,:] = np.mean(ptsInClust,axis = 0)

return centroids,clusterAssment

data = loadDataSet('testSet.txt')

muCentroids, clusterAssing = kMeans(data,4)

fig = plt.figure(0)

ax = fig.add_subplot(111)

ax.scatter(data[:,0],data[:,1],c = clusterAssing[:,0].A)

plt.show()

print(clusterAssing)

2、二分k均值算法

K均值算法可能会收敛到局部最小值，而非全局最小。一种用于度量聚类效果的指标为误差平方和(SSE)。因为取了平方，更加重视原理中心的点。为了克服k均值算法可能会收敛到局部最小值的问题，有人提出来二分k均值算法。

首先将所有点作为一个簇，然后将该簇一分为二，然后选择所有簇中对其划分能够最大程度减低SSE的值的簇，直到满足指定簇数为止。

伪代码

将所有点看成一个簇

计算SSE

while 当簇数目小于k时：

for 每一个簇：

计算总误差

在给定的簇上进行k均值聚类(k=2)

计算将该簇一分为二的总误差

选择使得误差最小的那个簇进行划分操作

python实现

import numpy as np

import matplotlib.pyplot as plt

def loadDataSet(fileName):

dataMat = []

with open(fileName) as f:

for line in f.readlines():

line = line.strip().split('\t')

dataMat.append(line)

dataMat = np.array(dataMat).astype(np.float32)

return dataMat

def distEclud(vecA,vecB):

return np.sqrt(np.sum(np.power((vecA-vecB),2)))

def randCent(dataSet,k):

m = np.shape(dataSet)[1]

center = np.mat(np.ones((k,m)))

for i in range(m):

centmin = min(dataSet[:,i])

centmax = max(dataSet[:,i])

center[:,i] = centmin + (centmax - centmin) * np.random.rand(k,1)

return center

def kMeans(dataSet,k,distMeans = distEclud,createCent = randCent):

m = np.shape(dataSet)[0]

clusterAssment = np.mat(np.zeros((m,2)))

centroids = createCent(dataSet,k)

clusterChanged = True

while clusterChanged:

clusterChanged = False

for i in range(m):

minDist = np.inf

minIndex = -1

for j in range(k):

distJI = distMeans(dataSet[i,:],centroids[j,:])

if distJI < minDist:

minDist = distJI

minIndex = j

if clusterAssment[i,0] != minIndex:

clusterChanged = True

clusterAssment[i,:] = minIndex,minDist**2

for cent in range(k):

ptsInClust = dataSet[np.nonzero(clusterAssment[:,0].A == cent)[0]]

centroids[cent,:] = np.mean(ptsInClust,axis = 0)

return centroids,clusterAssment

def biKmeans(dataSet,k,distMeans = distEclud):

m = np.shape(dataSet)[0]

clusterAssment = np.mat(np.zeros((m,2)))

centroid0 = np.mean(dataSet,axis=0).tolist()

centList = [centroid0]

for j in range(m):

clusterAssment[j,1] = distMeans(dataSet[j,:],np.mat(centroid0))**2

while (len(centList)

lowestSSE = np.inf

for i in range(len(centList)):

ptsInCurrCluster = dataSet[np.nonzero(clusterAssment[:,0].A == i)[0],:]

centroidMat,splitClustAss = kMeans(ptsInCurrCluster,2,distMeans)

sseSplit = np.sum(splitClustAss[:,1])

sseNotSplit = np.sum(clusterAssment[np.nonzero(clusterAssment[:,0].A != i)[0],1])

if (sseSplit + sseNotSplit) < lowestSSE:

bestCentToSplit = i

bestNewCents = centroidMat.copy()

bestClustAss = splitClustAss.copy()

lowestSSE = sseSplit + sseNotSplit

print('the best cent to split is ',bestCentToSplit)

# print('the len of the bestClust')

bestClustAss[np.nonzero(bestClustAss[:,0].A == 1)[0],0] = len(centList)

bestClustAss[np.nonzero(bestClustAss[:,0].A == 0)[0],0] = bestCentToSplit

clusterAssment[np.nonzero(clusterAssment[:,0].A == bestCentToSplit)[0],:] = bestClustAss.copy()

centList[bestCentToSplit] = bestNewCents[0,:].tolist()[0]

centList.append(bestNewCents[1,:].tolist()[0])

return np.mat(centList),clusterAssment

data = loadDataSet('testSet2.txt')

muCentroids, clusterAssing = biKmeans(data,3)

fig = plt.figure(0)

ax = fig.add_subplot(111)

ax.scatter(data[:,0],data[:,1],c = clusterAssing[:,0].A,cmap=plt.cm.Paired)

ax.scatter(muCentroids[:,0],muCentroids[:,1])

plt.show()

print(clusterAssing)

print(muCentroids)

代码及数据集下载：K-means

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