机器学习算法与Python实践之（六）二分k均值聚类

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机器学习算法与Python实践之（六）二分k均值聚类

zouxy09@qq.com

http://blog.csdn.net/zouxy09

机器学习算法与Python实践这个系列主要是参考《机器学习实战》这本书。因为自己想学习Python，然后也想对一些机器学习算法加深下了解，所以就想通过Python来实现几个比较常用的机器学习算法。恰好遇见这本同样定位的书籍，所以就参考这本书的过程来学习了。

在上一个博文中，我们聊到了k-means算法。但k-means算法有个比较大的缺点就是对初始k个质心点的选取比较敏感。有人提出了一个二分k均值（bisecting k-means）算法，它的出现就是为了一定情况下解决这个问题的。也就是说它对初始的k个质心的选择不太敏感。那下面我们就来了解和实现下这个算法。

一、二分k均值（bisecting k-means）算法

二分k均值（bisecting k-means）算法的主要思想是：首先将所有点作为一个簇，然后将该簇一分为二。之后选择能最大程度降低聚类代价函数（也就是误差平方和）的簇划分为两个簇。以此进行下去，直到簇的数目等于用户给定的数目k为止。

以上隐含着一个原则是：因为聚类的误差平方和能够衡量聚类性能，该值越小表示数据点月接近于它们的质心，聚类效果就越好。所以我们就需要对误差平方和最大的簇进行再一次的划分，因为误差平方和越大，表示该簇聚类越不好，越有可能是多个簇被当成一个簇了，所以我们首先需要对这个簇进行划分。

二分k均值算法的伪代码如下：

***************************************************************

将所有数据点看成一个簇

当簇数目小于k时

对每一个簇

计算总误差

在给定的簇上面进行k-均值聚类（k=2）

计算将该簇一分为二后的总误差

选择使得误差最小的那个簇进行划分操作

***************************************************************

二、Python实现

我使用的Python是2.7.5版本的。附加的库有Numpy和Matplotlib。具体的安装和配置见前面的博文。在代码中已经有了比较详细的注释了。不知道有没有错误的地方，如果有，还望大家指正（每次的运行结果都有可能不同）。里面我写了个可视化结果的函数，但只能在二维的数据上面使用。直接贴代码：

biKmeans.py

[python] view plaincopy





#################################################

# kmeans: k-means cluster

# Author : zouxy

# Date : 2013-12-25

# HomePage : http://blog.csdn.net/zouxy09
# Email : zouxy09@qq.com

#################################################

from numpy import *

import time

import matplotlib.pyplot as plt

# calculate Euclidean distance

def euclDistance(vector1, vector2):

return sqrt(sum(power(vector2 - vector1, 2)))

# init centroids with random samples

def initCentroids(dataSet, k):

numSamples, dim = dataSet.shape

centroids = zeros((k, dim))

for i in range(k):

index = int(random.uniform(0, numSamples))

centroids[i, :] = dataSet[index, :]

return centroids

# k-means cluster

def kmeans(dataSet, k):

numSamples = dataSet.shape[0]

# first column stores which cluster this sample belongs to,

# second column stores the error between this sample and its centroid

clusterAssment = mat(zeros((numSamples, 2)))

clusterChanged = True

## step 1: init centroids

centroids = initCentroids(dataSet, k)

while clusterChanged:

clusterChanged = False

## for each sample

for i in xrange(numSamples):

minDist = 100000.0

minIndex = 0

## for each centroid

## step 2: find the centroid who is closest

for j in range(k):

distance = euclDistance(centroids[j, :], dataSet[i, :])

if distance < minDist:

minDist = distance

minIndex = j

## step 3: update its cluster

if clusterAssment[i, 0] != minIndex:

clusterChanged = True

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

## step 4: update centroids

for j in range(k):

pointsInCluster = dataSet[nonzero(clusterAssment[:, 0].A == j)[0]]

centroids[j, :] = mean(pointsInCluster, axis = 0)

print 'Congratulations, cluster using k-means complete!'

return centroids, clusterAssment

# bisecting k-means cluster

def biKmeans(dataSet, k):

numSamples = dataSet.shape[0]

# first column stores which cluster this sample belongs to,

# second column stores the error between this sample and its centroid

clusterAssment = mat(zeros((numSamples, 2)))

# step 1: the init cluster is the whole data set

centroid = mean(dataSet, axis = 0).tolist()[0]

centList = [centroid]

for i in xrange(numSamples):

clusterAssment[i, 1] = euclDistance(mat(centroid), dataSet[i, :])**2

while len(centList) < k:

# min sum of square error

minSSE = 100000.0

numCurrCluster = len(centList)

# for each cluster

for i in range(numCurrCluster):

# step 2: get samples in cluster i

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

# step 3: cluster it to 2 sub-clusters using k-means

centroids, splitClusterAssment = kmeans(pointsInCurrCluster, 2)

# step 4: calculate the sum of square error after split this cluster

splitSSE = sum(splitClusterAssment[:, 1])

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

currSplitSSE = splitSSE + notSplitSSE

# step 5: find the best split cluster which has the min sum of square error

if currSplitSSE < minSSE:

minSSE = currSplitSSE

bestCentroidToSplit = i

bestNewCentroids = centroids.copy()

bestClusterAssment = splitClusterAssment.copy()

# step 6: modify the cluster index for adding new cluster

bestClusterAssment[nonzero(bestClusterAssment[:, 0].A == 1)[0], 0] = numCurrCluster

bestClusterAssment[nonzero(bestClusterAssment[:, 0].A == 0)[0], 0] = bestCentroidToSplit

# step 7: update and append the centroids of the new 2 sub-cluster

centList[bestCentroidToSplit] = bestNewCentroids[0, :]

centList.append(bestNewCentroids[1, :])

# step 8: update the index and error of the samples whose cluster have been changed

clusterAssment[nonzero(clusterAssment[:, 0].A == bestCentroidToSplit), :] = bestClusterAssment

print 'Congratulations, cluster using bi-kmeans complete!'

return mat(centList), clusterAssment

# show your cluster only available with 2-D data

def showCluster(dataSet, k, centroids, clusterAssment):

numSamples, dim = dataSet.shape

if dim != 2:

print "Sorry! I can not draw because the dimension of your data is not 2!"

return 1

mark = ['or', 'ob', 'og', 'ok', '^r', '+r', 'sr', 'dr', '<r', 'pr']

if k > len(mark):

print "Sorry! Your k is too large! please contact Zouxy"

return 1

# draw all samples

for i in xrange(numSamples):

markIndex = int(clusterAssment[i, 0])

plt.plot(dataSet[i, 0], dataSet[i, 1], mark[markIndex])

mark = ['Dr', 'Db', 'Dg', 'Dk', '^b', '+b', 'sb', 'db', '<b', 'pb']

# draw the centroids

for i in range(k):

plt.plot(centroids[i, 0], centroids[i, 1], mark[i], markersize = 12)

plt.show()

三、测试结果

测试数据是二维的，共80个样本。有4个类。具体见上一个博文。

测试代码：

test_biKmeans.py

[python] view plaincopy





#################################################

# kmeans: k-means cluster

# Author : zouxy

# Date : 2013-12-25

# HomePage : http://blog.csdn.net/zouxy09
# Email : zouxy09@qq.com

#################################################

from numpy import *

import time

import matplotlib.pyplot as plt

## step 1: load data

print "step 1: load data..."

dataSet = []

fileIn = open('E:/Python/Machine Learning in Action/testSet.txt')

for line in fileIn.readlines():

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

dataSet.append([float(lineArr[0]), float(lineArr[1])])

## step 2: clustering...

print "step 2: clustering..."

dataSet = mat(dataSet)

k = 4

centroids, clusterAssment = biKmeans(dataSet, k)

## step 3: show the result

print "step 3: show the result..."

showCluster(dataSet, k, centroids, clusterAssment)

这里贴出两次的运行结果：



不同的类用不同的颜色来表示，其中的大菱形是对应类的均值质心点。

事实上，这个算法在初始质心选择不同时运行效果也会不同。我没有看初始的论文，不确定它究竟是不是一定会收敛到全局最小值。《机器学习实战》这本书说是可以的，但因为每次运行的结果不同，所以我有点怀疑，自己去找资料也没找到相关的说明。对这个算法有了解的还望您不吝指点下，谢谢。