机器学习算法与Python实践之（六）二分k均值聚类

http://blog.csdn.net/zouxy09/article/details/17590137

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

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

#################################################
# 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('	')
    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。