K-均值聚类算法

聚类是一种无监督的学习算法，它将相似的数据归纳到同一簇中。K-均值是因为它可以按照k个不同的簇来分类，并且不同的簇中心采用簇中所含的均值计算而成。

K-均值算法

算法思想

K-均值是把数据集按照k个簇分类，其中k是用户给定的，其中每个簇是通过质心来计算簇的中心点。

主要步骤：

  • 随机确定k个初始点作为质心
  • 对数据集中的每个数据点找到距离最近的簇
  • 对于每一个簇，计算簇中所有点的均值并将均值作为质心
  • 重复步骤2，直到任意一个点的簇分配结果不变

    具体实现

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from numpy import *
import matplotlib
import matplotlib.pyplot as plt

def loadDataSet(fileName):      #general function to parse tab -delimited floats
    dataMat = []                #assume last column is target value
    fr = open(fileName)
    for line in fr.readlines():
        curLine = line.strip().split('\t')
        fltLine = map(float,curLine) #map all elements to float()
        dataMat.append(fltLine)
    return dataMat

def distEclud(vecA, vecB):
    return sqrt(sum(power(vecA - vecB, 2))) #la.norm(vecA-vecB)

def randCent(dataSet, k):
    n = shape(dataSet)[1]
    centroids = mat(zeros((k,n)))#create centroid mat
    for j in range(n):#create random cluster centers, within bounds of each dimension
        minJ = min(dataSet[:,j]) 
        rangeJ = float(max(dataSet[:,j]) - minJ)
        centroids[:,j] = mat(minJ + rangeJ * random.rand(k,1))
    return centroids
    
def kMeans(dataSet, k, distMeas=distEclud, createCent=randCent):
    m = shape(dataSet)[0]
    clusterAssment = mat(zeros((m,2)))#create mat to assign data points 
                                      #to a centroid, also holds SE of each point
    centroids = createCent(dataSet, k)
    clusterChanged = True
    while clusterChanged:
        clusterChanged = False
        for i in range(m):#for each data point assign it to the closest centroid
            minDist = inf; minIndex = -1
            for j in range(k):
                distJI = distMeas(centroids[j,:],dataSet[i,:])
                if distJI < minDist:
                    minDist = distJI; minIndex = j
            if clusterAssment[i,0] != minIndex: clusterChanged = True
            clusterAssment[i,:] = minIndex,minDist**2
        for cent in range(k):#recalculate centroids
            ptsInClust = dataSet[nonzero(clusterAssment[:,0].A==cent)[0]]#get all the point in this cluster
            centroids[cent,:] = mean(ptsInClust, axis=0) #assign centroid to mean 
            print ptsInClust
            print mean(ptsInClust, axis=0) 
            return
    return centroids, clusterAssment

def clusterClubs(numClust=5):
    datList = []
    for line in open('places.txt').readlines():
        lineArr = line.split('\t')
        datList.append([float(lineArr[4]), float(lineArr[3])])
    datMat = mat(datList)
    myCentroids, clustAssing = biKmeans(datMat, numClust, distMeas=distSLC)
    fig = plt.figure()
    rect=[0.1,0.1,0.8,0.8]
    scatterMarkers=['s', 'o', '^', '8', 'p', \
                    'd', 'v', 'h', '>', '<']
    axprops = dict(xticks=[], yticks=[])
    ax0=fig.add_axes(rect, label='ax0', **axprops)
    imgP = plt.imread('Portland.png')
    ax0.imshow(imgP)
    ax1=fig.add_axes(rect, label='ax1', frameon=False)
    for i in range(numClust):
        ptsInCurrCluster = datMat[nonzero(clustAssing[:,0].A==i)[0],:]
        markerStyle = scatterMarkers[i % len(scatterMarkers)]
        ax1.scatter(ptsInCurrCluster[:,0].flatten().A[0], ptsInCurrCluster[:,1].flatten().A[0], marker=markerStyle, s=90)
    ax1.scatter(myCentroids[:,0].flatten().A[0], myCentroids[:,1].flatten().A[0], marker='+', s=300)
    plt.show()

结果

K均值

算法收敛

设目标函数为

$$J(c, \mu) = \sum _{i=1}^m (x_i - \mu_{c_{(i)}})^2$$

Kmeans算法是将J调整到最小，每次调整质心，J值也会减小，同时c和$\mu$也会收敛。由于该函数是一个非凸函数，所以不能保证得到全局最优，智能确保局部最优解。

二分K均值算法

为了克服K均值算法收敛于局部最小值的问题，提出了二分K均值算法。

算法思想

该算法首先将所有点作为一个簇，然后将该簇一分为2，之后选择其中一个簇继续进行划分，划分规则是按照最大化SSE（目标函数）的值。

主要步骤：

  • 将所有点看成一个簇
  • 计算每一个簇的总误差
  • 在给定的簇上进行K均值聚类，计算将簇一分为二的总误差
  • 选择使得误差最小的那个簇进行再次划分
  • 重复步骤2，直到簇的个数满足要求

具体实现

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def biKMeans(dataSet, k, distMeans=distEclud):
    m, n = shape(dataSet)
    clusterAssment = mat(zeros((m, 2))) # init all data for index 0
    centroid = mean(dataSet, axis=0).tolist()
    centList = [centroid]
    for i in range(m):
        clusterAssment[i, 1] = distMeans(mat(centroid), dataSet[i, :]) ** 2
    while len(centList) < k:
        lowestSSE = inf
        for i in range(len(centList)):
            cluster = dataSet[nonzero(clusterAssment[:, 0].A == i)[0], :] # get the clust data of i
            centroidMat, splitCluster = kMeans(cluster, 2, distMeans)
            sseSplit = sum(splitCluster[:, 1]) #all sse
            sseNotSplit = sum(clusterAssment[nonzero(clusterAssment[:, 0].A != i)[0], 1]) # error sse
            #print sseSplit, sseNotSplit
            if sseSplit + sseNotSplit < lowestSSE:
                bestCentToSplit = i
                bestNewCent = centroidMat
                bestClust = splitCluster.copy()
                lowerSEE = sseSplit + sseNotSplit
        print bestClust
        bestClust[nonzero(bestClust[:, 0].A == 1)[0], 0] = len(centList)
        bestClust[nonzero(bestClust[:, 0].A == 0)[0], 0] = bestCentToSplit
        print bestClust
        print 'the bestCentToSplit is: ',bestCentToSplit
        print 'the len of bestClustAss is: ', len(bestClust)
        centList[bestCentToSplit] = bestNewCent[0, :].tolist()[0]
        centList.append(bestNewCent[1, :].tolist()[0])
        print clusterAssment
        clusterAssment[nonzero(clusterAssment[:, 0].A == bestCentToSplit)[0], :] = bestClust
        print clusterAssment
    return mat(centList), clusterAssment

结果

二分K均值