K-均值聚类算法

K-均值聚类算法

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

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

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

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