K-均值聚類算法

K-均值聚類算法

K-均值算法

算法思想

K-均值是把數據集按照k個簇分類，其中k是用戶給定的，其中每個簇是通過質心來計算簇的中心點。

主要步驟：

• 隨機確定k個初始點作為質心
• 對數據集中的每個數據點找到距離最近的簇
• 對于每一個簇，計算簇中所有點的均值并將均值作為質心

• 重復步驟2，直到任意一個點的簇分配結果不變

  具體實現

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()

結果

算法收斂

設目標函數為

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

Kmeans算法是將J調整到最小，每次調整質心，J值也會減小，同時c和$\mu$也會收斂。由于該函數是一個非凸函數，所以不能保證得到全局最優，智能確保局部最優解。

二分K均值算法

為了克服K均值算法收斂于局部最小值的問題，提出了二分K均值算法。

算法思想

該算法首先將所有點作為一個簇，然后將該簇一分為2，之后選擇其中一個簇繼續進行劃分，劃分規則是按照最大化SSE（目標函數）的值。

主要步驟：

• 將所有點看成一個簇
• 計算每一個簇的總誤差
• 在給定的簇上進行K均值聚類，計算將簇一分為二的總誤差
• 選擇使得誤差最小的那個簇進行再次劃分
• 重復步驟2，直到簇的個數滿足要求

具體實現

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

結果