k-均值聚類是非監督學習的一種，輸入必須指定聚簇中心個數k。k均值是基於相似度的聚類，為沒有標簽的一簇實例分為一類。
一 經典的k-均值聚類
思路：
1 隨機創建k個質心（k必須指定，二維的很容易確定，可視化數據分布，直觀確定即可）；
2 遍歷數據集的每個實例，計算其到每個質心的相似度，這里也就是歐氏距離；把每個實例都分配到距離最近的質心的那一類，用一個二維數組數據結構保存，第一列是最近質心序號，第二列是距離；
3 根據二維數組保存的數據，重新計算每個聚簇新的質心；
4 迭代2 和 3，直到收斂，即質心不再變化；
from numpy import * 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 print centroids 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 return centroids, clusterAssment
經典的k均值聚類有很大的缺點就是很容易收斂到局部最優，為了避免這種局部最優，我們引入了二分k-均值算法。
二 二分k-均值聚類算法
二分k-均值聚類算法是基於經典k-均值算法實現的；里面調用經典k-均值（k=2），把一個聚簇分成兩個，迭代到分成k個停止；
具體思路：
1 把整個數據集看成一個聚簇，計算質心；並用同樣的數據結構二維數組保存每個實例到質心的距離；
2 對每一個聚簇進行2-均值聚類划分；
3 計算划分后的誤差，選擇所有被划分的聚簇中總誤差最小的划分保存；
4 迭代2 和 3 直到聚簇數目達到k停止；
def biKmeans(dataSet, k, distMeas=distEclud): m = shape(dataSet)[0] clusterAssment = mat(zeros((m,2))) centroid0 = mean(dataSet, axis=0).tolist()[0] centList =[centroid0] #create a list with one centroid for j in range(m):#calc initial Error clusterAssment[j,1] = distMeas(mat(centroid0), dataSet[j,:])**2 while (len(centList) < k): lowestSSE = inf for i in range(len(centList)): ptsInCurrCluster = dataSet[nonzero(clusterAssment[:,0].A==i)[0],:]#get the data points currently in cluster i centroidMat, splitClustAss = kMeans(ptsInCurrCluster, 2, distMeas) sseSplit = sum(splitClustAss[:,1])#compare the SSE to the currrent minimum sseNotSplit = sum(clusterAssment[nonzero(clusterAssment[:,0].A!=i)[0],1]) print "sseSplit, and notSplit: ",sseSplit,'--',sseNotSplit if (sseSplit + sseNotSplit) < lowestSSE: bestCentToSplit = i bestNewCents = centroidMat bestClustAss = splitClustAss.copy() lowestSSE = sseSplit + sseNotSplit bestClustAss[nonzero(bestClustAss[:,0].A == 1)[0],0] = len(centList) #change 1 to 3,4, or whatever bestClustAss[nonzero(bestClustAss[:,0].A == 0)[0],0] = bestCentToSplit print 'the bestCentToSplit is: ',bestCentToSplit print 'the len of bestClustAss is: ', len(bestClustAss) centList[bestCentToSplit] = bestNewCents[0,:].tolist()[0]#replace a centroid with two best centroids centList.append(bestNewCents[1,:].tolist()[0]) clusterAssment[nonzero(clusterAssment[:,0].A == bestCentToSplit)[0],:]= bestClustAss#reassign new clusters, and SSE return mat(centList), clusterAssment
三 地理位置聚簇實例
地理位置的經緯度正好是二維的，可以可視化出來，所以很適合聚類算法確定質心個數k值；值得注意的是，球面計算距離，不能簡單的用歐式距離，而需要用球面距離公式，見函數distSLC；
代碼的含義給定n個俱樂部地址名稱，然后使用urllib包，調用yahoo地圖的API返回經緯度，調用我們上面實現的k均值聚類算法，找到聚簇的中心，最后利用matplotlib工具可視化出來；
import urllib import json def geoGrab(stAddress, city): apiStem = 'http://where.yahooapis.com/geocode?' #create a dict and constants for the goecoder params = {} params['flags'] = 'J'#JSON return type params['appid'] = 'aaa0VN6k' params['location'] = '%s %s' % (stAddress, city) url_params = urllib.urlencode(params) yahooApi = apiStem + url_params #print url_params print yahooApi c=urllib.urlopen(yahooApi) return json.loads(c.read()) from time import sleep def massPlaceFind(fileName): fw = open('places.txt', 'w') for line in open(fileName).readlines(): line = line.strip() lineArr = line.split('\t') retDict = geoGrab(lineArr[1], lineArr[2]) if retDict['ResultSet']['Error'] == 0: lat = float(retDict['ResultSet']['Results'][0]['latitude']) lng = float(retDict['ResultSet']['Results'][0]['longitude']) print "%s\t%f\t%f" % (lineArr[0], lat, lng) fw.write('%s\t%f\t%f\n' % (line, lat, lng)) else: print "error fetching" sleep(1) fw.close() def distSLC(vecA, vecB):#Spherical Law of Cosines a = sin(vecA[0,1]*pi/180) * sin(vecB[0,1]*pi/180) b = cos(vecA[0,1]*pi/180) * cos(vecB[0,1]*pi/180) * \ cos(pi * (vecB[0,0]-vecA[0,0]) /180) return arccos(a + b)*6371.0 #pi is imported with numpy import matplotlib import matplotlib.pyplot as plt 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()
四 總結
優點：易實現；
缺點：可能收斂到局部最小值，在大數據集上收斂較慢；
適用數據類型：數值型；