這一章開始聚類算法的總結,聚類算法是無監督學習的一種
無監督學習中,類似分類和迴歸中的目標變量事先是不存在的
所謂聚類就是在這些不知目標變量的情況下,找尋數據之間的關係,可以如何分類,分爲多少數據簇
聚類會把相似對象歸爲同一個簇中,簇內對象越相似,聚類效果越好
所謂k均值聚類,就是分爲k個簇,也就是k個分類
算法描述
優缺點
- 優點:容易實現
- 缺點:可能收斂到局部最小值,在大規模數據集上收斂較慢
- 適用數據類型:數值型數據
一般流程
- 收集數據:任意方法
- 準備數據:需要數值型數據來計算距離,或標稱型數據映射爲二值型數據再用在距離計算
- 分析數據:使用任意方法
- 訓練算法:不適用於無監督學習,即無監督學習沒有訓練過程
- 測試算法:應用聚類算法、觀察結果。可以使用量化的誤差指標如誤差平方和
- 使用算法:可以用於所希望的任何應用
算法僞代碼
k-均值聚類算法的流程是:首先,隨機確定k個初始點作爲質心。然後,將數據集中的每個點分配到一個簇中(利用歐式距離法求得最近的簇)。之後,每個簇的質心更新爲該簇中所有點的平均值。算法思路簡單易懂,僞代碼如下:
創建k個點作爲起始質心(一般是隨機選擇)
當任意一個點的簇分配結果發生改變時
對數據集中的每個數據點
對每個質心
計算質心於數據點之間的距離(這裏用歐式距離)
將數據點分配到距離其最近的簇
對i一個簇,計算簇中所有點的均值並將均值作爲質心
適用後處理提高聚類性能
- 用戶如何保證k的選擇是否合理?對於這個問題,我們採用SSE(誤差平方和)來度量聚類效果,當然,SSE值越小,數據點就越接近他們的質心,聚類效果就越好。
- 如何對算法進行改進?對於這個問題,可以對生成的簇進行後處理,一種方法是將具有最大的SSE值的簇劃分爲兩個簇,具體做法是將最大簇的點單獨過濾出來運行k-均值聚類算法。
- 如何對高維數據進行分析和處理?對於這個問題,有兩種可以量化的方法:1.合併最近的質心,或者合併兩個使得SSE增幅最小的質心(可以用暴力法計算所有兩個質心的距離然後合併距離最近的兩個點實現);2.合併兩個簇然後計算總SSE值。
二分k-均值算法
爲克服k-均值算法收斂於局部最小值的問題,有人提出一個稱爲二分k-均值的算法。該算法將所有的點歸爲一個簇,然後將這個簇分爲兩份,然後再選擇其中一個簇繼續分爲兩個簇,具體如何選擇簇需要取決於其劃分是否可以最大程度江都SSE的值。
僞代碼
將所有的點看成一個簇
當簇數目小於k時
對於每一個簇
計算總誤差
在給定的簇上進行K-均值聚類(k=2)
計算將該簇一份爲二後的總誤差
選擇使得誤差最小的那個簇進行劃分操作
一個栗子
# coding:utf-8
from numpy import *
def loadDataSet(fileName): #加載文件
dataMat = []
fr = open(fileName)
for line in fr.readlines():
curLine = line.strip().split('\t') #tab分割符
fltLine = map(float,curLine) #映射所有元素爲float浮點型
dataMat.append(fltLine)
return dataMat
def distEclud(vecA, vecB):
return sqrt(sum(power(vecA - vecB, 2))) #歐式距離
def randCent(dataSet, k):
n = shape(dataSet)[1]#set1中的n=2
centroids = mat(zeros((k,n)))#創建質心矩陣
print(centroids)
for j in range(n):#在規定範圍內創建隨機質心矩陣
minJ = min(dataSet[:,j])
rangeJ = float(max(dataSet[:,j]) - minJ)
centroids[:,j] = mat(minJ + rangeJ * random.rand(k,1))
return centroids
dataMat=mat(loadDataSet('testSet.txt'))#列表轉爲矩陣型
print(randCent(dataMat,2))
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
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
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()