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
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(' ')
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(' ')
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
