机器学习中有两类的大问题,一个是分类,一个是聚类。 分类是根据一些给定的已知类别标号的样本,训练某种学习机器,使它能够对未知类别的样本进行分类。这属于supervised learning(监督学习)。 而聚类指事先并不知道任何样本的类别标号,希望通过某种算法来把一组未知类别的样本划分成若干类别,这在机器学习中被称作 unsupervised learning (无监督学习)。 k均值(k-means)算法就是一种比较简单的聚类算法。
K-means算法是聚类分析中使用最广泛的算法之一。它把n个对象根据他们的属性分为k个聚类以便使得所获得的聚类满足:同一聚类中的对象相似度较高;而不同聚类中的对象相似度较小。 比如下图中的n个点,就可以分为3个聚类,用不同的颜色表示。
image1.jpg
k-means算法的基础是最小误差平方和准则。其代价函数是:
formula1.png
式中,μc(i)表示第i个聚类的均值。我们希望代价函数最小,直观的来说,各类内的样本越相似,其与该类均值间的误差平方越小,对所有类所得到的误差平方求和,即可验证分为k类时,各聚类是否是最优的。
上式的代价函数无法用解析的方法最小化,只能有迭代的方法。k-means算法是将样本聚类成 k个簇(cluster),其中k是用户给定的,其求解过程非常直观简单,具体算法描述如下:
(1)随机选取 k个聚类质心点 (2)重复下面过程直到收敛
{ 对于每一个样例 i,计算其应该属于的类:
formula2.png
对于每一个类 j,重新计算该类的质心:
formula3.png
}
下图从(a)到(f)演示了对n个样本点进行K-means聚类的过程和效果,这里k取2。
image2.jpg
创建k个点作为初始的质心点(随机选择)
当任意一个点的簇分配结果发生改变时
对数据集中的每一个数据点
对每一个质心
计算质心与数据点的距离
将数据点分配到距离最近的簇
对每一个簇,计算簇中所有点的均值,并将均值作为质心
编写此程序使用的是python 3,并且需要安装Numpy和matplotlib库。 安装方法可参考 https://www.jianshu.com/p/717521015940
为了方便理解,咱们第一次只准备了四组数据,放在testSet.txt里
1.658985 4.285136
-3.453687 3.424321
4.838138 -1.151539
-5.379713 -3.362104
在程序里,可以逐步打印出执行结果
from numpy import *
import matplotlib.pyplot as plt
# 计算欧氏距离的平方
def euclDistance(vector1, vector2):
return sum(power(vector2 - vector1, 2))
# 用随机样本初始化中心centroids
def initCentroids(dataSet, k):
numSamples, dim = dataSet.shape
centroids = zeros((k + 1, dim))
print("\n")
s = set()
for i in range(1, k + 1):
while True:
index = int(random.uniform(0, numSamples))
#这里为了方便查看结果,下面四行代码强制将两次随机数分别置为1和0
#若能理解此程序,则下面四行代码要删掉,才是真正的随机数
if 1 == i:
index = 1
elif 2 == i:
index = 0
#去重操作
if index not in s:
s.add(index)
break
print ("random index: %d" % index)
centroids[i, :] = dataSet[index, :]
print("\ncentroids:")
print(centroids)
return centroids
# 与中心的距离平方和,即最小误差平方和
def getTotalDistance(clusterAssment):
len = clusterAssment.shape[0]
Sum = 0.0
for i in range(len):
Sum = Sum + clusterAssment[i, 1]
return Sum
# k-means主算法
def kmeans(dataSet, k):
numSamples = dataSet.shape[0]
# 第一列存这个样本点属于哪个簇
# 第二列存这个样本点和样本中心的误差
clusterAssment = mat(zeros((numSamples, 2)))
for i in range(numSamples):
clusterAssment[i, 0] = -1
print("\nInitial clusterAssment:")
print(clusterAssment)
clusterChanged = True
# step 1: 初始化中心centroids
centroids = initCentroids(dataSet, k)
# 如果收敛完毕,则clusterChanged为False
while clusterChanged:
clusterChanged = False
# 对于每个样本点
print("\n")
for i in range(numSamples):
minDist = 100000.0
minIndex = 0
# 对于每个样本中心
# step 2: 找到最近的样本中心
for j in range(1, k + 1):
distance = euclDistance(centroids[j, :], dataSet[i, :])
print("i = %d, j = %d, distance = %s" % (i, j, distance))
if distance < minDist:
minDist = distance
minIndex = j
print("minIndex = %d, minDist = %f" % (minIndex, minDist))
# step 3: 更新样本点与中心点的分配关系
if clusterAssment[i, 0] != minIndex:
clusterChanged = True
clusterAssment[i, :] = minIndex, minDist
else:
clusterAssment[i, 1] = minDist
print ("clusterAssment:\n %s \n" % clusterAssment)
# step 4: 更新样本中心
for j in range(1, k + 1):
# 改变中心心位置
pointsInCluster = dataSet[nonzero(clusterAssment[:, 0].A == j)[0]]
print("\nPointsInCluster:\n%s\n" % pointsInCluster)
centroids[j, :] = mean(pointsInCluster, axis=0)
print("\ncentroids:\n%s\n" % centroids)
print ('Congratulations, cluster complete!')
return centroids, clusterAssment
# 以2D形式可视化数据
def showCluster(dataSet, k, centroids, clusterAssment):
numSamples, dim = dataSet.shape
if dim != 2:
print ("Sorry! I can not draw because the dimension of your data is not 2!")
return 1
mark = ['or', 'ob', 'og', 'ok', '^r', '+r', 'sr', 'dr', '<r', 'pr']
if k > len(mark):
print ("Sorry! Your k is too large!")
return 1
# 绘制所有非中心样本点
for i in range(numSamples):
markIndex = int(clusterAssment[i, 0])
plt.plot(dataSet[i, 0], dataSet[i, 1], mark[markIndex - 1])
mark = ['Dr', 'Db', 'Dg', 'Dk', '^b', '+b', 'sb', 'db', '<b', 'pb']
# 绘制中心点
for i in range(1, k + 1):
plt.plot(centroids[i, 0], centroids[i, 1], mark[i - 1], markersize=12)
plt.show()
# step 1: 载入数据
print ("step 1: load data...")
dataSet = []
fileIn = open('./testSet.txt')
for line in fileIn.readlines():
lineArr = line.strip().split('\t')
dataSet.append([float(lineArr[0]), float(lineArr[1])])
# step 2: 开始聚合...
print ("step 2: clustering...")
dataSet = mat(dataSet)
print ("dataSet:")
print (dataSet)
k = 2
centroids, clusterAssment = kmeans(dataSet, k)
#
print ("Final centroids:")
print (centroids)
print ("Final clusterAssment:")
print (clusterAssment)
print ("Total distance:")
print (getTotalDistance(clusterAssment))
# step 3: 显示结果
print ("step 3: show the result...")
showCluster(dataSet, k, centroids, clusterAssment)
运行结果:
result1.png
result2.png
为了观察到更好的效果,这次准备了多一点(80行)数据,放在testSet2.txt里
1.658985 4.285136
-3.453687 3.424321
4.838138 -1.151539
-5.379713 -3.362104
0.972564 2.924086
-3.567919 1.531611
0.450614 -3.302219
-3.487105 -1.724432
2.668759 1.594842
-3.156485 3.191137
3.165506 -3.999838
-2.786837 -3.099354
4.208187 2.984927
-2.123337 2.943366
0.704199 -0.479481
-0.392370 -3.963704
2.831667 1.574018
-0.790153 3.343144
2.943496 -3.357075
-3.195883 -2.283926
2.336445 2.875106
-1.786345 2.554248
2.190101 -1.906020
-3.403367 -2.778288
1.778124 3.880832
-1.688346 2.230267
2.592976 -2.054368
-4.007257 -3.207066
2.257734 3.387564
-2.679011 0.785119
0.939512 -4.023563
-3.674424 -2.261084
2.046259 2.735279
-3.189470 1.780269
4.372646 -0.822248
-2.579316 -3.497576
1.889034 5.190400
-0.798747 2.185588
2.836520 -2.658556
-3.837877 -3.253815
2.096701 3.886007
-2.709034 2.923887
3.367037 -3.184789
-2.121479 -4.232586
2.329546 3.179764
-3.284816 3.273099
3.091414 -3.815232
-3.762093 -2.432191
3.542056 2.778832
-1.736822 4.241041
2.127073 -2.983680
-4.323818 -3.938116
3.792121 5.135768
-4.786473 3.358547
2.624081 -3.260715
-4.009299 -2.978115
2.493525 1.963710
-2.513661 2.642162
1.864375 -3.176309
-3.171184 -3.572452
2.894220 2.489128
-2.562539 2.884438
3.491078 -3.947487
-2.565729 -2.012114
3.332948 3.983102
-1.616805 3.573188
2.280615 -2.559444
-2.651229 -3.103198
2.321395 3.154987
-1.685703 2.939697
3.031012 -3.620252
-4.599622 -2.185829
4.196223 1.126677
-2.133863 3.093686
4.668892 -2.562705
-2.793241 -2.149706
2.884105 3.043438
-2.967647 2.848696
4.479332 -1.764772
-4.905566 -2.911070
代码与上面的几乎一样,只是做了少量修改
from numpy import *
import matplotlib.pyplot as plt
# 计算欧氏距离的平方
def euclDistance(vector1, vector2):
return sum(power(vector2 - vector1, 2))
# 用随机样本初始化中心centroids
def initCentroids(dataSet, k):
numSamples, dim = dataSet.shape
centroids = zeros((k + 1, dim))
print("\n")
s = set()
for i in range(1, k + 1):
while True:
index = int(random.uniform(0, numSamples))
#去重操作
if index not in s:
s.add(index)
break
centroids[i, :] = dataSet[index, :]
return centroids
# 与中心的距离平方和,即最小误差平方和
def getTotalDistance(clusterAssment):
len = clusterAssment.shape[0]
Sum = 0.0
for i in range(len):
Sum = Sum + clusterAssment[i, 1]
return Sum
# k-means主算法
def kmeans(dataSet, k):
numSamples = dataSet.shape[0]
# 第一列存这个样本点属于哪个簇
# 第二列存这个样本点和样本中心的误差
clusterAssment = mat(zeros((numSamples, 2)))
for i in range(numSamples):
clusterAssment[i, 0] = -1
clusterChanged = True
# step 1: 初始化中心centroids
centroids = initCentroids(dataSet, k)
# 如果收敛完毕,则clusterChanged为False
while clusterChanged:
clusterChanged = False
# 对于每个样本点
for i in range(numSamples):
minDist = 100000.0
minIndex = 0
# 对于每个样本中心
# step 2: 找到最近的样本中心
for j in range(1, k + 1):
distance = euclDistance(centroids[j, :], dataSet[i, :])
if distance < minDist:
minDist = distance
minIndex = j
# step 3: 更新样本点与中心点的分配关系
if clusterAssment[i, 0] != minIndex:
clusterChanged = True
clusterAssment[i, :] = minIndex, minDist
else:
clusterAssment[i, 1] = minDist
# step 4: 更新样本中心
for j in range(1, k + 1):
# 改变中心心位置
pointsInCluster = dataSet[nonzero(clusterAssment[:, 0].A == j)[0]]
centroids[j, :] = mean(pointsInCluster, axis=0)
print ('Congratulations, cluster complete!')
return centroids, clusterAssment
# 以2D形式可视化数据
def showCluster(dataSet, k, centroids, clusterAssment):
numSamples, dim = dataSet.shape
if dim != 2:
print ("Sorry! I can not draw because the dimension of your data is not 2!")
return 1
mark = ['or', 'ob', 'og', 'ok', '^r', '+r', 'sr', 'dr', '<r', 'pr']
if k > len(mark):
print ("Sorry! Your k is too large!")
return 1
# 绘制所有非中心样本点
for i in range(numSamples):
markIndex = int(clusterAssment[i, 0])
plt.plot(dataSet[i, 0], dataSet[i, 1], mark[markIndex - 1])
mark = ['Dr', 'Db', 'Dg', 'Dk', '^b', '+b', 'sb', 'db', '<b', 'pb']
# 绘制中心点
for i in range(1, k + 1):
plt.plot(centroids[i, 0], centroids[i, 1], mark[i - 1], markersize=12)
plt.show()
# step 1: 载入数据
print ("step 1: load data...")
dataSet = []
fileIn = open('./testSet2.txt')
for line in fileIn.readlines():
lineArr = line.strip().split('\t')
dataSet.append([float(lineArr[0]), float(lineArr[1])])
# step 2: 开始聚合...
print ("step 2: clustering...")
dataSet = mat(dataSet)
print ("dataSet:")
print (dataSet)
k = 4
centroids, clusterAssment = kmeans(dataSet, k)
#
print ("Final centroids:")
print (centroids)
print ("Final clusterAssment:")
print (clusterAssment)
print ("Total distance:")
print (getTotalDistance(clusterAssment))
# step 3: 显示结果
print ("step 3: show the result...")
showCluster(dataSet, k, centroids, clusterAssment)
运行结果:
result3.png
result4.png
https://github.com/zhenghaishu/MachineLearning/tree/master/KMeans
(1)http://blog.csdn.net/zouxy09/article/details/17589329
(2)http://blog.csdn.net/eventqueue/article/details/73133617