From d3619b47811aa2a1a435328c44fdf6493c44d88b Mon Sep 17 00:00:00 2001 From: wangyangting Date: Thu, 7 Sep 2017 15:02:07 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=20KMeans=20=E8=81=9A?= =?UTF-8?q?=E7=B1=BB=E7=AE=97=E6=B3=95=E6=96=87=E6=A1=A3?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- docs/10.k-means聚类.md | 214 ++++++++++++++++-- .../apachecn-bikmeans-run-result-1.jpg | Bin 0 -> 50917 bytes .../apachecn-k-means-run-result-1.jpg | Bin 0 -> 52550 bytes images/10.KMeans/apachecn-k-means-term-1.jpg | Bin 0 -> 75941 bytes input/10.KMeans/testSet2.txt | 60 +++++ src/python/10.kmeans/kMeans.py | 72 +++++- 6 files changed, 316 insertions(+), 30 deletions(-) create mode 100644 images/10.KMeans/apachecn-bikmeans-run-result-1.jpg create mode 100644 images/10.KMeans/apachecn-k-means-run-result-1.jpg create mode 100644 images/10.KMeans/apachecn-k-means-term-1.jpg create mode 100644 input/10.KMeans/testSet2.txt diff --git a/docs/10.k-means聚类.md b/docs/10.k-means聚类.md index 33d0a0f1..2cd50d0b 100644 --- a/docs/10.k-means聚类.md +++ b/docs/10.k-means聚类.md @@ -1,21 +1,195 @@ -# 10) k-means聚类 -* 聚类介绍 - * 聚类是一种无监督的学习,它将相似的对象归到一个簇中。 - * 聚类分析试图将相似对象归入同一簇,将不相似对象归到不同簇。相似这一概念取决于所选择的相似度计算方法。 -* K-均值聚类算法 - * 优点 : 容易实现 - * 缺点 : 可能收敛到局部最小值,在大规模数据集上收敛较慢。 - * 使用数据类型 : 数值型数据。 -* K-均值算法工作流程 - * 首先,随机确定 k 个初始点作为质心。 - * 然后,将数据集中的每个点分配到一个簇中。(为每个点找到距其最近的质心,并将其分配给该质心所对应的簇) - * 最后,每个簇的质心更新为该簇所有点的平均值。 -* K-均值算法伪代码如下 - * 创建 k 个点作为起始质心(通常是随机选择) - * 当任意一个点的簇分配结果发生改变时 - * 对数据集中的每个数据点 - * 对每个质心 - * 计算质心与数据点之间的距离 - * 将数据点分配到距其最近的簇 - * 对每一个簇,计算簇中所有点的均值并将均值作为质心 +# 第 10 章 K-Means(K-均值)聚类算法 + +## K-Means 算法 +聚类是一种无监督的学习, 它将相似的对象归到一个簇中, 将不相似对象归到不同簇中. +相似这一概念取决于所选择的相似度计算方法. +K-Means 是发现给定数据集的 K 个簇的聚类算法, 之所以称之为 `K-均值` 是因为它可以发现 K 个不同的簇, 且每个簇的中心采用簇中所含值的均值计算而成. +簇个数 K 是用户指定的, 每一个簇通过其质心(centroid), 即簇中所有点的中心来描述. +聚类与分类算法的最大区别在于, 分类的目标类别已知, 而聚类的目标类别是未知的. + +``` +优点: 容易实现 +缺点:可能收敛到局部最小值, 在大规模数据集上收敛较慢 +使用数据类型 : 数值型数据 +``` + +### K-Means 场景 +主要用来聚类, 但是类别是未知的. +例如: 对地图上的点进行聚类. + +### K-Means 术语 + +* 簇: 簇中的对象是相似的. +* 质心: 簇中所有点的中心(计算所有点的均值而来). +* SSE: Sum of Sqared Error(平方误差和), SSE 值越小,表示越接近它们的质心. 由于对误差取了平方,因此更加注重那么远离中心的点. + +有关 `簇` 和 `质心` 术语更形象的介绍, 请参考下图: + +![K-Means 术语图](../images/10.KMeans/apachecn-k-means-term-1.jpg) + +### K-Means 工作流程 +1. 首先, 随机确定 K 个初始点作为质心. +2. 然后将数据集中的每个点分配到一个簇中, 具体来讲, 就是为每个点找到距其最近的质心, 并将其分配该质心所对应的簇. 这一步完成之后, 每个簇的质心更新为该簇说有点的平均值. + +上述过程的 `伪代码` 如下: + +* 创建 k 个点作为起始质心(通常是随机选择) +* 当任意一个点的簇分配结果发生改变时 + * 对数据集中的每个数据点 + * 对每个质心 + * 计算质心与数据点之间的距离 + * 将数据点分配到距其最近的簇 + * 对每一个簇, 计算簇中所有点的均值并将均值作为质心 + +### K-Means 开发流程 + +``` +收集数据:使用任意方法 +准备数据:需要数值型数据类计算距离, 也可以将标称型数据映射为二值型数据再用于距离计算 +分析数据:使用任意方法 +训练算法:此步骤不适用于 K-Means 算法 +测试算法:应用聚类算法、观察结果.可以使用量化的误差指标如误差平方和(后面会介绍)来评价算法的结果. +使用算法:可以用于所希望的任何应用.通常情况下, 簇质心可以代表整个簇的数据来做出决策. +``` + +### K-Means 聚类算法函数 + +#### 从文件加载数据集 + +```python +# 从文本中构建矩阵,加载文本文件,然后处理 +def loadDataSet(fileName): # 通用函数,用来解析以 tab 键分隔的 floats(浮点数),例如: 1.658985 4.285136 + dataMat = [] + fr = open(fileName) + for line in fr.readlines(): + curLine = line.strip().split('\t') + fltLine = map(float,curLine) # 映射所有的元素为 float(浮点数)类型 + dataMat.append(fltLine) + return dataMat +``` + +#### 计算两个向量的欧氏距离 + +```python +# 计算两个向量的欧式距离(可根据场景选择) +def distEclud(vecA, vecB): + return sqrt(sum(power(vecA - vecB, 2))) # la.norm(vecA-vecB) +``` + +#### 构建一个包含 K 个随机质心的集合 + +```python +# 为给定数据集构建一个包含 k 个随机质心的集合。随机质心必须要在整个数据集的边界之内,这可以通过找到数据集每一维的最小和最大值来完成。然后生成 0~1.0 之间的随机数并通过取值范围和最小值,以便确保随机点在数据的边界之内。 +def randCent(dataSet, k): + n = shape(dataSet)[1] # 列的数量 + centroids = mat(zeros((k,n))) # 创建k个质心矩阵 + for j in range(n): # 创建随机簇质心,并且在每一维的边界内 + minJ = min(dataSet[:,j]) # 最小值 + rangeJ = float(max(dataSet[:,j]) - minJ) # 范围 = 最大值 - 最小值 + centroids[:,j] = mat(minJ + rangeJ * random.rand(k,1)) # 随机生成 + return centroids +``` + +#### K-Means 聚类算法 + +```python +# k-means 聚类算法 +# 该算法会创建k个质心,然后将每个点分配到最近的质心,再重新计算质心。 +# 这个过程重复数次,知道数据点的簇分配结果不再改变位置。 +# 运行结果(多次运行结果可能会不一样,可以试试,原因为随机质心的影响,但总的结果是对的, 因为数据足够相似,也可能会陷入局部最小值) +def kMeans(dataSet, k, distMeas=distEclud, createCent=randCent): + m = shape(dataSet)[0] # 行数 + clusterAssment = mat(zeros((m, 2))) # 创建一个与 dataSet 行数一样,但是有两列的矩阵,用来保存簇分配结果 + centroids = createCent(dataSet, k) # 创建质心,随机k个质心 + clusterChanged = True + while clusterChanged: + clusterChanged = False + for i in range(m): # 循环每一个数据点并分配到最近的质心中去 + minDist = inf; minIndex = -1 + for j in range(k): + distJI = distMeas(centroids[j,:],dataSet[i,:]) # 计算数据点到质心的距离 + if distJI < minDist: # 如果距离比 minDist(最小距离)还小,更新 minDist(最小距离)和最小质心的 index(索引) + minDist = distJI; minIndex = j + if clusterAssment[i, 0] != minIndex: # 簇分配结果改变 + clusterChanged = True # 簇改变 + clusterAssment[i, :] = minIndex,minDist**2 # 更新簇分配结果为最小质心的 index(索引),minDist(最小距离)的平方 + print centroids + for cent in range(k): # 更新质心 + ptsInClust = dataSet[nonzero(clusterAssment[:, 0].A==cent)[0]] # 获取该簇中的所有点 + centroids[cent,:] = mean(ptsInClust, axis=0) # 将质心修改为簇中所有点的平均值,mean 就是求平均值的 + return centroids, clusterAssment +``` + +#### 测试函数 +1. 测试一下以上的基础函数是否可以如预期运行, 请看: +2. 测试一下 kMeans 函数是否可以如预期运行, 请看: + +参考运行结果如下: +![K-Means 运行结果1](../images/10.KMeans/apachecn-k-means-run-result-1.jpg) + +> 在 kMeans 的函数测试中,可能偶尔会陷入局部最小值(局部最优的结果,但不是全局最优的结果). + +### K-Means 聚类算法的缺陷 +在 kMeans 的函数测试中,可能偶尔会陷入局部最小值(局部最优的结果,但不是全局最优的结果). +所以为了客户 KMeans 算法收敛于局部最小值的问题,有更厉害的大佬提出了另一个称为二分K-均值(bisecting K-Means)的算法. + +### 二分 K-Means 聚类算法 +该算法首先将说有点作为一个簇,然后将该簇一分为二。 +之后选择其中一个簇继续进行划分,选择哪一个簇进行划分取决于对其划分时候可以最大程度降低 SSE(平方和误差)的值。 +上述基于 SSE 的划分过程不断重复,直到得到用户指定的簇数目为止。 + +#### 二分 K-Means 聚类算法伪代码 +* 将所有点看成一个簇 +* 当簇数目小雨 k 时 +* 对于每一个簇 + * 计算总误差 + * 在给定的簇上面进行 KMeans 聚类(k=2) + * 计算将该簇一分为二之后的总误差 +* 选择使得误差最小的那个簇进行划分操作 + +另一种做法是选择 SSE 最大的簇进行划分,直到簇数目达到用户指定的数目位置。 +接下来主要介绍该做法。 + +#### 二分 K-Means 聚类算法代码 + +```python +# 二分 KMeans 聚类算法, 基于 kMeans 基础之上的优化,以避免陷入局部最小值 +def biKMeans(dataSet, k, distMeas=distEclud): + m = shape(dataSet)[0] + clusterAssment = mat(zeros((m,2))) # 保存每个数据点的簇分配结果和平方误差 + centroid0 = mean(dataSet, axis=0).tolist()[0] # 质心初始化为所有数据点的均值 + centList =[centroid0] # 初始化只有 1 个质心的 list + for j in range(m): # 计算所有数据点到初始质心的距离平方误差 + clusterAssment[j,1] = distMeas(mat(centroid0), dataSet[j,:])**2 + while (len(centList) < k): # 当质心数量小于 k 时 + lowestSSE = inf + for i in range(len(centList)): # 对每一个质心 + ptsInCurrCluster = dataSet[nonzero(clusterAssment[:,0].A==i)[0],:] # 获取当前簇 i 下的所有数据点 + centroidMat, splitClustAss = kMeans(ptsInCurrCluster, 2, distMeas) # 将当前簇 i 进行二分 kMeans 处理 + sseSplit = sum(splitClustAss[:,1]) # 将二分 kMeans 结果中的平方和的距离进行求和 + sseNotSplit = sum(clusterAssment[nonzero(clusterAssment[:,0].A!=i)[0],1]) # 将未参与二分 kMeans 分配结果中的平方和的距离进行求和 + print "sseSplit, and notSplit: ",sseSplit,sseNotSplit + if (sseSplit + sseNotSplit) < lowestSSE: + bestCentToSplit = i + bestNewCents = centroidMat + bestClustAss = splitClustAss.copy() + lowestSSE = sseSplit + sseNotSplit + # 找出最好的簇分配结果 + bestClustAss[nonzero(bestClustAss[:,0].A == 1)[0],0] = len(centList) # 调用二分 kMeans 的结果,默认簇是 0,1. 当然也可以改成其它的数字 + bestClustAss[nonzero(bestClustAss[:,0].A == 0)[0],0] = bestCentToSplit # 更新为最佳质心 + print 'the bestCentToSplit is: ',bestCentToSplit + print 'the len of bestClustAss is: ', len(bestClustAss) + # 更新质心列表 + centList[bestCentToSplit] = bestNewCents[0,:].tolist()[0] # 更新原质心 list 中的第 i 个质心为使用二分 kMeans 后 bestNewCents 的第一个质心 + centList.append(bestNewCents[1,:].tolist()[0]) # 添加 bestNewCents 的第二个质心 + clusterAssment[nonzero(clusterAssment[:,0].A == bestCentToSplit)[0],:]= bestClustAss # 重新分配最好簇下的数据(质心)以及SSE + return mat(centList), clusterAssment +``` + +#### 测试二分 KMeans 聚类算法 +* 测试一下二分 KMeans 聚类算法,请看: + +上述函数可以运行多次,聚类会收敛到全局最小值,而原始的 kMeans() 函数偶尔会陷入局部最小值。 +运行参考结果如下: +![二分 K-Means 运行结果1](../images/10.KMeans/apachecn-bikmeans-run-result-1.jpg) \ No newline at end of file diff --git a/images/10.KMeans/apachecn-bikmeans-run-result-1.jpg b/images/10.KMeans/apachecn-bikmeans-run-result-1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..8863cc2df642f9e4733596d8c4e4f68506f7be64 GIT binary patch literal 50917 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