机器学习实战ByMatlab（4）：二分K-means算法-IT家园

摘要

其实K-means的缺点的根本原因就是：对K个质心的初始选取比较敏感。质心选取得不好很有可能就会陷入局部最小值。基于以上情况，有人提出了二分K-means算法来解决这种情况，也就是弱化初始质心的选取对

原文出处： Liu_LongPo的专栏（@Liu_LongPo）

前面我们在是实现K-means算法的时候，提到了它本身存在的缺陷：

1.可能收敛到局部最小值
2.在大规模数据集上收敛较慢

对于上一篇博文最后说的，当陷入局部最小值的时候，处理方法就是多运行几次K-means算法，然后选择畸变函数J较小的作为最佳聚类结果。这样的说法显然不能让我们接受，我们追求的应该是一次就能给出接近最优的聚类结果。

其实K-means的缺点的根本原因就是：对K个质心的初始选取比较敏感。质心选取得不好很有可能就会陷入局部最小值。

基于以上情况，有人提出了二分K-means算法来解决这种情况，也就是弱化初始质心的选取对最终聚类效果的影响。

二分K-means算法

在介绍二分K-means算法之前我们先说明一个定义：SSE（Sum of Squared Error）,也就是误差平方和，它是用来度量聚类效果的一个指标。其实SSE也就是我们在K-means算法中所说的畸变函数：

SSE计算的就是一个cluster中的每个点到质心的平方差，它可以度量聚类的好坏。显然SSE越小，说明聚类效果越好。

二分K-means算法的主要思想：
首先将所有点作为一个簇，然后将该簇一分为二。之后选择能最大程度降低聚类代价函数（也就是误差平方和）的簇划分为两个簇。以此进行下去，直到簇的数目等于用户给定的数目k为止。

二分k均值算法的伪代码如下：

将所有数据点看成一个簇

当簇数目小于k时

对每一个簇

计算总误差

在给定的簇上面进行k-均值聚类（k=2）

计算将该簇一分为二后的总误差

选择使得误差最小的那个簇进行划分操作

Matlab 实现

function bikMeans
%%
clc
clear
close all
%%
biK = 4;
biDataSet = load('testSet.txt');
[row,col] = size(biDataSet);
% 存储质心矩阵
biCentSet = zeros(biK,col);
% 初始化设定cluster数量为1
numCluster = 1;
%第一列存储每个点被分配的质心，第二列存储点到质心的距离
biClusterAssume = zeros(row,2);
%初始化质心
biCentSet(1,:) = mean(biDataSet)
for i = 1:row
 biClusterAssume(i,1) = numCluster;
 biClusterAssume(i,2) = distEclud(biDataSet(i,:),biCentSet(1,:));
end
while numCluster < biK
 minSSE = 10000;
 %寻找对哪个cluster进行划分最好，也就是寻找SSE最小的那个cluster
 for j = 1:numCluster
 curCluster = biDataSet(find(biClusterAssume(:,1) == j),:);
 [spiltCentSet,spiltClusterAssume] = kMeans(curCluster,2);
 spiltSSE = sum(spiltClusterAssume(:,2));
 noSpiltSSE = sum(biClusterAssume(find(biClusterAssume(:,1)~=j),2));
 curSSE = spiltSSE + noSpiltSSE;
 fprintf('第%d个cluster被划分后的误差为：%f \n' , [j, curSSE])
 if (curSSE < minSSE)
 minSSE = curSSE;
 bestClusterToSpilt = j;
 bestClusterAssume = spiltClusterAssume;
 bestCentSet = spiltCentSet;
 end
 end
 bestClusterToSpilt
 bestCentSet
 %更新cluster的数目
 numCluster = numCluster + 1;
 bestClusterAssume(find(bestClusterAssume(:,1) == 1),1) = bestClusterToSpilt;
 bestClusterAssume(find(bestClusterAssume(:,1) == 2),1) = numCluster;
 % 更新和添加质心坐标
 biCentSet(bestClusterToSpilt,:) = bestCentSet(1,:);
 biCentSet(numCluster,:) = bestCentSet(2,:);
 biCentSet
 % 更新被划分的cluster的每个点的质心分配以及误差
 biClusterAssume(find(biClusterAssume(:,1) == bestClusterToSpilt),:) = bestClusterAssume;
end
figure
%scatter(dataSet(:,1),dataSet(:,2),5)
for i = 1:biK
 pointCluster = find(biClusterAssume(:,1) == i);
 scatter(biDataSet(pointCluster,1),biDataSet(pointCluster,2),5)
 hold on
end
%hold on
scatter(biCentSet(:,1),biCentSet(:,2),300,'+')
hold off
end
% 计算欧式距离
function dist = distEclud(vecA,vecB)
 dist = sum(power((vecA-vecB),2));
end
% K-means算法
function [centSet,clusterAssment] = kMeans(dataSet,K)
[row,col] = size(dataSet);
% 存储质心矩阵
centSet = zeros(K,col);
% 随机初始化质心
for i= 1:col
 minV = min(dataSet(:,i));
 rangV = max(dataSet(:,i)) - minV;
 centSet(:,i) = repmat(minV,[K,1]) + rangV*rand(K,1);
end
% 用于存储每个点被分配的cluster以及到质心的距离
clusterAssment = zeros(row,2);
clusterChange = true;
while clusterChange
 clusterChange = false;
 % 计算每个点应该被分配的cluster
 for i = 1:row
 % 这部分可能可以优化
 minDist = 10000;
 minIndex = 0;
 for j = 1:K
 distCal = distEclud(dataSet(i,:) , centSet(j,:));
 if (distCal < minDist)
 minDist = distCal;
 minIndex = j;
 end
 end
 if minIndex ~= clusterAssment(i,1) 
 clusterChange = true;
 end
 clusterAssment(i,1) = minIndex;
 clusterAssment(i,2) = minDist;
 end
% 更新每个cluster 的质心
 for j = 1:K
 simpleCluster = find(clusterAssment(:,1) == j);
 centSet(j,:) = mean(dataSet(simpleCluster',:));
 end
end
end

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function bikMeans

clc

clear

close all

biK = 4;

biDataSet = load('testSet.txt');

[row,col] = size(biDataSet);

% 存储质心矩阵

biCentSet = zeros(biK,col);

% 初始化设定cluster数量为1

numCluster = 1;

%第一列存储每个点被分配的质心，第二列存储点到质心的距离

biClusterAssume = zeros(row,2);

%初始化质心

biCentSet(1,:) = mean(biDataSet)

for i = 1:row

biClusterAssume(i,1) = numCluster;

biClusterAssume(i,2) = distEclud(biDataSet(i,:),biCentSet(1,:));

end

while numCluster < biK

minSSE = 10000;

%寻找对哪个cluster进行划分最好，也就是寻找SSE最小的那个cluster

for j = 1:numCluster

curCluster = biDataSet(find(biClusterAssume(:,1) == j),:);

[spiltCentSet,spiltClusterAssume] = kMeans(curCluster,2);

spiltSSE = sum(spiltClusterAssume(:,2));

noSpiltSSE = sum(biClusterAssume(find(biClusterAssume(:,1)~=j),2));

curSSE = spiltSSE + noSpiltSSE;

fprintf('第%d个cluster被划分后的误差为：%f \n' , [j, curSSE])

if (curSSE < minSSE)

minSSE = curSSE;

bestClusterToSpilt = j;

bestClusterAssume = spiltClusterAssume;

bestCentSet = spiltCentSet;

end

bestClusterToSpilt

bestCentSet

%更新cluster的数目

numCluster = numCluster + 1;

bestClusterAssume(find(bestClusterAssume(:,1) == 1),1) = bestClusterToSpilt;

bestClusterAssume(find(bestClusterAssume(:,1) == 2),1) = numCluster;

% 更新和添加质心坐标

biCentSet(bestClusterToSpilt,:) = bestCentSet(1,:);

biCentSet(numCluster,:) = bestCentSet(2,:);

biCentSet

% 更新被划分的cluster的每个点的质心分配以及误差

biClusterAssume(find(biClusterAssume(:,1) == bestClusterToSpilt),:) = bestClusterAssume;

end

figure

%scatter(dataSet(:,1),dataSet(:,2),5)

for i = 1:biK

pointCluster = find(biClusterAssume(:,1) == i);

scatter(biDataSet(pointCluster,1),biDataSet(pointCluster,2),5)

hold on

end

%hold on

scatter(biCentSet(:,1),biCentSet(:,2),300,'+')

hold off

end

% 计算欧式距离

function dist = distEclud(vecA,vecB)

dist = sum(power((vecA-vecB),2));

end

% K-means算法

function [centSet,clusterAssment] = kMeans(dataSet,K)

[row,col] = size(dataSet);

% 存储质心矩阵

centSet = zeros(K,col);

% 随机初始化质心

for i= 1:col

minV = min(dataSet(:,i));

rangV = max(dataSet(:,i)) - minV;

centSet(:,i) = repmat(minV,[K,1]) + rangV*rand(K,1);

end

% 用于存储每个点被分配的cluster以及到质心的距离

clusterAssment = zeros(row,2);

clusterChange = true;

while clusterChange

clusterChange = false;

% 计算每个点应该被分配的cluster

for i = 1:row

% 这部分可能可以优化

minDist = 10000;

minIndex = 0;

for j = 1:K

distCal = distEclud(dataSet(i,:) , centSet(j,:));

if (distCal < minDist)

minDist = distCal;

minIndex = j;

end

if minIndex ~= clusterAssment(i,1)

clusterChange = true;

end

clusterAssment(i,1) = minIndex;

clusterAssment(i,2) = minDist;

end

% 更新每个cluster 的质心

for j = 1:K

simpleCluster = find(clusterAssment(:,1) == j);

centSet(j,:) = mean(dataSet(simpleCluster',:));

end

算法迭代过程如下

biCentSet =

-0.1036 0.0543
0 0
0 0
0 0

第1个cluster被划分后的误差为：792.916857

bestClusterToSpilt =

bestCentSet =

-0.2897 -2.8394
0.0825 2.9480

biCentSet =

-0.2897 -2.8394
0.0825 2.9480
0 0
0 0

第1个cluster被划分后的误差为：409.871545
第2个cluster被划分后的误差为：532.999616

bestClusterToSpilt =

bestCentSet =

-3.3824 -2.9473
2.8029 -2.7315

biCentSet =

-3.3824 -2.9473
0.0825 2.9480
2.8029 -2.7315
0 0

第1个cluster被划分后的误差为：395.669052
第2个cluster被划分后的误差为：149.954305
第3个cluster被划分后的误差为：393.431098

bestClusterToSpilt =

bestCentSet =

2.6265 3.1087
-2.4615 2.7874

biCentSet =

-3.3824 -2.9473
2.6265 3.1087
2.8029 -2.7315
-2.4615 2.7874

最终效果图

运用二分K-means算法进行聚类的时候，不同的初始质心聚类结果还是会稍微有点不同，因为实际上这也只是弱化随机质心对聚类结果的影响而已，并不能消除其影响，不过最终还是能收敛到全局最小。

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