C均值聚类算法及其实现
C-均值聚类算法是动态聚类算法的一种
动态聚类——兼顾系统聚类和分解聚类
下文文字描述居多，公式较少放心食用。
首先给出聚类准则函数是误差平方和准则：
J c = ∑ j = 1 c ∑ k = 1 n j ∣ ∣ x k − m j ∣ ∣ 2 J_{c}=\sum_{j=1}^{c}\sum_{k=1}^{n_{j}}||x_{k}-m_{j}||^{2} Jc=∑j=1c∑k=1nj∣∣xk−mj∣∣2
其中 c c c代表类数， n j n_{j} nj代表 j j j类的样本数
（1）C－均值算法（一）
1.从混合样本中抽取c个初始聚合中心。
2.计算每个样本到每个聚合中心的距离（这里用欧式距离 D e ( x , y ) = ∑ i = 1 d ∣ x i − y i ∣ 2 D_{e}(x,y)=\sqrt{\sum_{i=1}^{d}|x_{i}-y_{i}|^{2}} De(x,y)=∑i=1d∣xi−yi∣2），距离哪个中心近就归入哪类（这很好理解）
3.计算 c c c个新的聚类中心(由于有新的样本进入聚类)，计算公式就是求均值。（这里可以每个样本分类后更新聚类中心，也可以全部样本分类后统一更新聚类中心，分别为逐个处理法与批处理法，针对（一）算法实现用批处理法）
4.判断所有聚类中心如果与上次分类后的中心不同则返回第二步，否则算法结束。（这里是因为如果经过计算分类后聚类中心有变化，说明又改进，就不结束，如果已经收敛，没有改进了，就结束。）
（2）C－均值算法（二）
均值算法（二）是对（一）的改进
1.执行C－均值算法（一）的1，2，3步。（这里实现也采用批处理）
2.计算误差平方和（也就是准则函数），计算：
ρ i i = n i n i − 1 ∣ ∣ x k ( i ) − Z i ( I ) ∣ ∣ 2 , i = 1 , 2 , . . . , c \rho_{ii}=\frac{n_{i}}{n_{i}-1}||x_{k}^{(i)}-Z_{i}(I)||^{2},i=1,2,...,c ρii=ni−1ni∣∣xk(i)−Zi(I)∣∣2,i=1,2,...,c
ρ i i \rho_{ii} ρii表示使准则函数减小的部分
ρ i j = n j n j + 1 ∣ ∣ x k ( i ) − Z j ( I ) ∣ ∣ 2 , j = 1 , 2 , . . . , c j ≠ i \rho_{ij}=\frac{n_{j}}{n_{j}+1}||x_{k}^{(i)}-Z_{j}(I)||^{2},j=1,2,...,c \qquad j\ne i ρij=nj+1nj∣∣xk(i)−Zj(I)∣∣2,j=1,2,...,cj=i
ρ i j \rho_{ij} ρij表示使准则函数增加的部分
其中 i i i代表第 i i i类， j j j代表第 j j j类， n n n代表聚类中样本数， x k ( i ) x_{k}^{(i)} xk(i)代表第 i i i类第 k k k个样本， Z Z Z代表聚类中心， I I I代表第 I I I次迭代值。
找出 ρ i j \rho_{ij} ρij中的最小值 ρ i l \rho_{il} ρil，如果此最小值小于 ρ i i \rho_{ii} ρii，说明存在能够使聚类准则函数减小的操作。即把样本 x k ( i ) x_{k}^{(i)} xk(i)移动到聚类中心 w l w_{l} wl中。
3.经过移动操作后修改聚合中心与准则函数 J c J_{c} Jc的值。
Z i ( I + 1 ) = Z i ( I ) + 1 n i − 1 [ Z i ( I ) − x k ( i ) ] Z_{i}(I+1)=Z_{i}(I)+\frac{1}{n_{i}-1}\left[Z_{i}(I)-x_{k}^{(i)}\right] Zi(I+1)=Zi(I)+ni−11[Zi(I)−xk(i)]
Z l ( I + 1 ) = Z j ( I ) − 1 n l + 1 [ Z l ( I ) − x k ( i ) ] Z_{l}(I+1)=Z_{j}(I)-\frac{1}{n_{l}+1}\left[Z_{l}(I)-x_{k}^{(i)}\right] Zl(I+1)=Zj(I)−nl+11[Zl(I)−xk(i)]
J c ( I + 1 ) = J c ( I ) − ( ρ i i − ρ i l ) J_{c}(I+1)=J_{c}(I)-\left(\rho_{i i}-\rho_{i l}\right) Jc(I+1)=Jc(I)−(ρii−ρil)
4.若 J c ( I + 1 ) < J c ( I ) J_{c}(I+1)
太懒了，搁置了好久了，好多细节都忘了，所以直接上代码：
代码是分类IRIS数据集的，语言MATLAB R2017a，分类精度在89.33%，第一种均值算法相较于第二种算法更容易陷入局部最优（精度在66.67%）。
%% C均值算法对于IRIS数据集的分类(一)
%% author:mym NJUST
clear;
clc;
[number,txt,raw]=xlsread('iris.csv');
%对数据集乱序
index=randperm(150);
rand_number=number(index,:);
kind=3;%类别数
prime_index=sort(randperm(150,3));
m=number(prime_index,2:5);%初始随机选择的三个聚类中心
kind1=0;%记录每类的数目
kind2=0;
kind3=0;
m1=zeros(3,4);
while ~isequal(m,m1)
m1=m;
for i=1:150
D1=norm(rand_number(i,2:5)-m(1,:));
D2=norm(rand_number(i,2:5)-m(2,:));
D3=norm(rand_number(i,2:5)-m(3,:));%计算样本到聚类中心的欧式距离
vec=[D1 D2 D3];
[M,I]=min(vec);
if I==1
kind1=kind1+1;
Kone(kind1)=rand_number(i,1);
elseif I==2
kind2=kind2+1;
Ktwo(kind2)=rand_number(i,1);
else
kind3=kind3+1;
Kthree(kind3)=rand_number(i,1);%将样本放入聚类
end
end
if kind1~=0
m(1,:)=sum(number(Kone,2:5),1)/kind1;
end
if kind2~=0
m(2,:)=sum(number(Ktwo,2:5),1)/kind2;
end
if kind3~=0
m(3,:)=sum(number(Kthree,2:5),1)/kind3;%更新聚类中心
end
kind1=0;
kind2=0;
kind3=0;
Kone_store=Kone;
Ktwo_store=Ktwo;
Kthree_store=Kthree;%分类结果
clear Kone Ktwo Kthree;
end
JC=ERR(number,Kone_store,m(1,:))+ERR(number,Ktwo_store,m(2,:))+ERR(number,Kthree_store,m(3,:));
%% 计算准确率
count=0;
count1=0;
count2=0;
count3=0;
[~,n]=size(Kone_store);
for i=1:n
if Kone_store(i)<=50
count1=count1+1;
elseif Kone_store(i)>100
count2=count2+1;
else
count3=count3+1;
end
end
flag=[count1 count2 count3];
[a,b]=max(flag);
count=count+a;
count1=0;
count2=0;
count3=0;
[~,n]=size(Ktwo_store);
for i=1:n
if Ktwo_store(i)<=50
count1=count1+1;
elseif Ktwo_store(i)>100
count2=count2+1;
else
count3=count3+1;
end
end
flag=[count1 count2 count3];
[a,b]=max(flag);
count=count+a;
count1=0;
count2=0;
count3=0;
[~,n]=size(Kthree_store);
for i=1:n
if Kthree_store(i)<=50
count1=count1+1;
elseif Kthree_store(i)>100
count2=count2+1;
else
count3=count3+1;
end
end
flag=[count1 count2 count3];
[a,b]=max(flag);
count=count+a;
acc=count/150;
%% C均值算法对于IRIS数据集的分类(二)
%% author:mym NJUST
clear;
clc;
[number,txt,raw]=xlsread('iris.csv');
%对数据集乱序
index=randperm(150);
rand_number=number(index,:);
kind=3;%类别数
prime_index=randperm(150,3);
m=number(prime_index,2:5);%初始随机选择的三个聚类中心
kind1=0;%记录每类的数目
kind2=0;
kind3=0;
%% 第一步：初步分为三类
for i=1:150
D1=norm(rand_number(i,2:5)-m(1,:));
D2=norm(rand_number(i,2:5)-m(2,:));
D3=norm(rand_number(i,2:5)-m(3,:));%计算样本到聚类中心的欧式距离
vec=[D1 D2 D3];
[M,I]=min(vec);
if I==1
kind1=kind1+1;
Kone(kind1)=rand_number(i,1);
elseif I==2
kind2=kind2+1;
Ktwo(kind2)=rand_number(i,1);
else
kind3=kind3+1;
Kthree(kind3)=rand_number(i,1);%将样本放入聚类
end
end
%分类后重新计算聚类中心
if kind1~=0
m(1,:)=sum(number(Kone,2:5),1)/kind1;
end
if kind2~=0
m(2,:)=sum(number(Ktwo,2:5),1)/kind2;
end
if kind3~=0
m(3,:)=sum(number(Kthree,2:5),1)/kind3;%更新聚类中心
end
%% 修正聚类
%计算误差平方和
JC=zeros(1,150);
JC(1)=ERR(number,Kone,m(1,:))+ERR(number,Ktwo,m(2,:))+ERR(number,Kthree,m(3,:));
J_number=1;
J_last=0;
p=zeros(3,3);
while J_last~=JC(J_number)
J_last=JC(J_number);
[row1,col1]=size(Kone);
count1=1;
%对聚类（一）中的样本进行修正
while count1<=col1
p(1,1)=kind1/(kind1-1)*(norm(number(Kone(count1),2:5)-m(1,:))^2);
p(1,2)=kind2/(kind2+1)*(norm(number(Kone(count1),2:5)-m(2,:))^2);
p(1,3)=kind3/(kind3+1)*(norm(number(Kone(count1),2:5)-m(3,:))^2);
[min_value,min_index]=min(p(1,2:3));
if min_value<p(1,1)
m(1,:)=m(1,:)+1/(kind1-1)*(m(1,:)-number(Kone(count1),2:5));
if min_index==1
m(2,:)=m(2,:)-1/(kind2+1)*(m(2,:)-number(Kone(count1),2:5));
kind2=kind2+1;
Ktwo=Add(Ktwo,Kone(count1));
J_number=J_number+1;
JC(J_number)=JC(J_number-1)-(p(1,1)-p(1,2));
else
m(3,:)=m(3,:)-1/(kind3+1)*(m(3,:)-number(Kone(count1),2:5));
kind3=kind3+1;
Kthree=Add(Kthree,Kone(count1));
J_number=J_number+1;
JC(J_number)=JC(J_number-1)-(p(1,1)-p(1,3));
end
kind1=kind1-1;
Kone=Remove(Kone,count1);
col1=col1-1;
else
count1=count1+1;
J_number=J_number+1;
JC(J_number)=JC(J_number-1);
end
end
count2=1;
%对聚类（二）中的样本进行修正
[row2,col2]=size(Ktwo);
while count2<=col2
p(2,1)=kind1/(kind1+1)*(norm(number(Ktwo(count2),2:5)-m(1,:))^2);
p(2,2)=kind2/(kind2-1)*(norm(number(Ktwo(count2),2:5)-m(2,:))^2);
p(2,3)=kind3/(kind3+1)*(norm(number(Ktwo(count2),2:5)-m(3,:))^2);
[min_value,min_index]=min([p(2,1) p(2,3)]);
if min_value<p(2,2)
m(2,:)=m(2,:)+1/(kind2-1)*(m(2,:)-number(Ktwo(count2),2:5));
if min_index==1
m(1,:)=m(1,:)-1/(kind1+1)*(m(1,:)-number(Ktwo(count2),2:5));
kind1=kind1+1;
Kone=Add(Kone,Ktwo(count2));
J_number=J_number+1;
JC(J_number)=JC(J_number-1)-(p(2,2)-p(2,1));
else
m(3,:)=m(3,:)-1/(kind3+1)*(m(3,:)-number(Ktwo(count2),2:5));
kind3=kind3+1;
Kthree=Add(Kthree,Ktwo(count2));
J_number=J_number+1;
JC(J_number)=JC(J_number-1)-(p(2,2)-p(2,3));
end
kind2=kind2-1;
Ktwo=Remove(Ktwo,count2);
col2=col2-1;
else
count2=count2+1;
J_number=J_number+1;
JC(J_number)=JC(J_number-1);
end
end
%对聚类（三）中的样本进行修正
[row3,col3]=size(Kthree);
count3=1;
while count3<=col3
p(3,1)=kind1/(kind1+1)*(norm(number(Kthree(count3),2:5)-m(1,:))^2);
p(3,2)=kind2/(kind2+1)*(norm(number(Kthree(count3),2:5)-m(2,:))^2);
p(3,3)=kind3/(kind3-1)*(norm(number(Kthree(count3),2:5)-m(3,:))^2);
[min_value,min_index]=min([p(3,1) p(3,2)]);
if min_value<p(3,3)
m(3,:)=m(3,:)+1/(kind3-1)*(m(3,:)-number(Kthree(count3),2:5));
if min_index==1
m(1,:)=m(1,:)-1/(kind1+1)*(m(1,:)-number(Kthree(count3),2:5));
kind1=kind1+1;
Kone=Add(Kone,Kthree(count3));
J_number=J_number+1;
JC(J_number)=JC(J_number-1)-(p(3,3)-p(3,1));
else
m(2,:)=m(2,:)-1/(kind2+1)*(m(2,:)-number(Kthree(count3),2:5));
kind2=kind2+1;
Ktwo=Add(Ktwo,Kthree(count3));
J_number=J_number+1;
JC(J_number)=JC(J_number-1)-(p(3,3)-p(3,2));
end
kind3=kind3-1;
Kthree=Remove(Kthree,count3);
col3=col3-1;
else
count3=count3+1;
J_number=J_number+1;
JC(J_number)=JC(J_number-1);
end
end
end
%% 计算准确率
count=0;
count1=0;
count2=0;
count3=0;
[~,n]=size(Kone);
for i=1:n
if Kone(i)<=50
count1=count1+1;
elseif Kone(i)>100
count2=count2+1;
else
count3=count3+1;
end
end
flag=[count1 count2 count3];
[a,b]=max(flag);
count=count+a;
count1=0;
count2=0;
count3=0;
[~,n]=size(Ktwo);
for i=1:n
if Ktwo(i)<=50
count1=count1+1;
elseif Ktwo(i)>100
count2=count2+1;
else
count3=count3+1;
end
end
flag=[count1 count2 count3];
[a,b]=max(flag);
count=count+a;
count1=0;
count2=0;
count3=0;
[~,n]=size(Kthree);
for i=1:n
if Kthree(i)<=50
count1=count1+1;
elseif Kthree(i)>100
count2=count2+1;
else
count3=count3+1;
end
end
flag=[count1 count2 count3];
[a,b]=max(flag);
count=count+a;
acc=count/150;
function [ Knumber_out ] = Add(Knumber,add_number)
%输入参数为聚类向量与需要添加的数
%输出为添加后的聚类向量
Knumber_out=[Knumber add_number];
end
function [Knumber_out] = Remove(Knumber,Kindex)
%输入为聚类向量与需要去除的聚类索引
%输出为去除后的聚类向量
[row,col]=size(Knumber);
Knumber_out=[Knumber(1:Kindex-1) Knumber(Kindex+1:col)];
end
function [J] = ERR(number,index,m)
%输入为数字样本，聚类样本的索引以及聚类中心
%输出为误差平方和
[row,n]=size(index);
J=0;
for i=1:n
J=J+norm(number(index(i),2:5)-m)^2;
end
end