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：
$J_c={\sum }_{j=1}^c{\sum }_{k=1}^{n_j}||x_k-m_j|{|}^2$
其中$c$代表类数，$n_j$代表$j$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$D_e(x,y)=\sqrt{{\sum }_{i=1}^d|x_i-y_i{|}^2}$），距离哪个中心近就归入哪类（这很好理解）
3.计算$c$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：
${\rho }_{ii}=\frac{n_i}{n_i-1}||x_k^{(i)}-Z_i(I)|{|}^2,i=1,2,...,c$
${\rho }_{ii}$表示使准则函数减小的部分
${\rho }_{ij}=\frac{n_j}{n_j+1}||x_k^{(i)}-Z_j(I)|{|}^2,j=1,2,...,cj\ne i$
${\rho }_{ij}$表示使准则函数增加的部分
其中$i$代表第$i$类，$j$代表第$j$类，$n$代表聚类中样本数，$x_k^{(i)}$代表第$i$类第$k$个样本，$Z$代表聚类中心，$I$代表第$I$次迭代值。
找出${\rho }_{ij}$中的最小值${\rho }_{il}$，如果此最小值小于${\rho }_{ii}$，说明存在能够使聚类准则函数减小的操作。即把样本$x_k^{(i)}$移动到聚类中心$w_l$中。
3.经过移动操作后修改聚合中心与准则函数$J_c$的值。
$Z_i(I+1)=Z_i(I)+\frac{1}{n_i-1}\left[Z_i(I)-x_k^{(i)}\right]$
$Z_l(I+1)=Z_j(I)-\frac{1}{n_l+1}\left[Z_l(I)-x_k^{(i)}\right]$
$J_c(I+1)=J_c(I)-\left({\rho }_{ii}-{\rho }_{il}\right)$
4.若$J_c(I+1)<J_c(I)$则$I=I+1$返回2，否则算法结束。

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