K-mean clustering 算法

K-MEANS算法:

　　k-means 算法接受输入量 k ；然后将n个数据对象划分为 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。聚类相似度是利用各聚类中对象的均值所获得一个“中心对象”（引力中心）来进行计算的。

　　k-means 算法的工作过程说明如下：首先从n个数据对象任意选择 k 个对象作为初始聚类中心；而对于所剩下其它对象，则根据它们与这些聚类中心的相似度（距离），分别将它们分配给与其最相似的（聚类中心所代表的）聚类；然后再计算每个所获新聚类的聚类中心（该聚类中所有对象的均值）；不断重复这一过程直到标准测度函数开始收敛为止。一般都采用均方差作为标准测度函数. k个聚类具有以下特点：各聚类本身尽可能的紧凑，而各聚类之间尽可能的分开。 

　　补充一个Matlab实现方法：

　　function [cid,nr,centers] = cskmeans(x,k,nc)

　　% CSKMEANS K-Means clustering - general method.

　　% 

　　% This implements the more general k-means algorithm, where 

　　% HMEANS is used to find the initial partition and then each

　　% observation is examined for further improvements in minimizing

　　% the within-group sum of squares.

　　%

　　% [CID,NR,CENTERS] = CSKMEANS(X,K,NC) Performs K-means

　　% clustering using the data given in X. 

　　% 

　　% INPUTS: X is the n x d matrix of data,

　　% where each row indicates an observation. K indicates

　　% the number of desired clusters. NC is a k x d matrix for the

　　% initial cluster centers. If NC is not specified, then the

　　% centers will be randomly chosen from the observations.

　　%

　　% OUTPUTS: CID provides a set of n indexes indicating cluster

　　% membership for each point. NR is the number of observations

　　% in each cluster. CENTERS is a matrix, where each row

　　% corresponds to a cluster center.

　　%

　　% See also CSHMEANS

　　% W. L. and A. R. Martinez, 9/15/01

　　% Computational Statistics Toolbox 

　　warning off

　　[n,d] = size(x);

　　if nargin < 3

　　% Then pick some observations to be the cluster centers.

　　ind = ceil(n*rand(1,k));

　　% We will add some noise to make it interesting.

　　nc = x(ind,:) + randn(k,d);

　　end

　　% set up storage

　　% integer 1,...,k indicating cluster membership

　　cid = zeros(1,n); 

　　% Make this different to get the loop started.

　　oldcid = ones(1,n);

　　% The number in each cluster.

　　nr = zeros(1,k); 

　　% Set up maximum number of iterations.

　　maxiter = 100;

　　iter = 1;

　　while ~isequal(cid,oldcid) & iter < maxiter

　　% Implement the hmeans algorithm

　　% For each point, find the distance to all cluster centers

　　for i = 1:n

　　dist = sum((repmat(x(i,:),k,1)-nc).^2,2);

　　[m,ind] = min(dist); % assign it to this cluster center

　　cid(i) = ind;

　　end

　　% Find the new cluster centers

　　for i = 1:k

　　% find all points in this cluster

　　ind = find(cid==i);

　　% find the centroid

　　nc(i,:) = mean(x(ind,:));

　　% Find the number in each cluster;

　　nr(i) = length(ind);

　　end

　　iter = iter + 1;

　　end

　　% Now check each observation to see if the error can be minimized some more. 

　　% Loop through all points.

　　maxiter = 2;

　　iter = 1;

　　move = 1;

　　while iter < maxiter & move ~= 0 

　　move = 0;

　　% Loop through all points.

　　for i = 1:n

　　% find the distance to all cluster centers

　　dist = sum((repmat(x(i,:),k,1)-nc).^2,2);

　　r = cid(i); % This is the cluster id for x

　　%%nr,nr+1;

　　dadj = nr./(nr+1).*dist'; % All adjusted distances

　　[m,ind] = min(dadj); % minimum should be the cluster it belongs to

　　if ind ~= r % if not, then move x

　　cid(i) = ind;

　　ic = find(cid == ind);

　　nc(ind,:) = mean(x(ic,:));

　　move = 1;

　　end

　　end

　　iter = iter+1;

　　end

　　centers = nc;

　　if move == 0

　　disp('No points were moved after the initial clustering procedure.')

　　else

　　disp('Some points were moved after the initial clustering procedure.')

　　end

　　warning on