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
