KMeans和KMedoid算法是聚类算法中比较普遍的方法,本文讲了其原理和matlab中实现的代码。
1.目标:
找出一个分割,使得距离平方和最小
2.K-Means算法:
1. 将数据分为k个非空子集
2. 计算每个类中心点(k-means中用所有点的平均值,K-medoid用离该平均值最近的一个点)center
3. 将每个object聚类到最近的center
4. 返回2,当聚类结果不再变化的时候stop
复杂度:
O(kndt)
-计算两点间距离:d
-指定类:O(kn) ,k是类数
-迭代次数上限:t
3.K-Medoids算法:
1. 随机选择k个点作为初始medoid
2.将每个object聚类到最近的medoid
3. 更新每个类的medoid,计算objective function
4. 选择最佳参数
4. 返回2,当各类medoid不再变化的时候stop
复杂度:
O((n^2)d)
-计算各点间两两距离O((n^2)d)
-指定类:O(kn) ,k是类数
4.特点:
-聚类结果与初始点有关(因为是做steepest descent from a random initial starting oint)
-是局部最优解
-在实际做的时候,随机选择多组初始点,最后选择拥有最低TSD(Total Squared Distance)的那组
Kmeans KMedoid Implementation with matlab:
===================
下面是我用matlab上的实现:
说明:fea为训练样本数据,gnd为样本标号。算法中的思想和上面写的一模一样,在最后的判断accuracy方面,由于聚类和分类不同,只是得到一些 cluster ,而并不知道这些 cluster 应该被打上什么标签,或者说。由于我们的目的是衡量聚类算法的 performance ,因此直接假定这一步能实现最优的对应关系,将每个 cluster 对应到一类上去。一种办法是枚举所有可能的情况并选出最优解,另外,对于这样的问题,我们还可以用 Hungarian algorithm 来求解。具体的Hungarian代码我放在了资源里,调用方法已经写在下面函数中了。下面给出Kmeans&Kmedoid主函数。
Kmeans.m 函数:
function [ accuracy,MIhat ] = KMeans( K,mode )% Artificial Intelligence & Data Mining - KMeans & K-Medoids Clustering% Author: Rachel Zhang @ ZJU% CreateTime: 2012-11-18% Function: Clustering% -K: number of clusters% -mode: % 1: use kmeans cluster algorithm in matlab% 2: k_medroid algorithm: use data points as k centers% 3: k_means algorithm: use average as k centersglobal N_features;global N_samples;global fea;global gnd;switch (mode) case 1 %call system function KMeans label = kmeans(fea,K); [label,accuracy] = cal_accuracy(gnd,label); case 2%use kmedroid method for testcase = 1:10% do 10 times to get rid of the influence from Initial_center K_center = Initial_center(fea,K); %select initial points randomly changed_label = N_samples; label = zeros(1,N_samples); iteration_times = 0; while changed_label~=0 cls_label = cell(1,K); for i = 1: N_samples for j = 1 : K D(j) = dis(fea(i,:),K_center(j,:)); end [~,label(i)] = min(D); cls_label{label(i)} = [cls_label{label(i)} i]; end changed_label = 0; cls_center = zeros(K,N_features); for i = 1 : K cls_center(i,:) = mean(fea(cls_label{i},:)); D1 = []; for j = 1:size(cls_label{i},2)%number of samples clsutered in i-th class D1(j) = dis(cls_center(i,:),fea(cls_label{i}(j),:)); end [~,min_ind] = min(D1); if ~isequal(K_center(i,:),fea(cls_label{i}(min_ind),:)) K_center(i,:) = fea(cls_label{i}(min_ind),:); changed_label = changed_label+1; end end iteration_times = iteration_times+1; end [label,acc(testcase)] = cal_accuracy(gnd,label); end accuracy = max(acc); case 3%use k-means method for testcase = 1:10% do 10 times to get rid of the influence from Initial_center K_center = Initial_center(fea,K); %select initial points randomly changed_label = N_samples; label = zeros(1,N_samples); label_new = zeros(1,N_samples); while changed_label~=0 cls_label = cell(1,K); changed_label = 0; for i = 1: N_samples for j = 1 : K D(j) = dis(fea(i,:),K_center(j,:)); end [~,label_new(i)] = min(D); if(label_new(i)~=label(i)) changed_label = changed_label+1; end; cls_label{label_new(i)} = [cls_label{label_new(i)} i]; end label = label_new; for i = 1 : K %recalculate k centroid K_center(i,:) = mean(fea(cls_label{i},:)); end end [label,acc(testcase)] = cal_accuracy(gnd,label); end accuracy = max(acc);endMIhat = MutualInfo(gnd,label); function center = Initial_center(X,K) rnd_Idx = randperm(N_samples,K); center = X(rnd_Idx,:); end function res = dis(X1,X2) res = norm(X1-X2); end function [res,acc] = cal_accuracy(gnd,estimate_label) res = bestMap(gnd,estimate_label); acc = length(find(gnd == res))/length(gnd); endend
实验结果分析:
对上面得到的accuracy进行画图,横坐标为10个数据集,纵坐标为在其上进行聚类的准确率。
其中,auto为matlab内部kmeans函数。
画图:
function [ ] = Plot( A,B,C )%PLOT Summary of this function goes here% Detailed explanation goes herefigure;k = 1:10;plot(k,A,'-r',k,B,'-b',k,C,'-g');legend('auto','medoid','means');end
结果:
5类聚类:
7类聚类:
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