谱聚类算法 matlab

1、谱聚类算法步骤公式
（1）整理数据集，使数据集中数据在0-1之间。假设数据集m行n列。
（2）求邻接矩阵W。元素值为每一点到其他点之间距离，即权重。
（3）求相似度矩阵S，相似度矩阵的定义。相似度矩阵由权值矩阵得到，实践中一般用高斯核函数（也称径向基函数核）计算相似度，距离越大，代表其相似度越小。（这里我认为相似度矩阵与邻接矩阵一个概念，邻接矩阵即在图中表示顶点之间相邻关系的矩阵，因此这里用相似度矩阵表示该图顶点之间的相互关系）

（4）求度矩阵D。把的每一列元素加起来得到个数，然后把它们放在对角线上（其它地方都是零），组成一个的对角矩阵，记为度矩阵。
（5）求拉普拉斯矩阵L。使用Normalized相似变换 L=I-D-(1/2)SD-(1/2) (I单位矩阵)
（6）求出L的最小的k个特征值和对应的特征向量V

（7）将特征向量V进行kmeans聚类(少量的特征向量)，得到m行1列的矩阵，即分类结果。

2、谱聚类算法matlab实现
function C = spectral(W,sigma, num_clusters)
% 谱聚类算法
% 使用Normalized相似变换
% 输入  : W              : N-by-N 矩阵, 即连接矩阵
%        sigma          : 高斯核函数,sigma值不能为0
%        num_clusters   : 分类数
%
% 输出  : C : N-by-1矩阵 聚类结果，标签值
%
    format long
    m = size(W, 1);
    %计算相似度矩阵  相似度矩阵由权值矩阵得到，实践中一般用高斯核函数
    W = W.*W;   %平方
    W = -W/(2*sigma*sigma);
    S = full(spfun(@exp, W)); % 在这里S即为相似度矩阵，也就是这不在以邻接矩阵计算，而是采用相似度矩阵

    %获得度矩阵D
    D = full(sparse(1:m, 1:m, sum(S))); %所以此处D为相似度矩阵S中一列元素加起来放到对角线上，得到度矩阵D

    % 获得拉普拉斯矩阵 Do laplacian, L = D^(-1/2) * S * D^(-1/2)
    L = eye(m)-(D^(-1/2) * S * D^(-1/2)); %拉普拉斯矩阵

    % 求特征向量 V
    %  eigs 'SM';绝对值最小特征值
    [V, ~] = eigs(L, num_clusters, 'SM');
    % 对特征向量求k-means
    C=kmeans(V,num_clusters);
end

调用

clc
clf
clear
%twoCircles数据集
load('twoCircles.mat');
dataSet=twoCircles;
dataSet=dataSet/max(max(abs(dataSet)));
num_clusters=2;
sigma=0.1;
%XOR数据集
% load('XOR.mat');
% dataSet=XOR;
% dataSet=dataSet/max(max(abs(dataSet)));
% num_clusters=4;
% sigma=0.1;
Z=pdist(dataSet);
W=squareform(Z);
C = spectral(W,sigma, num_clusters);
plot(dataSet(C==1,1),dataSet(C==1,2),'r.', dataSet(C==2,1),dataSet(C==2,2),'b.', dataSet(C==3,1),dataSet(C==3,2),'g.', dataSet(C==4,1),dataSet(C==4,2),'m.');

3、结果
使用twoCircles数据集结果如下

使用XOR数据集结果如下

4、数据集
twoCircles数据集
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0.118163138199935 -0.00666648250514589
0.110780653575040 0.0265235804271568
0.100737917234786 0.0156260544496707
0.126536424113384 0.0560221149798287
0.130698280785180 0.0684661165422064
0.104542664983812 0.0809424119400617
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0.0641053278934059 0.0947081798114114
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0.185553922917037 -0.491627279645423
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0.219204028289945 -0.424033207830737
0.245742156083010 -0.391214678624873
0.271531239374451 -0.415838279344467
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0.361675847978310 -0.159811109806494
0.395587569270693 -0.163709904402880
0.360873026344104 -0.155004205758110
0.402373685593358 -0.107735146318242
0.402966624140516 -0.126363800088237
0.373867567207756 -0.0896191882457839

XOR数据集
11.7019498550828 11.9864146917909
11.2854197541034 9.61106025805058
12.9293997424851 7.77139077294215
11.0678743631211 10.4674361646635
10.2184969161492 10.0020184568202
13.0269044940811 9.51316600308935
11.8515238069851 8.48249559804150
13.1609598294686 10.3732294253150
10.9263007068675 9.96449067911523
12.9089787789130 10.9301847688819
12.9750456160589 11.9504775516646
10.9553894933476 9.91018786085430
12.1339163659016 9.87242404255941
11.2520226218202 11.0674309284256
13.5473713727711 11.9613951483563
11.9684417926901 11.5074099651376
12.6895038389352 11.1186184934507
10.2832003244487 11.5415772866841
12.4229965264968 11.1877890494122
10.1643053902437 10.5663656603295
11.2048464152179 9.71682231158900
11.0806034399502 10.1652955523462
12.2846926890514 9.78779540958933
10.1824985132917 12.8606632258936
12.0979543910706 11.9424610915524
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13.9315067764341 3.53992252705657
10.0194033879627 0.920979800389733
12.6752472116720 4.27580101528130

5、参考
谱聚类算法(Spectral Clustering) http://www.cnblogs.com/sparkwen/p/3155850.html
从拉普拉斯矩阵说到谱聚类 http://blog.csdn.net/v_july_v/article/details/40738211

6、源码及数据
https://github.com/Keeplingshi/MachineLearning/tree/master/matlab/spectralClustering