转：谱聚类

谱聚类

 

广义上来说，任何在算法中用到SVD/特征值分解的，都叫Spectral Algorithm。顺便说一下，对于任意矩阵只存在奇异值分解，不存在特征值分解。对于正定的对称矩阵，奇异值就是特征值，奇异向量就是特征向量。

传统的聚类算法，如K-Means、EM算法都是建立在凸球形样本空间上，当样本空间不为凸时，算法会陷入局部最优，最终结果受初始参数的选择影响比较大。而谱聚类可以在任意形状的样本空间上聚类，且收敛于全局最优解。

谱聚类和CHAMELEON聚类很像，都是把样本点的相似度放到一个带权无向图中，采用“图划分”的方法进行聚类。只是谱聚类算法在进行图划分的时候发现计算量很大，转而求特征值去了，而且最后还在几个小特征向量组成的矩阵上进行了K-Means聚类。

Simply speaking，谱聚类算法分为3步：

1. 构造一个N×N的权值矩阵W，W_ij表示样本i和样本j的相似度，显然W是个对称矩阵。相似度的计算方法很多了，你可以用欧拉距离、街区距离、向量夹角、皮尔森相关系数等。并不是任意两个点间的相似度都要表示在图上，我们希望的权值图是比较稀疏的，有2种方法：权值小于阈值的认为是0；K最邻近方法，即每个点只和跟它最近的k个点连起来，CHAMELEON算法的第1阶段就是这么干的。再构造一个对角矩阵D，D_ii为W第i列元素之和。最后构造矩阵L=D-W。可以证明L是个半正定和对称矩阵。
2. 求L的前K小特征值对应的特征向量（这要用到奇异值分解了）。把K个特征向量放在一起构造一个N×K的矩阵M。
3. 把M的每一行当成一个新的样本点，对这N个新的样本点进行K-Means聚类。

从文件读入样本点，最终算得矩阵L

#include<math.h>

#include<string.h>

#include"matrix.h"

#include"svd.h"

 

#define N 19        //样本点个数

#define K 4         //K-Means算法中的K

#define T 0.1       //样本点之间相似度的阈值

 

double sample[N][2];    //存放所有样本点的坐标（2维的）

 

void readSample(char *filename){

    FILE *fp;

    if((fp=fopen(filename,"r"))==NULL){

        perror("fopen");

        exit(0);

    }

    char buf[50]={0};

    int i=0;

    while(fgets(buf,sizeof(buf),fp)!=NULL){

        char *w=strtok(buf," \t");

        double x=atof(w);

        w=strtok(NULL," \t");

        double y=atof(w);

        sample[i][0]=x;

        sample[i][1]=y;

        i++;

        memset(buf,0x00,sizeof(buf));

    }

    assert(i==N);

    fclose(fp);

}

 

double** getSimMatrix(){

    //为二维矩阵申请空间

    double **matrix=getMatrix(N,N);

    //计算样本点两两之间的相似度，得到矩阵W

    int i,j;

    for(i=0;i<N;i++){

        matrix[i][i]=1;

        for(j=i+1;j<N;j++){

            double dist=sqrt(pow(sample[i][0]-sample[j][0],2)+pow(sample[i][1]-sample[j][1],2));

            double sim=1.0/(1+dist);

            if(sim>T){

                matrix[j][i]=sim;

                matrix[i][j]=sim;

            }

        }

    }

    //计算L=D-W

    for(j=0;j<N;j++){

        double sum=0;

        for(i=0;i<N;i++){

            sum+=matrix[i][j];

            if(i!=j)

                matrix[i][j]=0-matrix[i][j];

        }

        matrix[j][j]=matrix[j][j]-sum;

    }

    return matrix;

}

 

int main(){

    char *file="/home/orisun/data";

    readSample(file);

    double **L=getSimMatrix();

    printMatrix(L,N,N);

     

    double **M=singleVector(L,N,N,5);

    printMatrix(M,N,5);

     

    freeMatrix(L,N);

 

    return 0;

}

L已是对称矩阵，直接奇异值分解的得到的就是特征向量

+ View Code

+ View Code

最后是运行KMeans的Java代码

package ai;

 

public class Global {

    //计算两个向量的欧氏距离

    public static double calEuraDist(double[] arr1,double[] arr2,int len){

        double result=0.0;

        for(int i=0;i<len;i++){

            result+=Math.pow(arr1[i]-arr2[i],2.0);

        }

        return Math.sqrt(result);

    }

}

package ai;

 

public class DataObject {

 

    String docname;

    double[] vector;

    int cid;   

    boolean visited;

     

    public DataObject(int len){

        vector=new double[len];

    }

 

    public String getName() {

        return docname;

    }

 

    public void setName(String docname) {

        this.docname = docname;

    }

 

    public double[] getVector() {

        return vector;

    }

 

    public void setVector(double[] vector) {

        this.vector = vector;

    }

 

    public int getCid() {

        return cid;

    }

 

    public void setCid(int cid) {

        this.cid = cid;

    }

 

    public boolean isVisited() {

        return visited;

    }

 

    public void setVisited(boolean visited) {

        this.visited = visited;

    }

 

}

package ai;

 

import java.io.BufferedReader;

import java.io.File;

import java.io.FileReader;

import java.io.IOException;

import java.util.ArrayList;

import java.util.Iterator;

public class DataSource {

 

    ArrayList<DataObject> objects;

    int row;

    int col;

 

    public void readMatrix(File dataFile) {

        try {

            FileReader fr = new FileReader(dataFile);

            BufferedReader br = new BufferedReader(fr);

            String line = br.readLine();

            String[] words = line.split("\\s+");

            row = Integer.parseInt(words[0]);

            // row=1000;

            col = Integer.parseInt(words[1]);

            objects = new ArrayList<DataObject>(row);

            for (int i = 0; i < row; i++) {

                DataObject object = new DataObject(col);

                line = br.readLine();

                words = line.split("\\s+");

                for (int j = 0; j < col; j++) {

                    object.getVector()[j] = Double.parseDouble(words[j]);

                }

                objects.add(object);

            }

            br.close();

        } catch (IOException e) {

            e.printStackTrace();

        }

    }

 

    public void readRLabel(File file) {

        try {

            FileReader fr = new FileReader(file);

            BufferedReader br = new BufferedReader(fr);

            String line = null;

            for (int i = 0; i < row; i++) {

                line = br.readLine();

                objects.get(i).setName(line.trim());

            }

        } catch (IOException e) {

            e.printStackTrace();

        }

    }

 

    public void printResult(ArrayList<DataObject> objects, int n) {

        //DBScan是从第1类开始，K-Means是从第0类开始

//      for (int i =0; i <n; i++) {

        for(int i=1;i<=n;i++){

            System.out.println("=============属于第"+i+"类的有：===========================");

            Iterator<DataObject> iter = objects.iterator();

            while (iter.hasNext()) {

                DataObject object = iter.next();

                int cid=object.getCid();

                if(cid==i){

                    System.out.println(object.getName());

//                  switch(Integer.parseInt(object.getName())/1000){

//                  case 0:

//                      System.out.println(0);

//                      break;

//                  case 1:

//                      System.out.println(1);

//                      break;

//                  case 2:

//                      System.out.println(2);

//                      break;

//                  case 3:

//                      System.out.println(3);

//                      break;

//                  case 4:

//                      System.out.println(4);

//                      break;

//                  case 5:

//                      System.out.println(5);

//                      break;

//                  default:

//                      System.out.println("Go Out");

//                      break;

//                  }              

                }

            }

        }

    }

}

package ai;

 

import java.io.File;

import java.util.ArrayList;

import java.util.Iterator;

import java.util.Random;

  

public class KMeans {

  

    int k; // 指定划分的簇数

    double mu; // 迭代终止条件，当各个新质心相对于老质心偏移量小于mu时终止迭代

    double[][] center; // 上一次各簇质心的位置

    int repeat; // 重复运行次数

    double[] crita; // 存放每次运行的满意度

  

    public KMeans(int k, double mu, int repeat, int len) {

        this.k = k;

        this.mu = mu;

        this.repeat = repeat;

        center = new double[k][];

        for (int i = 0; i < k; i++)

            center[i] = new double[len];

        crita = new double[repeat];

    }

  

    // 初始化k个质心，每个质心是len维的向量，每维均在left--right之间

    public void initCenter(int len, ArrayList<DataObject> objects) {

        Random random = new Random(System.currentTimeMillis());

        int[] count = new int[k]; // 记录每个簇有多少个元素

        Iterator<DataObject> iter = objects.iterator();

        while (iter.hasNext()) {

            DataObject object = iter.next();

            int id = random.nextInt(10000)%k;

            count[id]++;

            for (int i = 0; i < len; i++)

                center[id][i] += object.getVector()[i];

        }

        for (int i = 0; i < k; i++) {

            for (int j = 0; j < len; j++) {

                center[i][j] /= count[i];

            }

        }

    }

  

    // 把数据集中的每个点归到离它最近的那个质心

    public void classify(ArrayList<DataObject> objects) {

        Iterator<DataObject> iter = objects.iterator();

        while (iter.hasNext()) {

            DataObject object = iter.next();

            double[] vector = object.getVector();

            int len = vector.length;

            int index = 0;

            double neardist = Double.MAX_VALUE;

            for (int i = 0; i < k; i++) {

                double dist = Global.calEuraDist(vector, center[i], len); // 使用欧氏距离

                if (dist < neardist) {

                    neardist = dist;

                    index = i;

                }

            }

            object.setCid(index);

        }

    }

  

    // 重新计算每个簇的质心，并判断终止条件是否满足，如果不满足更新各簇的质心,如果满足就返回true.len是数据的维数

    public boolean calNewCenter(ArrayList<DataObject> objects, int len) {

        boolean end = true;

        int[] count = new int[k]; // 记录每个簇有多少个元素

        double[][] sum = new double[k][];

        for (int i = 0; i < k; i++)

            sum[i] = new double[len];

        Iterator<DataObject> iter = objects.iterator();

        while (iter.hasNext()) {

            DataObject object = iter.next();

            int id = object.getCid();

            count[id]++;

            for (int i = 0; i < len; i++)

                sum[id][i] += object.getVector()[i];

        }

        for (int i = 0; i < k; i++) {

            if (count[i] != 0) {

                for (int j = 0; j < len; j++) {

                    sum[i][j] /= count[i];

                }

            }

            // 簇中不包含任何点,及时调整质心

            else {

                int a=(i+1)%k;

                int b=(i+3)%k;

                int c=(i+5)%k;

                for (int j = 0; j < len; j++) {

                    center[i][j] = (center[a][j]+center[b][j]+center[c][j])/3;

                }

            }

        }

        for (int i = 0; i < k; i++) {

            // 只要有一个质心需要移动的距离超过了mu，就返回false

            if (Global.calEuraDist(sum[i], center[i], len) >= mu) {

                end = false;

                break;

            }

        }

        if (!end) {

            for (int i = 0; i < k; i++) {

                for (int j = 0; j < len; j++)

                    center[i][j] = sum[i][j];

            }

        }

        return end;

    }

  

    // 计算各簇内数据和方差的加权平均，得出本次聚类的满意度.len是数据的维数

    public double getSati(ArrayList<DataObject> objects, int len) {

        double satisfy = 0.0;

        int[] count = new int[k];

        double[] ss = new double[k];

        Iterator<DataObject> iter = objects.iterator();

        while (iter.hasNext()) {

            DataObject object = iter.next();

            int id = object.getCid();

            count[id]++;

            for (int i = 0; i < len; i++)

                ss[id] += Math.pow(object.getVector()[i] - center[id][i], 2.0);

        }

        for (int i = 0; i < k; i++) {

            satisfy += count[i] * ss[i];

        }

        return satisfy;

    }

  

    public double run(int round, DataSource datasource, int len) {

        System.out.println("第" + round + "次运行");

        initCenter(len,datasource.objects);

        classify(datasource.objects);

        while (!calNewCenter(datasource.objects, len)) {

            classify(datasource.objects);

        }

        datasource.printResult(datasource.objects, k);

        double ss = getSati(datasource.objects, len);

        System.out.println("加权方差：" + ss);

        return ss;

    }

  

    public static void main(String[] args) {

        DataSource datasource = new DataSource();

        datasource.readMatrix(new File("/home/orisun/test/dot.mat"));

        datasource.readRLabel(new File("/home/orisun/test/dot.rlabel"));

        int len = datasource.col;

        // 划分为4个簇，质心移动小于1E-8时终止迭代，重复运行7次

        KMeans km = new KMeans(4, 1E-10, 7, len);

        int index = 0;

        double minsa = Double.MAX_VALUE;

        for (int i = 0; i < km.repeat; i++) {

            double ss = km.run(i, datasource, len);

            if (ss < minsa) {

                minsa = ss;

                index = i;

            }

        }

        System.out.println("最好的结果是第" + index + "次。");

    }

}

原文来自:博客园（华夏35度）http://www.cnblogs.com/zhangchaoyang 作者:Orisun