聚类分析之基于密度的聚类算法(DBSCAN)

一  什么是基于密度的聚类算法

由于层次聚类算法和划分式聚类算往往只能发现凸形的聚类簇。为了弥补这一缺陷,发现各种任意形状的聚类簇,开发出基于密度的聚类算法。这类算法认为,在整个样本空间点中,各目标类簇是由一群的稠密样本点组成的,而这些稠密样本点被低密度区域(噪声)分割,而算法的目的就是要过滤低密度区域,发现稠密样本点。

二  DBSCANDensity-based Spatial Clustering of Applications with Noise

是一种基于高密度联通区域的聚类算法,它将类簇定义为高密度相连点的最大集合。它本身对噪声不敏感,并且能发现任意形状的类簇。

DBSCAN中的的几个定义:

Ε领域:给定对象半径Ε内的区域称为该对象的Ε领域

核心对象:如果给定对象Ε领域内的样本点数大于等于MinPts,则称该对象为核心对象

直接密度可达:对于样本集合D,如果样本点qpΕ领域内,并且p为核心对象,那么对象q从对象p直接密度可达

密度可达:对于样本集合D,给定一串样本点p1,p2….pnp= p1,q= pn,假如对象pipi-1直接密度可达,那么对象q从对象p密度可达

密度相连:对于样本集合D中的任意一点O,如果存在对象p到对象o密度可达,并且对象q到对象o密度可达,那么对象q到对象p密度相连

可以发现,密度可达是直接密度可达的传递闭包,并且这种关系是非对称的。密度相连是对称关系。DBSCAN目的是找到密度相连对象的最大集合。

Eg: 假设半径Ε=3MinPts=3,点pE领域中有点{m,p,p1,p2,o}, mE领域中有点{m,q,p,m1,m2},qE领域中有点{q,m},oE领域中有点{o,p,s},sE领域中有点{o,s,s1}.

那么核心对象有p,m,o,s(q不是核心对象,因为它对应的E领域中点数量等于2,小于MinPts=3)

m从点p直接密度可达,因为mpE领域内,并且p为核心对象;

q从点p密度可达,因为点q从点m直接密度可达,并且点m从点p直接密度可达;

q到点s密度相连,因为点q从点p密度可达,并且s从点p密度可达。

三  算法描述

算法:DBSCAN

输入:E — 半径

      MinPts — 给定点E领域内成为核心对象的最小领域点数

      D — 集合

输出:目标类簇集合

方法:repeat

1)       判断输入点是否为核心对象

2)       找出核心对象的E领域中的所有直接密度可达点

      util 所有输入点都判断完毕

      repeat

         针对所有核心对象的E领域所有直接密度可达点找到最大密度相连对象集合,

         中间涉及到一些密度可达对象的合并。

      Util 所有核心对象的E领域都遍历完毕



算法:DBSCAN

输入:E  半径

      MinPts  给定点在E领域内成为核心对象的最小领域点数

      D  集合

输出:目标类簇集合

方法:repeat

1)       判断输入点是否为核心对象

2)       找出核心对象的E领域中的所有直接密度可达点

      util 所有输入点都判断完毕

      repeat

         针对所有核心对象的E领域所有直接密度可达点找到最大密度相连对象集合,

         中间涉及到一些密度可达对象的合并。

      Util 所有核心对象的E领域都遍历完毕

四  算法实现

package com.dbscan;

public class DataPoint {
    private String dataPointName; // 样本点名
    private double dimensioin[]; // 样本点的维度
    private boolean isKey; //是否是核心对象

    public DataPoint(){

    }

    public DataPoint(double[] dimensioin,String dataPointName,boolean isKey){
         this.dataPointName=dataPointName;
         this.dimensioin=dimensioin;
         this.isKey=isKey;
    }

}

------------

package com.dbscan;

import java.util.ArrayList;
import java.util.List;



public class Cluster {
    private List<DataPoint> dataPoints = new ArrayList<DataPoint>(); // 类簇中的样本点
    private String clusterName; //簇名

    public List<DataPoint> getDataPoints() {
        return dataPoints;
    }

    public void setDataPoints(List<DataPoint> dataPoints) {
        this.dataPoints = dataPoints;
    }

    public String getClusterName() {
        return clusterName;
    }

    public void setClusterName(String clusterName) {
        this.clusterName = clusterName;
    }

}

------------

package com.dbscan;

import java.util.ArrayList;
import java.util.List;

public class ClusterAnalysis {

   
    public List<Cluster> doDbscanAnalysis(List<DataPoint> dataPoints,
            double radius, int ObjectNum) {
         List<Cluster> clusterList=new ArrayList<Cluster>();
         for(int i=0; i<dataPoints.size();i++){
             DataPoint dp=dataPoints.get(i);
             List<DataPoint> arrivableObjects=isKeyAndReturnObjects(dp,dataPoints,radius,ObjectNum);
             if(arrivableObjects!=null){
                  Cluster tempCluster=new Cluster();
                  tempCluster.setClusterName("Cluster "+i);
                  tempCluster.setDataPoints(arrivableObjects);
                  clusterList.add(tempCluster);
             }
         }

         for(int i=0;i<clusterList.size();i++){
             for(int j=0;j<clusterList.size();j++){
                  if(i!=j){
                      Cluster clusterA=clusterList.get(i);
                      Cluster clusterB=clusterList.get(j);

                      List<DataPoint> dpsA=clusterA.getDataPoints();
                      List<DataPoint> dpsB=clusterB.getDataPoints();

                      boolean flag=mergeList(dpsA,dpsB);
                      if(flag){
                          clusterList.set(j, new Cluster());
                      }
                  }
             }
         }

         return clusterList;
    }

   

    public void displayCluster(List<Cluster> clusterList){
        if(clusterList!=null){
            for(Cluster tempCluster:clusterList){
               if(tempCluster.getDataPoints()!=null&&tempCluster.getDataPoints().size()>0){
                    System.out.println("----------"+tempCluster.getClusterName()+"----------");
                    for(DataPoint dp:tempCluster.getDataPoints()){
                        System.out.println(dp.getDataPointName());
                    }
                }
            }
        }
    }

   
    private double getDistance(DataPoint dp1,DataPoint dp2){
        double distance=0.0;
        double[] dim1=dp1.getDimensioin();
        double[] dim2=dp2.getDimensioin();
        if(dim1.length==dim2.length){
            for(int i=0;i<dim1.length;i++){
                double temp=Math.pow((dim1[i]-dim2[i]), 2);
                distance=distance+temp;
            }
            distance=Math.pow(distance, 0.5);
            return distance;
        }
        return distance;
    }

   
   private List<DataPoint> isKeyAndReturnObjects(DataPoint dataPoint,List<DataPoint> dataPoints,double radius,int ObjectNum){
       List<DataPoint> arrivableObjects=new ArrayList<DataPoint>(); //用来存储所有直接密度可达对象

       for(DataPoint dp:dataPoints){
          double distance=getDistance(dataPoint,dp);
          if(distance<=radius){
              arrivableObjects.add(dp);
          }
       }

       if(arrivableObjects.size()>=ObjectNum){
           dataPoint.setKey(true);
           return arrivableObjects;
       }

       return null;
   }

  
   private boolean isContain(DataPoint dp,List<DataPoint> dps){
      boolean flag=false;
      String name=dp.getDataPointName().trim();
      for(DataPoint tempDp:dps){
         String tempName=tempDp.getDataPointName().trim();
         if(name.equals(tempName)){
             flag=true;
             break;
         }
      }

      return flag;
   }

  
   private boolean mergeList(List<DataPoint> dps1,List<DataPoint> dps2){
       boolean flag=false;

       if(dps1==null||dps2==null||dps1.size()==0||dps2.size()==0){
           return flag;
       }

       for(DataPoint dp:dps2){
          if(dp.isKey()&&isContain(dp,dps1)){
             flag=true;
             break;
          }
       }

       if(flag){
           for(DataPoint dp:dps2){
              if(!isContain(dp,dps1)){
                  DataPoint tempDp=new DataPoint(dp.getDimensioin(),dp.getDataPointName(),dp.isKey());
                  dps1.add(tempDp);
              }
           }
       }


       return flag;
   }

   public static void main(String[] args){
       ArrayList<DataPoint> dpoints = new ArrayList<DataPoint>();
      
       double[] a={2,3};
       double[] b={2,4};
       double[] c={1,4};
       double[] d={1,3};
       double[] e={2,2};
       double[] f={3,2};

       double[] g={8,7};
       double[] h={8,6};
       double[] i={7,7};
       double[] j={7,6};
       double[] k={8,5};

       double[] l={100,2};//孤立


       double[] m={8,20};
       double[] n={8,19};
       double[] o={7,18};
       double[] p={7,17};
       double[] q={8,21};

       dpoints.add(new DataPoint(a,"a",false));
       dpoints.add(new DataPoint(b,"b",false));
       dpoints.add(new DataPoint(c,"c",false));
       dpoints.add(new DataPoint(d,"d",false));
       dpoints.add(new DataPoint(e,"e",false));
       dpoints.add(new DataPoint(f,"f",false));

       dpoints.add(new DataPoint(g,"g",false));
       dpoints.add(new DataPoint(h,"h",false));
       dpoints.add(new DataPoint(i,"i",false));
       dpoints.add(new DataPoint(j,"j",false));
       dpoints.add(new DataPoint(k,"k",false));

       dpoints.add(new DataPoint(l,"l",false));

       dpoints.add(new DataPoint(m,"m",false));
       dpoints.add(new DataPoint(n,"n",false));
       dpoints.add(new DataPoint(o,"o",false));
       dpoints.add(new DataPoint(p,"p",false));
       dpoints.add(new DataPoint(q,"q",false));

       ClusterAnalysis ca=new ClusterAnalysis();
       List<Cluster> clusterList=ca.doDbscanAnalysis(dpoints, 2, 4);
       ca.displayCluster(clusterList);

   }
}

   
}

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