K-Means 聚类算法

K-Means 概念定义：

K-Means 是一种基于距离的排他的聚类划分方法。

上面的 K-Means 描述中包含了几个概念：

• 聚类（Clustering）：K-Means 是一种聚类分析（Cluster Analysis）方法。聚类就是将数据对象分组成为多个类或者簇 (Cluster)，使得在同一个簇中的对象之间具有较高的相似度，而不同簇中的对象差别较大。
• 划分（Partitioning）：聚类可以基于划分，也可以基于分层。划分即将对象划分成不同的簇，而分层是将对象分等级。
• 排他（Exclusive）：对于一个数据对象，只能被划分到一个簇中。如果一个数据对象可以被划分到多个簇中，则称为可重叠的（Overlapping）。
• 距离（Distance）：基于距离的聚类是将距离近的相似的对象聚在一起。基于概率分布模型的聚类是在一组对象中，找到能符合特定分布模型的对象的集合，他们不一定是距离最近的或者最相似的，而是能完美的呈现出概率分布模型所描述的模型。

K-Means 问题描述：

给定一个 n 个对象的数据集，它可以构建数据的 k 个划分，每个划分就是一个簇，并且 k ≤ n。同时还需满足：

1. 每个组至少包含一个对象。
2. 每个对象必须属于且仅属于一个簇。

Simply speaking, K-Means clustering is an algorithm to classify or to group your objects based on attributes/features, into K number of groups. K is a positive integer number. The grouping is done by minimizing the sum of squares of distances between data and the corresponding cluster centroid. Thus, the purpose of K-means clustering is to classify the data.

例如，有如下包含 10 条数据的集合。集合中每项描述了一个人的身高（Height: inches）和体重（Weight: kilograms）。

Height Weight
-------------
(73.0, 72.6) 
(61.0, 54.4) 
(67.0, 99.9) 
(68.0, 97.3) 
(62.0, 59.0) 
(75.0, 81.6) 
(74.0, 77.1) 
(66.0, 97.3) 
(68.0, 93.3) 
(61.0, 59.0)

通过按照身高和体重的聚类，可以将上述 10 条数据分组成 3 类。

Height Weight
-------------
(67.0, 99.9) 
(68.0, 97.3) 
(66.0, 97.3) 
(68.0, 93.3)

(73.0, 72.6) 
(75.0, 81.6) 
(74.0, 77.1)

(61.0, 54.4) 
(62.0, 59.0) 
(61.0, 59.0)

分类结果可以描述为：中等身高并且很重、很高并且中等体重、矮并且轻。如果用图形来观察分组状况则结果一目了然。

K-Means 算法实现：

由于 K-Means 算法值针对给定的完整数据集进行操作，不需要任何特殊的训练数据，所以 K-Means 是一种无监督的机器学习方法（Unsupervised Machine Learning Technique）。

K-Means 算法最常见的实现方式是使用迭代式精化启发法的 Lloyd's algorithm。

• 给定划分数量 k。创建一个初始划分，从数据集中随机地选择 k 个对象，每个对象初始地代表了一个簇中心（Cluster Centroid）。对于其他对象，计算其与各个簇中心的距离，将它们划入距离最近的簇。
• 采用迭代的重定位技术，尝试通过对象在划分间移动来改进划分。所谓重定位技术，就是当有新的对象加入簇或者已有对象离开簇的时候，重新计算簇的平均值，然后对对象进行重新分配。这个过程不断重复，直到各簇中对象不再变化为止。

randomly assign all data items to a cluster 
loop until no change in cluster assignments 
  compute centroids for each cluster 
  reassign each data item to cluster of closest centroid 
end

简洁点儿的表述即为：

initialize clustering 
loop 
  update centroids 
  update clustering 
end loop

应用 K-Means 算法到上述身高与体重的示例，聚类过程如下图所示。

K-Means 优缺点：

当结果簇是密集的，而且簇和簇之间的区别比较明显时，K-Means 的效果较好。对于大数据集，K-Means 是相对可伸缩的和高效的，它的复杂度是 O(nkt)，n 是对象的个数，k 是簇的数目，t 是迭代的次数，通常 k << n，且 t << n，所以算法经常以局部最优结束。

K-Means 的最大问题是要求先给出 k 的个数。k 的选择一般基于经验值和多次实验结果，对于不同的数据集，k 的取值没有可借鉴性。另外，K-Means 对孤立点数据是敏感的，少量噪声数据就能对平均值造成极大的影响。

Basic K-Means - Lloyd's algorithm C# 代码实现：

Code below referenced from Machine Learning Using C# Succinctly by James McCaffrey, and article K-Means Data Clustering Using C#.

using System;

namespace ClusterNumeric
{
  class ClusterNumProgram
  {
    static void Main(string[] args)
    {
      Console.WriteLine("\nBegin k-means clustering demo\n");

      double[][] rawData = new double[10][];
      rawData[0] = new double[] { 73, 72.6 };
      rawData[1] = new double[] { 61, 54.4 };
      rawData[2] = new double[] { 67, 99.9 };
      rawData[3] = new double[] { 68, 97.3 };
      rawData[4] = new double[] { 62, 59.0 };
      rawData[5] = new double[] { 75, 81.6 };
      rawData[6] = new double[] { 74, 77.1 };
      rawData[7] = new double[] { 66, 97.3 };
      rawData[8] = new double[] { 68, 93.3 };
      rawData[9] = new double[] { 61, 59.0 };

      Console.WriteLine("Raw unclustered height (in.) weight (kg.) data:\n");
      Console.WriteLine(" ID Height Weight");
      Console.WriteLine("---------------------");
      ShowData(rawData, 1, true, true);

      int numClusters = 3;
      Console.WriteLine("\nSetting numClusters to " + numClusters);

      Console.WriteLine("Starting clustering using k-means algorithm");
      Clusterer c = new Clusterer(numClusters);
      int[] clustering = c.Cluster(rawData);
      Console.WriteLine("Clustering complete\n");

      Console.WriteLine("Final clustering in internal form:\n");
      ShowVector(clustering, true);

      Console.WriteLine("Raw data by cluster:\n");
      Console.WriteLine(" ID Height Weight");
      ShowClustered(rawData, clustering, numClusters, 1);

      Console.WriteLine("\nEnd k-means clustering demo\n");
      Console.ReadLine();
    }

    static void ShowData(
      double[][] data, int decimals,
      bool indices, bool newLine)
    {
      for (int i = 0; i < data.Length; ++i)
      {
        if (indices == true)
          Console.Write(i.ToString().PadLeft(3) + " ");

        for (int j = 0; j < data[i].Length; ++j)
        {
          double v = data[i][j];
          Console.Write(v.ToString("F" + decimals) + "   ");
        }

        Console.WriteLine("");
      }

      if (newLine == true)
        Console.WriteLine("");
    }

    static void ShowVector(int[] vector, bool newLine)
    {
      for (int i = 0; i < vector.Length; ++i)
        Console.Write(vector[i] + " ");

      if (newLine == true)
        Console.WriteLine("\n");
    }

    static void ShowClustered(
      double[][] data, int[] clustering,
      int numClusters, int decimals)
    {
      for (int k = 0; k < numClusters; ++k)
      {
        Console.WriteLine("===================");
        for (int i = 0; i < data.Length; ++i)
        {
          int clusterID = clustering[i];
          if (clusterID != k) continue;
          Console.Write(i.ToString().PadLeft(3) + " ");
          for (int j = 0; j < data[i].Length; ++j)
          {
            double v = data[i][j];
            Console.Write(v.ToString("F" + decimals) + "   ");
          }
          Console.WriteLine("");
        }
        Console.WriteLine("===================");
      }
    }
  }

  public class Clusterer
  {
    private int numClusters; // number of clusters 
    private int[] clustering; // index = a tuple, value = cluster ID 
    private double[][] centroids; // mean (vector) of each cluster 
    private Random rnd; // for initialization 

    public Clusterer(int numClusters)
    {
      this.numClusters = numClusters;
      this.centroids = new double[numClusters][];
      this.rnd = new Random(0); // arbitrary seed 
    }

    public int[] Cluster(double[][] data)
    {
      int numTuples = data.Length;
      int numValues = data[0].Length;
      this.clustering = new int[numTuples];

      for (int k = 0; k < numClusters; ++k) // allocate each centroid 
        this.centroids[k] = new double[numValues];

      InitRandom(data);

      Console.WriteLine("\nInitial random clustering:");
      for (int i = 0; i < clustering.Length; ++i)
        Console.Write(clustering[i] + " ");
      Console.WriteLine("\n");

      bool changed = true; // change in clustering? 
      int maxCount = numTuples * 10; // sanity check 
      int ct = 0;
      while (changed == true && ct <= maxCount)
      {
        ++ct; // k-means typically converges very quickly 
        UpdateCentroids(data); // no effect if fail 
        changed = UpdateClustering(data); // no effect if fail 
      }

      int[] result = new int[numTuples];
      Array.Copy(this.clustering, result, clustering.Length);
      return result;
    }

    private void InitRandom(double[][] data)
    {
      int numTuples = data.Length;

      int clusterID = 0;
      for (int i = 0; i < numTuples; ++i)
      {
        clustering[i] = clusterID++;
        if (clusterID == numClusters)
          clusterID = 0;
      }
      for (int i = 0; i < numTuples; ++i)
      {
        int r = rnd.Next(i, clustering.Length);
        int tmp = clustering[r];
        clustering[r] = clustering[i];
        clustering[i] = tmp;
      }
    }

    private void UpdateCentroids(double[][] data)
    {
      int[] clusterCounts = new int[numClusters];
      for (int i = 0; i < data.Length; ++i)
      {
        int clusterID = clustering[i];
        ++clusterCounts[clusterID];
      }

      // zero-out this.centroids so it can be used as scratch 
      for (int k = 0; k < centroids.Length; ++k)
        for (int j = 0; j < centroids[k].Length; ++j)
          centroids[k][j] = 0.0;

      for (int i = 0; i < data.Length; ++i)
      {
        int clusterID = clustering[i];
        for (int j = 0; j < data[i].Length; ++j)
          centroids[clusterID][j] += data[i][j]; // accumulate sum 
      }

      for (int k = 0; k < centroids.Length; ++k)
        for (int j = 0; j < centroids[k].Length; ++j)
          centroids[k][j] /= clusterCounts[k]; // danger? 
    }

    private bool UpdateClustering(double[][] data)
    {
      // (re)assign each tuple to a cluster (closest centroid) 
      // returns false if no tuple assignments change OR 
      // if the reassignment would result in a clustering where 
      // one or more clusters have no tuples. 

      bool changed = false; // did any tuple change cluster? 

      int[] newClustering = new int[clustering.Length]; // proposed result 
      Array.Copy(clustering, newClustering, clustering.Length);

      double[] distances = new double[numClusters]; // from tuple to centroids

      for (int i = 0; i < data.Length; ++i) // walk through each tuple 
      {
        for (int k = 0; k < numClusters; ++k)
          distances[k] = Distance(data[i], centroids[k]);

        int newClusterID = MinIndex(distances); // find closest centroid 
        if (newClusterID != newClustering[i])
        {
          changed = true; // note a new clustering 
          newClustering[i] = newClusterID; // accept update 
        }
      }

      if (changed == false)
        return false; // no change so bail 

      // check proposed clustering cluster counts 
      int[] clusterCounts = new int[numClusters];
      for (int i = 0; i < data.Length; ++i)
      {
        int clusterID = newClustering[i];
        ++clusterCounts[clusterID];
      }

      for (int k = 0; k < numClusters; ++k)
        if (clusterCounts[k] == 0)
          return false; // bad clustering 

      Array.Copy(newClustering, clustering, newClustering.Length); // update 
      return true; // good clustering and at least one change 
    }

    // Euclidean distance between two vectors for UpdateClustering() 
    private static double Distance(double[] tuple, double[] centroid)
    {
      double sumSquaredDiffs = 0.0;
      for (int j = 0; j < tuple.Length; ++j)
        sumSquaredDiffs += (tuple[j] - centroid[j]) * (tuple[j] - centroid[j]);
      return Math.Sqrt(sumSquaredDiffs);
    }

    // helper for UpdateClustering() to find closest centroid 
    private static int MinIndex(double[] distances)
    {
      int indexOfMin = 0;
      double smallDist = distances[0];
      for (int k = 1; k < distances.Length; ++k)
      {
        if (distances[k] < smallDist)
        {
          smallDist = distances[k];
          indexOfMin = k;
        }
      }
      return indexOfMin;
    }
  }
}

运行结果如下：

参考资料

• 基于 Apache Mahout 构建社会化推荐引擎
• 探索推荐引擎内部的秘密，第 1 部分: 推荐引擎初探
• 探索推荐引擎内部的秘密，第 2 部分: 深入推荐引擎相关算法 - 协同过滤
• 探索推荐引擎内部的秘密，第 3 部分: 深入推荐引擎相关算法 - 聚类
• 浅谈Kmeans聚类
• Mahout学习——K-Means Clustering
• 基本Kmeans算法介绍及其实现
• 算法杂货铺——k均值聚类(K-means)
• K-Means Data Clustering Using C#
• K-Means Clustering Used in Intention Based Scoring Projects
• Multi-Threaded K-Means Clustering in .NET 4.0
• CS229 Lecture notes -- Andrew Ng
• Cluster analysis
• k-means clustering
• K-means++ clustering
• Lloyd's algorithm
• NP-hard
• Voronoi diagram
• K-Means 算法
• A Tutorial on Clustering Algorithms
• Unsupervised learning or Clustering – K-means Gaussian mixture models

本篇文章《K-Means 聚类算法》由 Dennis Gao 发表自博客园个人博客，未经作者本人同意禁止以任何的形式转载，任何自动的或人为的爬虫转载行为均为耍流氓。