机器学习（三）k均值聚类

k均值聚类

原文地址：/article/7649351.html

作者：hjimce

高斯混合模型和k均值聚类是聚类算法中的两种比较常用而简单的算法，这里先介绍k均值聚类算法。

一、K-means算法理论简介

K-means算法是硬聚类算法，是典型的基于原型的目标函数聚类方法的代表，它是数据点到原型的某种距离作为优化的目标函数，利用函数求极值的方法得到迭代运算的调整规则。K-means算法以欧式距离作为相似度测度，它是求对应某一初始聚类中心向量V最优分类，使得评价指标J最小。算法采用误差平方和准则函数作为聚类准则函数，总之就是要使得下面的公式最小化：



算法过程如下：

1）从N个文档随机选取K个文档作为质心

2）对剩余的每个文档测量其到每个质心的距离，并把它归到最近的质心的类

3）重新计算已经得到的各个类的质心

4）迭代2～3步直至新的质心与原质心相等或小于指定阈值，算法结束

二、K-means算法实现

K均值聚类算法实现分为四个步骤，设数据集为二维data，用matlab把数据绘制出来，如下所示：

绘制代码：



绘制结果：



现在假设要把该数据集分为4类，算法步骤如下：

1) 初始化聚类中心

由于聚类个数选择4，因此我们用matlab随机生成4个不重复的整数，且大小不超过数据点的个数，得到初始聚类中心A,B,C,D。

2) 计算每个点分别到4个聚类中心的距离，找出最小的那个。假设点p为数据集中的点，求出A、B、C、D中距离p点最近的那个点，设B距离P最近，那么就把p点聚类为B类。

3) 更新聚类中心。根据步骤2可得数据集的聚类结果，根据聚类结果，计算每个类的重心位置，作为更新的聚类中心。然后返回步骤2，重新进行聚类，如此循环步骤2与步骤3，直到迭代收敛。

最后贴一下代码：

close all;
clear;
clc;
% %生成高斯分布随机数1
% mu = [2 3];
% SIGMA = [1 0; 0 2];
% r1 = mvnrnd(mu,SIGMA,100);
% plot(r1(:,1),r1(:,2),'r+');
% hold on;
% %生成高斯分布随机数1
% mu = [7 8];
% SIGMA = [ 1 0; 0 2];
% r2 = mvnrnd(mu,SIGMA,100);
% plot(r2(:,1),r2(:,2),'*')
%
% data=[r1;r2]

data=importdata('data.txt');
% 算法流程
figure(1);
plot(data(:,1),data(:,2),'*');
figure(2);
[m n]=size(data);
%聚类个数为4，则4个不重复的整数
p=randperm(m);
kn=4;
d=p(1:kn);
center(1:kn,:)=data(d(:),:);
flag=zeros(1,m);
%计算每个点到4个质心点的最近的那个点
it=1;
while(it<30 for="" ii="1:m;" flag="" ii="" -1="" mindist="inf;" for="" jj="1:kn;" dst="norm(data(ii,:)-center(jj,:));" if="" mindist="">dst;
mindist=dst;
flag(ii)=jj;
end
end
end
%更新聚类中心
center=zeros(size(center));
countflag=zeros(1,kn);
for ii=1:m
for jj=1:kn
if(flag(ii)==jj)
center(jj,:)=center(jj,:)+data(ii,:);
countflag(jj)=countflag(jj)+1;
end
end
end
for jj=1:kn
center(jj,:)=center(jj,:)./countflag(jj);
end
it=it+1;
end
hold on;
for i=1:m;
if(flag(i)==1);
plot(data(i,1),data(i,2),'.y');
elseif flag(i)==2
plot(data(i,1),data(i,2),'.b');
elseif (flag(i)==3)
plot(data(i,1),data(i,2),'.k');
elseif (flag(i)==4)
plot(data(i,1),data(i,2),'.r');
end
end</30>

聚类结果如下：

opencv版kmeans：
声明：

enum
{
KMEANS_RANDOM_CENTERS=0, // Chooses random centers for k-Means initialization
KMEANS_PP_CENTERS=2,     // Uses k-Means++ algorithm for initialization
KMEANS_USE_INITIAL_LABELS=1 // Uses the user-provided labels for K-Means initialization
};
//! clusters the input data using k-Means algorithm
CV_EXPORTS_W double kmeans( InputArray data, int K, CV_OUT InputOutputArray bestLabels,
TermCriteria criteria, int attempts,
int flags, OutputArray centers=noArray() );

实现函数：

double kmeans( const Mat& data, int K, Mat& best_labels,
TermCriteria criteria, int attempts,
int flags, Mat* _centers )
{
const int SPP_TRIALS = 3;
int N = data.rows > 1 ? data.rows : data.cols;
int dims = (data.rows > 1 ? data.cols : 1)*data.channels();
int type = data.depth();
bool simd = checkHardwareSupport(CV_CPU_SSE);
attempts = std::max(attempts, 1);
CV_Assert( type == CV_32F && K > 0 );
Mat _labels;
if( flags & CV_KMEANS_USE_INITIAL_LABELS )
{
CV_Assert( (best_labels.cols == 1 || best_labels.rows == 1) &&
best_labels.cols*best_labels.rows == N &&
best_labels.type() == CV_32S &&
best_labels.isContinuous());
best_labels.copyTo(_labels);
}
else
{
if( !((best_labels.cols == 1 || best_labels.rows == 1) &&
best_labels.cols*best_labels.rows == N &&
best_labels.type() == CV_32S &&
best_labels.isContinuous()))
best_labels.create(N, 1, CV_32S);
_labels.create(best_labels.size(), best_labels.type());
}
int* labels = _labels.ptr<int>();
Mat centers(K, dims, type), old_centers(K, dims, type);
vector<int> counters(K);
vector<Vec2f> _box(dims);
Vec2f* box = &_box[0];
double best_compactness = DBL_MAX, compactness = 0;
RNG& rng = theRNG();
int a, iter, i, j, k;
if( criteria.type & TermCriteria::EPS )
criteria.epsilon = std::max(criteria.epsilon, 0.);
else
criteria.epsilon = FLT_EPSILON;
criteria.epsilon *= criteria.epsilon;
if( criteria.type & TermCriteria::COUNT )
criteria.maxCount = std::min(std::max(criteria.maxCount, 2), 100);
else
criteria.maxCount = 100;
if( K == 1 )
{
attempts = 1;
criteria.maxCount = 2;
}
const float* sample = data.ptr<float>(0);
for( j = 0; j < dims; j++ )
box[j] = Vec2f(sample[j], sample[j]);
for( i = 1; i < N; i++ )
{
sample = data.ptr<float>(i);
for( j = 0; j < dims; j++ )
{
float v = sample[j];
box[j][0] = std::min(box[j][0], v);
box[j][1] = std::max(box[j][1], v);
}
}
for( a = 0; a < attempts; a++ )
{
double max_center_shift = DBL_MAX;
for( iter = 0; iter < criteria.maxCount && max_center_shift > criteria.epsilon; iter++ )
{
swap(centers, old_centers);
if( iter == 0 && (a > 0 || !(flags & KMEANS_USE_INITIAL_LABELS)) )
{
if( flags & KMEANS_PP_CENTERS )
generateCentersPP(data, centers, K, rng, SPP_TRIALS);
else
{
for( k = 0; k < K; k++ )
generateRandomCenter(_box, centers.ptr<float>(k), rng);
}
}
else
{
if( iter == 0 && a == 0 && (flags & KMEANS_USE_INITIAL_LABELS) )
{
for( i = 0; i < N; i++ )
CV_Assert( (unsigned)labels[i] < (unsigned)K );
}

// compute centers
centers = Scalar(0);
for( k = 0; k < K; k++ )
counters[k] = 0;
for( i = 0; i < N; i++ )
{
sample = data.ptr<float>(i);
k = labels[i];
float* center = centers.ptr<float>(k);
for( j = 0; j <= dims - 4; j += 4 )
{
float t0 = center[j] + sample[j];
float t1 = center[j+1] + sample[j+1];
center[j] = t0;
center[j+1] = t1;
t0 = center[j+2] + sample[j+2];
t1 = center[j+3] + sample[j+3];
center[j+2] = t0;
center[j+3] = t1;
}
for( ; j < dims; j++ )
center[j] += sample[j];
counters[k]++;
}
if( iter > 0 )
max_center_shift = 0;
for( k = 0; k < K; k++ )
{
float* center = centers.ptr<float>(k);
if( counters[k] != 0 )
{
float scale = 1.f/counters[k];
for( j = 0; j < dims; j++ )
center[j] *= scale;
}
else
generateRandomCenter(_box, center, rng);

if( iter > 0 )
{
double dist = 0;
const float* old_center = old_centers.ptr<float>(k);
for( j = 0; j < dims; j++ )
{
double t = center[j] - old_center[j];
dist += t*t;
}
max_center_shift = std::max(max_center_shift, dist);
}
}
}
// assign labels
compactness = 0;
for( i = 0; i < N; i++ )
{
sample = data.ptr<float>(i);
int k_best = 0;
double min_dist = DBL_MAX;
for( k = 0; k < K; k++ )
{
const float* center = centers.ptr<float>(k);
double dist = distance(sample, center, dims, simd);
if( min_dist > dist )
{
min_dist = dist;
k_best = k;
}
}
compactness += min_dist;
labels[i] = k_best;
}
}
if( compactness < best_compactness )
{
best_compactness = compactness;
if( _centers )
centers.copyTo(*_centers);
_labels.copyTo(best_labels);
}
}
return best_compactness;
}
}

调用方法：

const int kMeansItCount = 10;  //迭代次数
const int kMeansType = cv::KMEANS_PP_CENTERS; //Use kmeans++ center initialization by Arthur and Vassilvitskii

cv::Mat bgdLabels, fgdLabels; //记录背景和前景的像素样本集中每个像素对应GMM的哪个高斯模型，论文中的kn
//kmeans中参数_bgdSamples为：每行一个样本
//kmeans的输出为bgdLabels，里面保存的是输入样本集中每一个样本对应的类标签（样本聚为componentsCount类后）
kmeans( _fgdSamples, GMM::componentsCount, fgdLabels,
cv::TermCriteria( CV_TERMCRIT_ITER, kMeansItCount, 0.0), 0, kMeansType );

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