K-means算法和矢量量化

语音信号的数字处理课程作业——矢量量化。这里采用了K-means算法，即假设量化种类是已知的，当然也可以采用LBG算法等，不过K-means比较简单。矢量是二维的，可以在平面上清楚的表示出来。

1. 算法描述

本次实验选择了K-means算法对数据进行矢量量化。算法主要包括以下几个步骤

• 初始化：载入训练数据，确定初始码本中心（4个）；
• 最近邻分类：对训练数据计算距离（此处采用欧式距离），按照距离最小分类；
• 码本更新：重新生成包腔对应的质心；
• 重复分类和码本更新步骤，知道达到最大迭代次数或满足一定停止准则；
• 利用上述步骤得到的码本对测试数据进行矢量量化，并求最小均方误差。

本实验准备使用MATLAB软件完成矢量量化任务，具体步骤实现如下

1. 将training.dat和to_be_quantized.dat置于当前工作文件夹内，采用load命令载入training.dat 。
2. 采用合适的规则选取初始的码本中心。如图 1所示。

图 1 码本中心选择

1. 计算训练数据和每一码本中心之间的距离。
2. 采用最近邻准则进行分类。
3. 重新计算质心，计算公式如下所示。
4. 重复3～5，直到满足最大迭代次数或是两次迭代结果没有发生改变时，此时结果为训练结果。
5. 利用训练结果对to_be_quantized.dat进行矢量量化。

2. 代码

MATLAB代码如下

 %% training
 load('training.dat');
 scatter(training(:,),training(:,));
 %初始中心选取
 x_max = max(training(:,));
 x_min = min(training(:,));
 y_max = max(training(:,));
 y_min = min(training(:,));
 z1 = [(*x_min+x_max)/ (*y_min+y_max)/];
 z2 = [(*x_max+x_min)/ (*y_min+y_max)/];
 z3 = [(*x_min+x_max)/ (*y_max+y_min)/];
 z4 = [(*x_max+x_min)/ (*y_max+y_min)/];
 z = [z1;z2;z3;z4];
 hold on;
 scatter(z(:,),z(:,));
 legend('训练数据','码本');grid on;
 hold off;
 for k = :
     %码本分类，欧式距离
     distancetoz1 = (training - repmat(z1,size(training,),)).^;
     distancetoz1 = sum(distancetoz1,);
     distancetoz2 = (training - repmat(z2,size(training,),)).^;
     distancetoz2 = sum(distancetoz2,);
     distancetoz3 = (training - repmat(z3,size(training,),)).^;
     distancetoz3 = sum(distancetoz3,);
     distancetoz4 = (training - repmat(z4,size(training,),)).^;
     distancetoz4 = sum(distancetoz4,);
     distance = [distancetoz1 distancetoz2 distancetoz3 distancetoz4];
     % 分类
     if(classification == (distance == repmat(min(distance,[],),,)))
         error = mean(min(distance,[],));
         break;      %如果两次迭代之间没有变化，结束迭代
     end;
     classification = (distance == repmat(min(distance,[],),,));
     c1 = training(classification(:,),:);
     c2 = training(classification(:,),:);
     c3 = training(classification(:,),:);
     c4 = training(classification(:,),:);
     figure;scatter(c1(:,),c1(:,));hold on;scatter(c2(:,),c2(:,));
     scatter(c3(:,),c3(:,));scatter(c4(:,),c4(:,));
     legend('类型1','类型2','类型3','类型4');grid on;hold off;
     % 码本更新
     z1 = mean(c1);
     z2 = mean(c2);
     z3 = mean(c3);
     z4 = mean(c4);
     z = [z1;z2;z3;z4];
 end
 %% Test
 load('to_be_quantized.dat')
 distancetoz1 = (to_be_quantized - repmat(z1,size(to_be_quantized,),)).^;
 distancetoz1 = sum(distancetoz1,);
 distancetoz2 = (to_be_quantized - repmat(z2,size(to_be_quantized,),)).^;
 distancetoz2 = sum(distancetoz2,);
 distancetoz3 = (to_be_quantized - repmat(z3,size(to_be_quantized,),)).^;
 distancetoz3 = sum(distancetoz3,);
 distancetoz4 = (to_be_quantized - repmat(z4,size(to_be_quantized,),)).^;
 distancetoz4 = sum(distancetoz4,);
 distance = [distancetoz1 distancetoz2 distancetoz3 distancetoz4];
 testerror = mean(min(distance,[],));

 classification = (distance == repmat(min(distance,[],),,));
 c1 = to_be_quantized(classification(:,),:);
 c2 = to_be_quantized(classification(:,),:);
 c3 = to_be_quantized(classification(:,),:);
 c4 = to_be_quantized(classification(:,),:);
 figure;scatter(c1(:,),c1(:,));hold on;scatter(c2(:,),c2(:,));
 scatter(c3(:,),c3(:,));scatter(c4(:,),c4(:,));
 legend('类型1','类型2','类型3','类型4');grid on;hold off;

3. 实验结果

图 2 训练码本分布

图 3第一次迭代结果                   图 4第四次迭代结果

图 5第八次迭代结果                  图 6第九次迭代结果

图 2展示了训练数据的分布，图 3～6是迭代过程中分类的变化情况，迭代完成后的码本为

• Z1 = [1.62060631541935 -0.108624145483871]
• Z2 = [7.96065094375000 -0.999061308437500]
• Z3 = [1.72161941468750 6.82121444062500]
• Z4 = [4.43652765757576 2.18874305151515]

4. 实验数据

training.dat

   8.4416189e+000 -7.9885975e-001
   1.1480908e+000  7.8735044e+000
   7.7380144e+000 -1.2165061e+000
   8.9727144e-001  7.3962468e+000
   7.5343823e+000 -1.1424504e+000
  -6.9234039e-001 -1.7096610e+000
   7.6418740e+000 -1.3563792e+000
   3.1091418e+000  6.3850541e+000
   2.3482174e+000  4.7553506e-001
  -1.3840364e+000 -2.5480394e+000
   8.2008897e+000 -1.1448387e+000
  -1.1392497e+000 -2.0809884e+000
   3.7970116e+000  1.6906469e+000
   3.4484200e+000  1.3980911e+000
   2.5701485e+000  5.3755044e+000
   8.3899076e+000 -6.6675309e-001
   2.0146545e+000  5.6984592e+000
   1.8853328e+000  5.2762628e-001
   5.6781432e+000  3.2588691e+000
   1.0102480e+000  5.8167707e+000
   7.7302763e+000 -1.2030348e+000
   4.2118845e+000  1.6527181e+000
   4.3920049e-001  6.7168970e+000
   8.1934984e-001 -5.1917945e-001
   4.3708769e+000  2.1613573e+000
   1.8569681e+000  4.8380565e+000
   3.4732504e+000  1.7953635e+000
   7.5822756e+000 -1.1521814e+000
   2.6434078e+000  6.3295690e+000
   1.9968582e+000  7.3529314e+000
   4.0833513e+000  1.4936002e+000
   3.6767894e+000  6.7446912e+000
   1.3524515e+000  6.8177858e+000
   3.9711504e+000  1.5452503e+000
   1.5594711e+000  6.3885281e+000
   3.4692089e+000  1.7118124e+000
   5.2575491e+000  2.5601553e+000
   7.8827882e+000 -6.8867840e-001
   4.8176593e+000  2.1684005e+000
   2.7402486e+000  8.3320174e+000
   2.2549011e+000  3.9393641e-001
   8.0840542e+000 -7.3155184e-001
   8.8753667e-001  6.1607892e+000
   1.8067727e+000 -2.1099454e-001
   6.8650914e+000  4.4228389e+000
   6.4174056e+000  3.7590081e+000
   4.0933273e+000  1.3598676e+000
   2.2882999e+000  5.1876795e-001
   7.9225523e+000 -1.1725456e+000
   4.3561335e+000  1.8976163e+000
   8.3279098e+000 -1.0232899e+000
   6.2551331e+000  3.3449949e+000
   3.1276024e+000  7.8463356e-001
   6.5241605e+000  3.4561490e+000
   4.1588140e-001  6.4974858e+000
   2.7379263e+000  6.4746080e+000
   7.2185639e+000 -1.3525589e+000
   7.5424890e+000 -1.5317814e+000
   3.7468423e+000  1.6110753e+000
   8.8708536e+000 -5.6439331e-001
   7.6960713e+000 -1.1960633e+000
   7.5979552e+000 -1.1469059e+000
   2.8220978e+000  1.0360184e+000
   3.8165165e+000  1.6082223e+000
   6.6799248e-002 -1.2910367e+000
   2.3054028e+000  2.8450986e-001
   4.2788715e+000  5.1995858e+000
   3.0006534e+000  9.1250414e-001
   7.6051326e+000 -1.1005476e+000
   2.5331653e+000  9.7428007e-001
   1.0743104e+000  6.0859296e+000
   6.7237149e-001  8.6117274e+000
   2.4333003e+000  7.1421389e-001
   1.7723473e+000  7.1841833e+000
   3.5762796e+000  1.5348648e+000
   2.7863558e+000  7.3565043e-001
   8.0284284e+000 -7.9636983e-001
   8.4672682e+000 -8.2062254e-001
   2.3519727e+000  8.1632796e-001
   7.4240720e+000  4.1800229e+000
   1.9724319e+000  4.4328699e-001
   7.7622621e+000 -1.3506605e+000
   2.3793018e+000 -4.3107386e-001
   3.2455220e+000  1.2697488e+000
   1.3644859e+000  5.9712644e+000
   5.4815655e+000  2.6608754e+000
  -1.2002073e+000 -2.1765731e+000
  -3.5558595e-001  6.4387512e+000
   3.9418185e+000  1.9858047e+000
   1.0533626e+000 -7.9068285e-001
   1.9560213e+000  6.2001316e+000
   7.5555203e+000 -1.2087337e+000
   1.7851705e+000  7.0073148e+000
   2.2736274e+000  7.9336349e-001
   7.6615799e+000 -1.0445564e+000
   2.7181608e+000  4.7615418e-001
   1.8291149e+000 -6.7261971e-001
   7.8640867e+000 -1.4296092e+000
   2.6362814e+000  5.8303048e-001
   3.7771102e+000  1.2928196e+000
   7.5360359e+000 -9.7942712e-001
   4.0257498e+000  1.2217666e+000
   8.4500853e+000 -7.6599648e-001
   3.0488646e+000  6.2159289e+000
   2.0954150e+000  2.5848825e-001
   1.6592148e+000  7.5650162e+000
   3.5535363e+000  1.3326217e+000
   4.3388636e+000  2.1235893e+000
   3.1233524e+000  1.3971470e+000
   7.6317385e+000 -1.0744610e+000
   8.5028402e-001 -3.2822876e-001
   8.6903131e+000 -2.6843242e-001
   4.4418011e+000  2.5676053e+000
   2.5119872e+000 -1.0521242e-001
   1.9613752e+000  7.0072931e+000
   3.2607143e+000  1.5432286e+000
   3.2830401e+000  1.0228031e+000
   8.0201528e+000 -7.0827461e-001
   3.1597313e+000  7.6750043e+000
   9.0059933e+000 -9.6130246e-001
   1.1037820e+000 -1.2980812e-001
   1.5334911e+000  7.4282719e+000
   6.0948533e-001  6.3861341e+000
   4.0065706e-001 -1.1015776e+000
   2.3451558e+000  8.6384057e+000
   1.4490876e+000  8.6646066e+000
   8.0421821e+000 -8.1100509e-001
   8.0175747e+000 -5.6119093e-001

to_be_quantized

   3.7682247e+000  8.3609865e-001
   2.6963398e+000  6.5766226e-001
   3.3438207e+000  1.2495321e+000
   1.3646195e+000 -6.3947640e-001
   7.8227583e+000 -8.8616996e-001
   1.3532508e+000  7.6607304e+000
   2.2741739e+000  6.9387226e+000
   3.5361382e+000  5.9729821e+000
   8.0409138e+000 -1.1234886e+000
   7.9630460e+000 -1.3032200e+000
   2.3478158e+000  6.9759690e+000
   3.2632942e+000  1.5675470e+000
   1.5241488e+000  7.1053147e+000
   5.7320838e+000  3.4042655e+000
   2.3339411e+000  6.9428434e+000
   6.5330392e+000  3.4415860e+000
   3.1068803e+000  8.0080363e+000
   7.4078126e+000 -1.3416027e+000
   1.9925474e+000 -2.7782790e-001
   5.0187915e+000  2.7058427e+000
   2.6535497e-001 -1.2622069e+000
   1.4960584e+000  6.3355004e+000
   3.1933474e-001  7.1467466e+000
   8.2821020e+000 -9.5178778e-001
   2.5653586e+000  6.9836115e+000
   3.6937139e+000  1.1535671e+000
   8.5390043e+000 -5.0678923e-001
   7.5436898e-001 -6.7669379e-001
   2.1638213e+000  7.6142401e+000
   4.8522826e+000  2.7079076e+000
   5.4890641e+000  3.3875394e+000
   4.2525899e+000  1.8861744e+000
   8.4088615e+000 -1.1920963e+000
   5.5396960e+000  2.9680110e+000
   3.3334381e+000  1.4384861e+000
   3.5212919e+000  1.0327602e+000
   4.6303492e+000  2.1627805e+000
   3.9385929e+000  1.0010804e+000
   8.4553633e+000 -7.2297277e-001
   1.8111095e+000  7.6132396e+000
   1.1240984e+000 -2.7029879e-001
  -3.3840083e-002 -1.5590834e+000
   7.1674870e+000 -1.5449905e+000
   8.5103026e+000 -9.8820393e-001
   7.7529857e+000 -1.4787432e+000
   1.8704913e+000  6.9370116e+000
   6.0271939e+000  3.2118915e+000
   2.8287461e+000  7.3399383e+000
   4.1568876e+000  1.5631238e+000
   8.2187067e-001 -5.8546437e-001
   3.1084965e+000  5.3512449e+000
   4.1581386e+000  2.1763345e+000
   3.2267474e+000  1.4105815e+000
   8.1564752e-001  7.2540175e+000
   8.0241402e+000 -8.2411742e-001
   6.2773554e+000  3.1729045e+000
   8.5460058e+000 -1.0330056e+000
   8.6215210e+000 -7.4057378e-001
   7.4872291e+000 -1.0113921e+000
   3.3155133e+000  9.7636038e-001
   2.1051593e+000  3.4894654e-001
   3.6776134e+000  1.5387928e+000
   2.9009105e+000  5.6931589e+000
   8.0567164e+000 -1.0000803e+000