K-means算法和矢量量化

浏览数：24 / 时间：2015年06月08日

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

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 。
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

计算训练数据和每一码本中心之间的距离。
采用最近邻准则进行分类。
重新计算质心，计算公式如下所示。 $技术分享$
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。
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

 1 %% training
 2 load(‘training.dat‘);
 3 scatter(training(:,1),training(:,2));
 4 %初始中心选取
 5 x_max = max(training(:,1));
 6 x_min = min(training(:,1));
 7 y_max = max(training(:,2));
 8 y_min = min(training(:,2));
 9 z1 = [(3*x_min+x_max)/4 (3*y_min+y_max)/4];
10 z2 = [(3*x_max+x_min)/4 (3*y_min+y_max)/4];
11 z3 = [(3*x_min+x_max)/4 (3*y_max+y_min)/4];
12 z4 = [(3*x_max+x_min)/4 (3*y_max+y_min)/4];
13 z = [z1;z2;z3;z4];
14 hold on;
15 scatter(z(:,1),z(:,2));
16 legend(‘训练数据‘,‘码本‘);grid on;
17 hold off;
18 for k = 1:20
19     %码本分类，欧式距离
20     distancetoz1 = (training - repmat(z1,size(training,1),1)).^2;
21     distancetoz1 = sum(distancetoz1,2);
22     distancetoz2 = (training - repmat(z2,size(training,1),1)).^2;
23     distancetoz2 = sum(distancetoz2,2);
24     distancetoz3 = (training - repmat(z3,size(training,1),1)).^2;
25     distancetoz3 = sum(distancetoz3,2);
26     distancetoz4 = (training - repmat(z4,size(training,1),1)).^2;
27     distancetoz4 = sum(distancetoz4,2);
28     distance = [distancetoz1 distancetoz2 distancetoz3 distancetoz4];
29     % 分类
30     if(classification == (distance == repmat(min(distance,[],2),1,4)))
31         error = mean(min(distance,[],2));
32         break;      %如果两次迭代之间没有变化，结束迭代
33     end;
34     classification = (distance == repmat(min(distance,[],2),1,4));
35     c1 = training(classification(:,1),:);
36     c2 = training(classification(:,2),:);
37     c3 = training(classification(:,3),:);
38     c4 = training(classification(:,4),:);
39     figure;scatter(c1(:,1),c1(:,2));hold on;scatter(c2(:,1),c2(:,2));
40     scatter(c3(:,1),c3(:,2));scatter(c4(:,1),c4(:,2));
41     legend(‘类型1‘,‘类型2‘,‘类型3‘,‘类型4‘);grid on;hold off;
42     % 码本更新
43     z1 = mean(c1);
44     z2 = mean(c2);
45     z3 = mean(c3);
46     z4 = mean(c4);
47     z = [z1;z2;z3;z4];
48 end
49 %% Test
50 load(‘to_be_quantized.dat‘)
51 distancetoz1 = (to_be_quantized - repmat(z1,size(to_be_quantized,1),1)).^2;
52 distancetoz1 = sum(distancetoz1,2);
53 distancetoz2 = (to_be_quantized - repmat(z2,size(to_be_quantized,1),1)).^2;
54 distancetoz2 = sum(distancetoz2,2);
55 distancetoz3 = (to_be_quantized - repmat(z3,size(to_be_quantized,1),1)).^2;
56 distancetoz3 = sum(distancetoz3,2);
57 distancetoz4 = (to_be_quantized - repmat(z4,size(to_be_quantized,1),1)).^2;
58 distancetoz4 = sum(distancetoz4,2);
59 distance = [distancetoz1 distancetoz2 distancetoz3 distancetoz4];
60 testerror = mean(min(distance,[],2));
61 
62 classification = (distance == repmat(min(distance,[],2),1,4));
63 c1 = to_be_quantized(classification(:,1),:);
64 c2 = to_be_quantized(classification(:,2),:);
65 c3 = to_be_quantized(classification(:,3),:);
66 c4 = to_be_quantized(classification(:,4),:);
67 figure;scatter(c1(:,1),c1(:,2));hold on;scatter(c2(:,1),c2(:,2));
68 scatter(c3(:,1),c3(:,2));scatter(c4(:,1),c4(:,2));
69 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

  1   8.4416189e+000 -7.9885975e-001
  2   1.1480908e+000  7.8735044e+000
  3   7.7380144e+000 -1.2165061e+000
  4   8.9727144e-001  7.3962468e+000
  5   7.5343823e+000 -1.1424504e+000
  6  -6.9234039e-001 -1.7096610e+000
  7   7.6418740e+000 -1.3563792e+000
  8   3.1091418e+000  6.3850541e+000
  9   2.3482174e+000  4.7553506e-001
 10  -1.3840364e+000 -2.5480394e+000
 11   8.2008897e+000 -1.1448387e+000
 12  -1.1392497e+000 -2.0809884e+000
 13   3.7970116e+000  1.6906469e+000
 14   3.4484200e+000  1.3980911e+000
 15   2.5701485e+000  5.3755044e+000
 16   8.3899076e+000 -6.6675309e-001
 17   2.0146545e+000  5.6984592e+000
 18   1.8853328e+000  5.2762628e-001
 19   5.6781432e+000  3.2588691e+000
 20   1.0102480e+000  5.8167707e+000
 21   7.7302763e+000 -1.2030348e+000
 22   4.2118845e+000  1.6527181e+000
 23   4.3920049e-001  6.7168970e+000
 24   8.1934984e-001 -5.1917945e-001
 25   4.3708769e+000  2.1613573e+000
 26   1.8569681e+000  4.8380565e+000
 27   3.4732504e+000  1.7953635e+000
 28   7.5822756e+000 -1.1521814e+000
 29   2.6434078e+000  6.3295690e+000
 30   1.9968582e+000  7.3529314e+000
 31   4.0833513e+000  1.4936002e+000
 32   3.6767894e+000  6.7446912e+000
 33   1.3524515e+000  6.8177858e+000
 34   3.9711504e+000  1.5452503e+000
 35   1.5594711e+000  6.3885281e+000
 36   3.4692089e+000  1.7118124e+000
 37   5.2575491e+000  2.5601553e+000
 38   7.8827882e+000 -6.8867840e-001
 39   4.8176593e+000  2.1684005e+000
 40   2.7402486e+000  8.3320174e+000
 41   2.2549011e+000  3.9393641e-001
 42   8.0840542e+000 -7.3155184e-001
 43   8.8753667e-001  6.1607892e+000
 44   1.8067727e+000 -2.1099454e-001
 45   6.8650914e+000  4.4228389e+000
 46   6.4174056e+000  3.7590081e+000
 47   4.0933273e+000  1.3598676e+000
 48   2.2882999e+000  5.1876795e-001
 49   7.9225523e+000 -1.1725456e+000
 50   4.3561335e+000  1.8976163e+000
 51   8.3279098e+000 -1.0232899e+000
 52   6.2551331e+000  3.3449949e+000
 53   3.1276024e+000  7.8463356e-001
 54   6.5241605e+000  3.4561490e+000
 55   4.1588140e-001  6.4974858e+000
 56   2.7379263e+000  6.4746080e+000
 57   7.2185639e+000 -1.3525589e+000
 58   7.5424890e+000 -1.5317814e+000
 59   3.7468423e+000  1.6110753e+000
 60   8.8708536e+000 -5.6439331e-001
 61   7.6960713e+000 -1.1960633e+000
 62   7.5979552e+000 -1.1469059e+000
 63   2.8220978e+000  1.0360184e+000
 64   3.8165165e+000  1.6082223e+000
 65   6.6799248e-002 -1.2910367e+000
 66   2.3054028e+000  2.8450986e-001
 67   4.2788715e+000  5.1995858e+000
 68   3.0006534e+000  9.1250414e-001
 69   7.6051326e+000 -1.1005476e+000
 70   2.5331653e+000  9.7428007e-001
 71   1.0743104e+000  6.0859296e+000
 72   6.7237149e-001  8.6117274e+000
 73   2.4333003e+000  7.1421389e-001
 74   1.7723473e+000  7.1841833e+000
 75   3.5762796e+000  1.5348648e+000
 76   2.7863558e+000  7.3565043e-001
 77   8.0284284e+000 -7.9636983e-001
 78   8.4672682e+000 -8.2062254e-001
 79   2.3519727e+000  8.1632796e-001
 80   7.4240720e+000  4.1800229e+000
 81   1.9724319e+000  4.4328699e-001
 82   7.7622621e+000 -1.3506605e+000
 83   2.3793018e+000 -4.3107386e-001
 84   3.2455220e+000  1.2697488e+000
 85   1.3644859e+000  5.9712644e+000
 86   5.4815655e+000  2.6608754e+000
 87  -1.2002073e+000 -2.1765731e+000
 88  -3.5558595e-001  6.4387512e+000
 89   3.9418185e+000  1.9858047e+000
 90   1.0533626e+000 -7.9068285e-001
 91   1.9560213e+000  6.2001316e+000
 92   7.5555203e+000 -1.2087337e+000
 93   1.7851705e+000  7.0073148e+000
 94   2.2736274e+000  7.9336349e-001
 95   7.6615799e+000 -1.0445564e+000
 96   2.7181608e+000  4.7615418e-001
 97   1.8291149e+000 -6.7261971e-001
 98   7.8640867e+000 -1.4296092e+000
 99   2.6362814e+000  5.8303048e-001
100   3.7771102e+000  1.2928196e+000
101   7.5360359e+000 -9.7942712e-001
102   4.0257498e+000  1.2217666e+000
103   8.4500853e+000 -7.6599648e-001
104   3.0488646e+000  6.2159289e+000
105   2.0954150e+000  2.5848825e-001
106   1.6592148e+000  7.5650162e+000
107   3.5535363e+000  1.3326217e+000
108   4.3388636e+000  2.1235893e+000
109   3.1233524e+000  1.3971470e+000
110   7.6317385e+000 -1.0744610e+000
111   8.5028402e-001 -3.2822876e-001
112   8.6903131e+000 -2.6843242e-001
113   4.4418011e+000  2.5676053e+000
114   2.5119872e+000 -1.0521242e-001
115   1.9613752e+000  7.0072931e+000
116   3.2607143e+000  1.5432286e+000
117   3.2830401e+000  1.0228031e+000
118   8.0201528e+000 -7.0827461e-001
119   3.1597313e+000  7.6750043e+000
120   9.0059933e+000 -9.6130246e-001
121   1.1037820e+000 -1.2980812e-001
122   1.5334911e+000  7.4282719e+000
123   6.0948533e-001  6.3861341e+000
124   4.0065706e-001 -1.1015776e+000
125   2.3451558e+000  8.6384057e+000
126   1.4490876e+000  8.6646066e+000
127   8.0421821e+000 -8.1100509e-001
128   8.0175747e+000 -5.6119093e-001

View Code

to_be_quantized

 1   3.7682247e+000  8.3609865e-001
 2   2.6963398e+000  6.5766226e-001
 3   3.3438207e+000  1.2495321e+000
 4   1.3646195e+000 -6.3947640e-001
 5   7.8227583e+000 -8.8616996e-001
 6   1.3532508e+000  7.6607304e+000
 7   2.2741739e+000  6.9387226e+000
 8   3.5361382e+000  5.9729821e+000
 9   8.0409138e+000 -1.1234886e+000
10   7.9630460e+000 -1.3032200e+000
11   2.3478158e+000  6.9759690e+000
12   3.2632942e+000  1.5675470e+000
13   1.5241488e+000  7.1053147e+000
14   5.7320838e+000  3.4042655e+000
15   2.3339411e+000  6.9428434e+000
16   6.5330392e+000  3.4415860e+000
17   3.1068803e+000  8.0080363e+000
18   7.4078126e+000 -1.3416027e+000
19   1.9925474e+000 -2.7782790e-001
20   5.0187915e+000  2.7058427e+000
21   2.6535497e-001 -1.2622069e+000
22   1.4960584e+000  6.3355004e+000
23   3.1933474e-001  7.1467466e+000
24   8.2821020e+000 -9.5178778e-001
25   2.5653586e+000  6.9836115e+000
26   3.6937139e+000  1.1535671e+000
27   8.5390043e+000 -5.0678923e-001
28   7.5436898e-001 -6.7669379e-001
29   2.1638213e+000  7.6142401e+000
30   4.8522826e+000  2.7079076e+000
31   5.4890641e+000  3.3875394e+000
32   4.2525899e+000  1.8861744e+000
33   8.4088615e+000 -1.1920963e+000
34   5.5396960e+000  2.9680110e+000
35   3.3334381e+000  1.4384861e+000
36   3.5212919e+000  1.0327602e+000
37   4.6303492e+000  2.1627805e+000
38   3.9385929e+000  1.0010804e+000
39   8.4553633e+000 -7.2297277e-001
40   1.8111095e+000  7.6132396e+000
41   1.1240984e+000 -2.7029879e-001
42  -3.3840083e-002 -1.5590834e+000
43   7.1674870e+000 -1.5449905e+000
44   8.5103026e+000 -9.8820393e-001
45   7.7529857e+000 -1.4787432e+000
46   1.8704913e+000  6.9370116e+000
47   6.0271939e+000  3.2118915e+000
48   2.8287461e+000  7.3399383e+000
49   4.1568876e+000  1.5631238e+000
50   8.2187067e-001 -5.8546437e-001
51   3.1084965e+000  5.3512449e+000
52   4.1581386e+000  2.1763345e+000
53   3.2267474e+000  1.4105815e+000
54   8.1564752e-001  7.2540175e+000
55   8.0241402e+000 -8.2411742e-001
56   6.2773554e+000  3.1729045e+000
57   8.5460058e+000 -1.0330056e+000
58   8.6215210e+000 -7.4057378e-001
59   7.4872291e+000 -1.0113921e+000
60   3.3155133e+000  9.7636038e-001
61   2.1051593e+000  3.4894654e-001
62   3.6776134e+000  1.5387928e+000
63   2.9009105e+000  5.6931589e+000
64   8.0567164e+000 -1.0000803e+000