UFLDL教程之（三）PCA and Whitening exercise

系统管理员 2021-07-16 17:30 407阅读 0赞

Exercise:PCA and Whitening

第0步：数据准备

UFLDL下载的文件中，包含数据集IMAGES_RAW，它是一个512*512*10的矩阵，也就是10幅512*512的图像

（a）载入数据

利用sampleIMAGESRAW函数，从IMAGES_RAW中提取numPatches个图像块儿，每个图像块儿大小为patchSize，并将提取到的图像块儿按列存放，分别存放在在矩阵patches的每一列中，即patches(:,i)存放的是第i个图像块儿的所有像素值

（b）数据去均值化处理

将每一个图像块儿的所有像素值都减去该图像块儿的平均像素值，实现数据的去均值化

下图是显示的随机选取的图像块儿

271503287651852.jpg

第一步：执行PCA

该部分分为两部分

（1）进行PCA计算，这里仅仅对数据x进行旋转得到xrot，而不进行主成分的提取

具体地：

①计算数据x的协方差矩阵sigma

  1.          ![271519180789570.png][]

②对sigma进行特征分解，利用matlab的eig函数，从而得到sigma的特征向量构成的矩阵U

[U,S,V]=eig(sigma);

U=[u1,…,ui,…,un]，它的每一列分别是sigma的特征向量，n是输入数据的特征维数

S=diag([λ1,…λi,…,λn])是由sigma的特征值作为对角元素的对角阵，ui和λi相对应；

为了后续的计算，这里要将U的各列次序进行调换，使得调换后的各列所对应的特征值大小依次递减;

调换后的矩阵仍记作U，相应的特征值对角阵仍即为S，即：

U=[u1,…,ui,…,un]，S=diag([λ1,…λi,…,λn])，满足：λ1>=…>=λi>=…>=λn

③利用矩阵U对数据x进行旋转，得到xrot，即xrot=U‘*x

（2）对旋转后的数据求解协方差矩阵covar，并将其可视化，观察得到的选择后的数据是否正确

PCA保证选择后的数据的协方差矩阵是一个对角阵，如果covar是正确的

那么它的图像应该是一个蓝色背景，并且在对角线位置有一斜线

这里显示协方差矩阵covar利用了matlab的imagesc，该函数真的很强大呀

imagesc(covar)的作用是：把矩阵covar以图像形式显示出来，矩阵中不同的数值会被赋予不同的颜色

得到的协方差矩阵的图像如下：可以看到，图像处了对角线位置外，其余部分颜色都相同

271537486569025.jpg　　

第二步：满足条件的主成分个数

本部分，找到满足条件的主成分的个数k

也就是找到最小的k值，使得(λ1+…+ λk)/(λ1+…+ λn)>某个百分数，如99%

第三步：利用找到的主成分个数，对数据进行降维

在第二步，已经找到了数字k，也就是，保留数据的k个主成分就满足了要求

在该步，将对数据x进行降维，只留下k个主成分，得到xTidle

同时，为了观察降维后的数据的好坏，在利用U(:,k)将降维后的数据变换会原来的维数，也就是得到了原数据的近似恢复数据

并利用网格将恢复出的图像显示出，与原图像进行比较，下面第一幅图是由降维后的图像恢复出的原数据，下图是相应的原数据，可以发现，降维后的数据基本可以恢复出于原数据非常相近的数据

271552062971408.jpg271552158445057.jpg

第四步：PCA白化+正则化

该部分分为两步

（1）执行具有白化和正则化的PCA

首先，对数据进行旋转（利用特征矩阵U）

然后，利用特征值对旋转后的数据进行缩放，实现白化

同时，在利用特征值缩放时，利用参数ε对特征值进行微调，实现正则化

271631124694609.png

（b）计算百化后的数据的协方差矩阵，观察该协方差矩阵

如果加入了正则化项，则该协方差矩阵的对角线元素都小于1

如果没有加入正则项（即仅有旋转+白化），则该协方差矩阵的对角线元素都为1（实际上，是令ε为一个极小的数）

下图是白化后数据的协方差矩阵对应的图像，上图是加入正则化后的结果，下图是没有加入正则化后的结果

271656099842208.jpg

271657432976249.jpg

第五步：ZCA白化

ZCA白化，就是在PCA白化的基础上做了一个旋转，即

271923492655269.png

下面的第一幅图是ZCA白化后的结果图，第二幅图是相应的原始图像

可以看到，ZCA白化的结果图似乎是原始图像的边缘

271926024063210.jpg271926112978960.jpg

下面，是该部分的pca_gen的代码

  1. clc
  2. clear
  3. close all
  5. %%================================================================
  6. %% Step 0a: Load data
  7. %  Here we provide the code to load natural image data into x.
  8. %  x will be a 144 * 10000 matrix, where the kth column x(:, k) corresponds to
  9. %  the raw image data from the kth 12x12 image patch sampled.
  10. %  You do not need to change the code below.
  11. x = sampleIMAGESRAW();%从IMAGES_RAW中读取一些图像patches
  12. figure('name','Raw images');%显示一个figure，标题为raw images
  13. randsel = randi(size(x,2),200,1); % A random selection of samples for visualization
  14. display_network(x(:,randsel));%显示随机选取的图像块儿
  15. %% Step 0b: Zero-mean the data (by row)
  16. %  You can make use of the mean and repmat/bsxfun functions.
  17. % -------------------- YOUR CODE HERE -------------------- 
  18. x=x-repmat(mean(x),size(x,1),1);%x的每一列的所有元素都减去该列的均值
  21. %%================================================================
  22. %% Step 1a: Implement PCA to obtain xRot
  23. %  Implement PCA to obtain xRot, the matrix in which the data is expressed
  24. %  with respect to the eigenbasis of sigma, which is the matrix U.
  25. % -------------------- YOUR CODE HERE -------------------- 
  26. xRot = zeros(size(x)); % You need to compute this
  27. % 计算协方差矩阵并进行特征值分解
  28. m=size(x,2);% 输入的样本个数
  29. sigma=x*x'/m;%输入数据的协方差矩阵
  30. [U,S,V]=eig(sigma);%对协方差矩阵进行特征值分解
  31. [S_Value,S_Index]=sort(diag(S),'descend');%提取S的对角线元素，将其按降序排列，sIndex是排序后的编号
  32. U=U(:,S_Index);
  33. S=diag(S_Value);
  34. % 对数据进行旋转
  35. xRot=U'*x;
  37. %% Step 1b: Check your implementation of PCA
  38. %  The covariance matrix for the data expressed with respect to the basis U
  39. %  should be a diagonal matrix with non-zero entries only along the main
  40. %  diagonal. We will verify this here.
  41. %  Write code to compute the covariance matrix, covar. 
  42. %  When visualised as an image, you should see a straight line across the
  43. %  diagonal (non-zero entries) against a blue background (zero entries).
  44. % -------------------- YOUR CODE HERE -------------------- 
  45. covar = zeros(size(x, 1)); % You need to compute this
  46. covar=xRot*xRot'/m;%旋转数据后的数据对应的协方差矩阵
  47. % Visualise the covariance matrix. You should see a line across the
  48. % diagonal against a blue background.
  49. figure('name','Visualisation of covariance matrix');
  50. imagesc(covar);
  52. %%================================================================
  53. %% Step 2: Find k, the number of components to retain
  54. %  Write code to determine k, the number of components to retain in order
  55. %  to retain at least 99% of the variance.
  56. % -------------------- YOUR CODE HERE -------------------- 
  57. k = 0; % Set k accordingly
  58. S_diag=diag(S);
  59. S_sum=sum(S_diag);
  60. for k=1:size(x,1)
  61.     Sk_sum=sum(S_diag(1:k));
  62.     if Sk_sum/S_sum>=0.99
  63.         break;
  64.     end
  65. end
  67. %%================================================================
  68. %% Step 3: Implement PCA with dimension reduction
  69. %  Now that you have found k, you can reduce the dimension of the data by
  70. %  discarding the remaining dimensions. In this way, you can represent the
  71. %  data in k dimensions instead of the original 144, which will save you
  72. %  computational time when running learning algorithms on the reduced
  73. %  representation.
  74. % 
  75. %  Following the dimension reduction, invert the PCA transformation to produce 
  76. %  the matrix xHat, the dimension-reduced data with respect to the original basis.
  77. %  Visualise the data and compare it to the raw data. You will observe that
  78. %  there is little loss due to throwing away the principal components that
  79. %  correspond to dimensions with low variation.
  81. % -------------------- YOUR CODE HERE -------------------- 
  82. % 对数据进行降维
  83. xTidle=U(:,1:k)'*x;
  84. % 利用降维后的数据xTidle对数据进行恢复
  85. xHat = zeros(size(x));  % You need to compute this
  86. xHat = U*[xTidle;zeros(m-k,size(x,2))];
  88. % Visualise the data, and compare it to the raw data
  89. % You should observe that the raw and processed data are of comparable quality.
  90. % For comparison, you may wish to generate a PCA reduced image which
  91. % retains only 90% of the variance.
  93. figure('name',['PCA processed images ',sprintf('(%d / %d dimensions)', k, size(x, 1)),'']);
  94. display_network(xHat(:,randsel));
  95. figure('name','Raw images');
  96. display_network(x(:,randsel));
  98. %%================================================================
  99. %% Step 4a: Implement PCA with whitening and regularisation
  100. %  Implement PCA with whitening and regularisation to produce the matrix
  101. %  xPCAWhite. 
  102. epsilon =0.000001;
  103. % -------------------- YOUR CODE HERE -------------------- 
  104. xPCAWhite = zeros(size(x));
  105. xPCAWhite=diag(1./sqrt(S_diag+epsilon))*xRot;
  106. %% Step 4b: Check your implementation of PCA whitening 
  107. %  Check your implementation of PCA whitening with and without regularisation. 
  108. %  PCA whitening without regularisation results a covariance matrix 
  109. %  that is equal to the identity matrix. PCA whitening with regularisation
  110. %  results in a covariance matrix with diagonal entries starting close to 
  111. %  1 and gradually becoming smaller. We will verify these properties here.
  112. %  Write code to compute the covariance matrix, covar. 
  113. %
  114. %  Without regularisation (set epsilon to 0 or close to 0), 
  115. %  when visualised as an image, you should see a red line across the
  116. %  diagonal (one entries) against a blue background (zero entries).
  117. %  With regularisation, you should see a red line that slowly turns
  118. %  blue across the diagonal, corresponding to the one entries slowly
  119. %  becoming smaller.
  120. % -------------------- YOUR CODE HERE -------------------- 
  121. covar=xPCAWhite*xPCAWhite'/m;
  122. % Visualise the covariance matrix. You should see a red line across the
  123. % diagonal against a blue background.
  124. figure('name','Visualisation of covariance matrix');
  125. imagesc(covar);
  127. %%================================================================
  128. %% Step 5: Implement ZCA whitening
  129. %  Now implement ZCA whitening to produce the matrix xZCAWhite. 
  130. %  Visualise the data and compare it to the raw data. You should observe
  131. %  that whitening results in, among other things, enhanced edges.
  132. xZCAWhite = zeros(size(x));
  133. % -------------------- YOUR CODE HERE -------------------- 
  134. xZCAWhite=U*xPCAWhite;%ZCA白化即在PCA白化基础上做了一个旋转
  135. % Visualise the data, and compare it to the raw data.
  136. % You should observe that the whitened images have enhanced edges.
  137. figure('name','ZCA whitened images');
  138. display_network(xZCAWhite(:,randsel));
  139. figure('name','Raw images');
  140. display_network(x(:,randsel));

发表评论

表情：
评论列表 （有 0 条评论，407人围观）

还没有评论，来说两句吧...

相关阅读