Deep learning：十一(PCA和whitening在二维数据中的练习)

本文转载http://www.cnblogs.com/tornadomeet/archive/2013/03/21/2973631.html

前言：

　　这节主要是练习下PCA，PCA Whitening以及ZCA Whitening在2D数据上的使用，2D的数据集是45个数据点，每个数据点是2维的。参考的资料是：Exercise:PCA in 2D。结合前面的博文Deep learning：十(PCA和whitening)理论知识，来进一步理解PCA和Whitening的作用。

 

　　matlab某些函数：

　　scatter:

　　scatter(X,Y,<S>,<C>,’<type>’);
　　<S> – 点的大小控制，设为和X，Y同长度一维向量，则值决定点的大小；设为常数或缺省，则所有点大小统一。
　　<C> – 点的颜色控制，设为和X，Y同长度一维向量，则色彩由值大小线性分布；设为和X，Y同长度三维向量，则按colormap RGB值定义每点颜色，[0,0,0]是黑色，[1,1,1]是白色。缺省则颜色统一。
　　<type> – 点型：可选filled指代填充，缺省则画出的是空心圈。

　　plot:

　　plot可以用来画直线，比如说plot([1 2],[0 4])是画出一条连接(1,0)到(2,4)的直线，主要点坐标的对应关系。

 

　　实验过程：

　　一、首先download这些二维数据，因为数据是以文本方式保存的，所以load的时候是以ascii码读入的。然后对输入样本进行协方差矩阵计算，并计算出该矩阵的SVD分解，得到其特征值向量，在原数据点上画出2条主方向，如下图所示：

 　　

　　二、将经过PCA降维后的新数据在坐标中显示出来，如下图所示：

 　　

　　三、用新数据反过来重建原数据，其结果如下图所示:

 　　

　　四、使用PCA whitening的方法得到原数据的分布情况如：

 　　

　　五、使用ZCA whitening的方法得到的原数据的分布如下所示：

 　　

　　PCA whitening和ZCA whitening不同之处在于处理后的结果数据的方差不同，尽管不同维度的方差是相等的。

 

　　实验代码：

close all%%================================================================%% Step 0: Load data%  We have provided the code to load data from pcaData.txt into x.%  x is a 2 * 45 matrix, where the kth column x(:,k) corresponds to%  the kth data point.Here we provide the code to load natural image data into x.%  You do not need to change the code below.x = load('pcaData.txt','-ascii');figure(1);scatter(x(1, :), x(2, :));title('Raw data');%%================================================================%% Step 1a: Implement PCA to obtain U %  Implement PCA to obtain the rotation matrix U, which is the eigenbasis%  sigma. % -------------------- YOUR CODE HERE -------------------- u = zeros(size(x, 1)); % You need to compute this[n m] = size(x);%x = x-repmat(mean(x,2),1,m);%预处理，均值为0sigma = (1.0/m)*x*x';[u s v] = svd(sigma);% -------------------------------------------------------- hold onplot([0 u(1,1)], [0 u(2,1)]);%画第一条线plot([0 u(1,2)], [0 u(2,2)]);%第二条线scatter(x(1, :), x(2, :));hold off%%================================================================%% Step 1b: Compute xRot, the projection on to the eigenbasis%  Now, compute xRot by projecting the data on to the basis defined%  by U. Visualize the points by performing a scatter plot.% -------------------- YOUR CODE HERE -------------------- xRot = zeros(size(x)); % You need to compute thisxRot = u'*x;% -------------------------------------------------------- % Visualise the covariance matrix. You should see a line across the% diagonal against a blue background.figure(2);scatter(xRot(1, :), xRot(2, :));title('xRot');%%================================================================%% Step 2: Reduce the number of dimensions from 2 to 1. %  Compute xRot again (this time projecting to 1 dimension).%  Then, compute xHat by projecting the xRot back onto the original axes %  to see the effect of dimension reduction% -------------------- YOUR CODE HERE -------------------- k = 1; % Use k = 1 and project the data onto the first eigenbasisxHat = zeros(size(x)); % You need to compute thisxHat = u*([u(:,1),zeros(2,1)]'*x);% -------------------------------------------------------- figure(3);scatter(xHat(1, :), xHat(2, :));title('xHat');%%================================================================%% Step 3: PCA Whitening%  Complute xPCAWhite and plot the results.epsilon = 1e-5;% -------------------- YOUR CODE HERE -------------------- xPCAWhite = zeros(size(x)); % You need to compute thisxPCAWhite = diag(1./sqrt(diag(s)+epsilon))*u'*x;% -------------------------------------------------------- figure(4);scatter(xPCAWhite(1, :), xPCAWhite(2, :));title('xPCAWhite');%%================================================================%% Step 3: ZCA Whitening%  Complute xZCAWhite and plot the results.% -------------------- YOUR CODE HERE -------------------- xZCAWhite = zeros(size(x)); % You need to compute thisxZCAWhite = u*diag(1./sqrt(diag(s)+epsilon))*u'*x;% -------------------------------------------------------- figure(5);scatter(xZCAWhite(1, :), xZCAWhite(2, :));title('xZCAWhite');%% Congratulations! When you have reached this point, you are done!%  You can now move onto the next PCA exercise. :)

 

 

　　参考资料：

     Exercise:PCA in 2D

     Deep learning：十(PCA和whitening)