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　　前言：

　　這節主要是練習下PCA，PCA Whitening以及ZCA Whitening在2D數據上的使用，2D的數據集是45個數據點，每個數據點是2維的。參考的資料是：Exercise:PCA in 2D。結合前面的博文Deep learning：十(PCA和whitening)理論知識，來進一步理解PCA和Whitening的作用。

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　　matlab某些函數：

　　scatter:

　　scatter(X,Y,<S>,<C>,’<type>’);
　　<S> – 點的大小控制，設為和X，Y同長度一維向量，則值決定點的大小；設為常數或缺省，則所有點大小統一。
　　<C> – 點的顏色控制，設為和X，Y同長度一維向量，則色彩由值大小線性分布；設為和X，Y同長度三維向量，則按colormap RGB值定義每點顏色，[0,0,0]是黑色，[1,1,1]是白色。缺省則顏色統一。
　　<type> – 點型：可選filled指代填充，缺省則畫出的是空心圈。

　　plot:

　　plot可以用來畫直線，比如說plot([1 2],[0 4])是畫出一條連接(1,0)到(2,4)的直線，主要點坐標的對應關系。

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　　實驗過程：

　　一、首先download這些二維數據，因為數據是以文本方式保存的，所以load的時候是以ascii碼讀入的。然后對輸入樣本進行協方差矩陣計算，并計算出該矩陣的SVD分解，得到其特征值向量，在原數據點上畫出2條主方向，如下圖所示：

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　　二、將經過PCA降維后的新數據在坐標中顯示出來，如下圖所示：

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　　三、用新數據反過來重建原數據，其結果如下圖所示:

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　　四、使用PCA whitening的方法得到原數據的分布情況如：

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　　五、使用ZCA whitening的方法得到的原數據的分布如下所示：

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　　PCA whitening和ZCA whitening不同之處在于處理后的結果數據的方差不同，盡管不同維度的方差是相等的。

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　　實驗代碼：

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);%預處理，均值為0 sigma = (1.0/m)*x*x'; [u s v] = svd(sigma);% -------------------------------------------------------- hold on plot([0 u(1,1)], [0 u(2,1)]);%畫第一條線 plot([0 u(1,2)], [0 u(2,2)]);%第二條線 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 this xRot = 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 eigenbasis xHat = zeros(size(x)); % You need to compute this xHat = u*([u(:,1),zeros(n,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 this xPCAWhite = 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 this xZCAWhite = 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. :)

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　　參考資料：

? ? ?Exercise:PCA in 2D

? ? ?Deep learning：十(PCA和whitening)

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轉載于:https://www.cnblogs.com/tornadomeet/archive/2013/03/21/2973631.html

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