在很多情況下，我們要處理的數據的維度很高，需要提取主要的特征進行分析這就是PCA（主成分分析），白化是為了減少各個特征之間的冗余，因為在許多自然數據中，各個特征之間往往存在着一種關聯，為了減少特征之間的關聯，需要用到所謂的白化（whitening）.
首先下載數據pcaData.rar，下面要對這里面包含的45個2維樣本點進行PAC和白化處理，數據中每一列代表一個樣本點。
第一步 畫出原始數據：
第二步：執行PCA，找到數據變化最大的方向：
第三步：將原始數據投射到上面找的兩個方向上：
第四步：降維，此例中把數據由2維降維到1維，畫出降維后的數據：
第五步：PCA白化處理：
第六步：ZCA白化處理：
下面是程序matlab源代碼：
1 close all;clear all;clc; 2
3 %%================================================================
4 %% Step 0: Load data 5 % We have provided the code to load data from pcaData.txt into x. 6 % x is a 2 * 45 matrix, where the kth column x(:,k) corresponds to 7 % the kth data point.Here we provide the code to load natural image data into x. 8 % You do not need to change the code below. 9
10 x = load('pcaData.txt','-ascii'); 11 figure(1); 12 scatter(x(1, :), x(2, :)); 13 title('Raw data'); 14
15
16 %%================================================================
17 %% Step 1a: Implement PCA to obtain U 18 % Implement PCA to obtain the rotation matrix U, which is the eigenbasis 19 % sigma. 20
21 % -------------------- YOUR CODE HERE --------------------
22 u = zeros(size(x, 1)); % You need to compute this
23
24 sigma = x * x'/ size(x, 2);
25 [u,S,V] = svd(sigma); 26
27
28
29 % --------------------------------------------------------
30 hold on 31 plot([0 u(1,1)], [0 u(2,1)]); 32 plot([0 u(1,2)], [0 u(2,2)]); 33 scatter(x(1, :), x(2, :)); 34 hold off 35
36 %%================================================================
37 %% Step 1b: Compute xRot, the projection on to the eigenbasis 38 % Now, compute xRot by projecting the data on to the basis defined 39 % by U. Visualize the points by performing a scatter plot. 40
41 % -------------------- YOUR CODE HERE --------------------
42 xRot = zeros(size(x)); % You need to compute this
43 xRot = u' * x;
44
45 % --------------------------------------------------------
46
47 % Visualise the covariance matrix. You should see a line across the 48 % diagonal against a blue background. 49 figure(2); 50 scatter(xRot(1, :), xRot(2, :)); 51 title('xRot'); 52
53 %%================================================================
54 %% Step 2: Reduce the number of dimensions from 2 to 1. 55 % Compute xRot again (this time projecting to 1 dimension). 56 % Then, compute xHat by projecting the xRot back onto the original axes 57 % to see the effect of dimension reduction 58
59 % -------------------- YOUR CODE HERE --------------------
60 k = 1; % Use k = 1 and project the data onto the first eigenbasis 61 xHat = zeros(size(x)); % You need to compute this
62 z = u(:, 1:k)' * x;
63 xHat = u(:,1:k) * z; 64
65 % --------------------------------------------------------
66 figure(3); 67 scatter(xHat(1, :), xHat(2, :)); 68 title('xHat'); 69
70
71 %%================================================================
72 %% Step 3: PCA Whitening 73 % Complute xPCAWhite and plot the results. 74
75 epsilon = 1e-5; 76 % -------------------- YOUR CODE HERE --------------------
77 xPCAWhite = zeros(size(x)); % You need to compute this
78
79 xPCAWhite = diag(1 ./ sqrt(diag(S) + epsilon)) * xRot; 80
81
82
83 % --------------------------------------------------------
84 figure(4); 85 scatter(xPCAWhite(1, :), xPCAWhite(2, :)); 86 title('xPCAWhite'); 87
88 %%================================================================
89 %% Step 3: ZCA Whitening 90 % Complute xZCAWhite and plot the results. 91
92 % -------------------- YOUR CODE HERE --------------------
93 xZCAWhite = zeros(size(x)); % You need to compute this
94
95 xZCAWhite = u * xPCAWhite; 96 % --------------------------------------------------------
97 figure(5); 98 scatter(xZCAWhite(1, :), xZCAWhite(2, :)); 99 title('xZCAWhite'); 100
101 %% Congratulations! When you have reached this point, you are done!
102 % You can now move onto the next PCA exercise. :)