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. :)