%%================================================================%% Step 0a: Load data%  Here we provide the code to load natural image data into x.%  x will be a 144 * 10000 matrix, where the kth column x(:, k) corresponds to%  the raw image data from the kth 12x12 image patch sampled.%  You do not need to change the code below.x = sampleIMAGESRAW();figure(‘name‘,‘Raw images‘);randsel = randi(size(x,2),200,1); % A random selection of samples for visualizationdisplay_network(x(:,randsel));%%================================================================%% Step 0b: Zero-mean the data (by row)%  You can make use of the mean and repmat/bsxfun functions.% -------------------- YOUR CODE HERE -------------------- x = x-repmat(mean(x,1),size(x,1),1);%%================================================================%% Step 1a: Implement PCA to obtain xRot%  Implement PCA to obtain xRot, the matrix in which the data is expressed%  with respect to the eigenbasis of sigma, which is the matrix U.% -------------------- YOUR CODE HERE -------------------- %xRot = zeros(size(x)); % You need to compute thissigma = x*x‘ ./ size(x,2);[u,s,v] = svd(sigma);xRot = u‘ * x;%%================================================================%% Step 1b: Check your implementation of PCA%  The covariance matrix for the data expressed with respect to the basis U%  should be a diagonal matrix with non-zero entries only along the main%  diagonal. We will verify this here.%  Write code to compute the covariance matrix, covar. %  When visualised as an image, you should see a straight line across the%  diagonal (non-zero entries) against a blue background (zero entries).% -------------------- YOUR CODE HERE -------------------- %covar = zeros(size(x, 1)); % You need to compute thiscovar = xRot*xRot‘ ./ size(x,2);% Visualise the covariance matrix. You should see a line across the% diagonal against a blue background.figure(‘name‘,‘Visualisation of covariance matrix‘);imagesc(covar);%%================================================================%% Step 2: Find k, the number of components to retain%  Write code to determine k, the number of components to retain in order%  to retain at least 99% of the variance.% -------------------- YOUR CODE HERE -------------------- %k = 0; % Set k accordinglyeigenvalue = diag(covar);total = sum(eigenvalue);tmpSum = 0;for k=1:size(x,1)    tmpSum = tmpSum+eigenvalue(k);    if(tmpSum / total >= 0.9)        break;    endend%%================================================================%% Step 3: Implement PCA with dimension reduction%  Now that you have found k, you can reduce the dimension of the data by%  discarding the remaining dimensions. In this way, you can represent the%  data in k dimensions instead of the original 144, which will save you%  computational time when running learning algorithms on the reduced%  representation.% %  Following the dimension reduction, invert the PCA transformation to produce %  the matrix xHat, the dimension-reduced data with respect to the original basis.%  Visualise the data and compare it to the raw data. You will observe that%  there is little loss due to throwing away the principal components that%  correspond to dimensions with low variation.% -------------------- YOUR CODE HERE -------------------- %xHat = zeros(size(x));  % You need to compute thisxRot(k+1:size(x,1), :) = 0;xHat = u * xRot;% Visualise the data, and compare it to the raw data% You should observe that the raw and processed data are of comparable quality.% For comparison, you may wish to generate a PCA reduced image which% retains only 90% of the variance.figure(‘name‘,[‘PCA processed images ‘,sprintf(‘(%d / %d dimensions)‘, k, size(x, 1)),‘‘]);display_network(xHat(:,randsel));figure(‘name‘,‘Raw images‘);display_network(x(:,randsel));%%================================================================%% Step 4a: Implement PCA with whitening and regularisation%  Implement PCA with whitening and regularisation to produce the matrix%  xPCAWhite. %epsilon = 0;epsilon = 0.1;%xPCAWhite = zeros(size(x));% -------------------- YOUR CODE HERE -------------------- xPCAWhite = diag(1 ./ sqrt(diag(s)+epsilon)) * u‘ * x;%%================================================================%% Step 4b: Check your implementation of PCA whitening %  Check your implementation of PCA whitening with and without regularisation. %  PCA whitening without regularisation results a covariance matrix %  that is equal to the identity matrix. PCA whitening with regularisation%  results in a covariance matrix with diagonal entries starting close to %  1 and gradually becoming smaller. We will verify these properties here.%  Write code to compute the covariance matrix, covar. %%  Without regularisation (set epsilon to 0 or close to 0), %  when visualised as an image, you should see a red line across the%  diagonal (one entries) against a blue background (zero entries).%  With regularisation, you should see a red line that slowly turns%  blue across the diagonal, corresponding to the one entries slowly%  becoming smaller.% -------------------- YOUR CODE HERE -------------------- covar = xPCAWhite * xPCAWhite‘ ./ size(x,2);% Visualise the covariance matrix. You should see a red line across the% diagonal against a blue background.figure(‘name‘,‘Visualisation of covariance matrix‘);imagesc(covar);%%================================================================%% Step 5: Implement ZCA whitening%  Now implement ZCA whitening to produce the matrix xZCAWhite. %  Visualise the data and compare it to the raw data. You should observe%  that whitening results in, among other things, enhanced edges.%xZCAWhite = zeros(size(x));xZCAWhite = u * xPCAWhite;% -------------------- YOUR CODE HERE -------------------- % Visualise the data, and compare it to the raw data.% You should observe that the whitened images have enhanced edges.figure(‘name‘,‘ZCA whitened images‘);display_network(xZCAWhite(:,randsel));figure(‘name‘,‘Raw images‘);display_network(x(:,randsel));