Step 0: Load data

The starter code contains code to load 45 2D data points. When plotted using the scatter function, the results should look like the following:

 

Step 1: Implement PCA

In this step, you will implement PCA to obtain x_rot, the matrix in which the data is "rotated" to the basis comprising made up of the principal components

 

Step 1a: Finding the PCA basis

Find and , and draw two lines in your figure to show the resulting basis on top of the given data points.

Step 1b: Check xRot

Compute xRot, and use the scatter function to check that xRot looks as it should, which should be something like the following:

 

Step 2: Dimension reduce and replot

In the next step, set k, the number of components to retain, to be 1

Step 3: PCA Whitening

Step 4: ZCA Whitening

Code

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‘); % 载入数据figure(1);scatter(x(1, :), x(2, :)); % 用圆圈绘制出数据分布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，均值为零 —— 2维，每一维减去该维上的均值sigma=(1.0/m)*x*x‘; % 协方差矩阵[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)]); % 画第二条线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(n,1)]‘*x); % 降维% 使特征点落在特征向量所指的方向上而不是原坐标系上% -------------------------------------------------------- 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;  % 每个特征除以对应的特征向量，以使每个特征有一致的方差% -------------------------------------------------------- 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. :)

Exercise: PCA in 2D