UFLDL Exercise:PCA in 2D

练习题网址：http://deeplearning.stanford.edu/wiki/index.php/Exercise:PCA_in_2D

pca_2d.m的代码如下：

clc;clear;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
avg = mean(x,2);
% x = x-repmat(avg,1,size(x,2));    % x is already zero mean,so it does not
% need to do that again.
sigma = x*x'/size(x,2);
[u,s,v] = svd(sigma);

% --------------------------------------------------------
hold on
plot([0 u(1,1)], [0 u(2,1)]);
plot([0 u(1,2)], [0 u(2,2)]);
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
xTilde = zeros(size(x)); % You need to compute this
xTilde(1:k,:) = xRot(1:k,:);
xHat = u * xTilde;


% -------------------------------------------------------- 
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 * xPCAWhite;

% -------------------------------------------------------- 
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. :)