UFLDL——Exercise: PCA in 2D 主成分分析

实验要求可以参考deeplearning的tutorial，
Exercise: PCA in 2D。

 

1. 实验描述：

实验在二维数据上进行PCA降维，PCA白化处理，以及ZCA白化处理，原理可以参考之间的博客，下面直接贴代码。

 

在实验中，我计算了每一次原始数据，PCA旋转，PCA白化处理，以及ZCA白化处理后的协方差矩阵，结果为：





计算协方差我使用了matlab自带的cov(x),它要求矩阵x的每一行代表一个数据，这个tutorial实验说明求得的协方差矩阵的结果不同，为matlab计算的时候需要除以n-1（n为数据的个数），但这些对最后的结果都是没有影响的。

2. 源代码

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');

original = cov(x')%*(size(x,2)-1)

%%================================================================
%% 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

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;

rotate = cov(xRot')%*(size(x,2)-1)

% --------------------------------------------------------

% 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
xHat = zeros(size(x)); % You need to compute this

xTiled = zeros(size(x));
xTiled(1:k,:) = xRot(1:k,:);
xHat = u*xTiled;

% --------------------------------------------------------
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
epsilon = 1e-5;
xPCAWhite = diag(1./sqrt(diag(s) + epsilon)) * xRot;

PCAWhite = cov(xPCAWhite')%*(size(x,2)-1)

% --------------------------------------------------------
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;

ZCAWhite = cov(xZCAWhite')%*(size(x,2)-1)

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