PCA和白化练习之处理二维数据

在很多情况下，我们要处理的数据的维度很高，需要提取主要的特征进行分析这就是PCA（主成分分析），白化是为了减少各个特征之间的冗余，因为在许多自然数据中，各个特征之间往往存在着一种关联，为了减少特征之间的关联，需要用到所谓的白化（whitening）.

首先下载数据pcaData.rar，下面要对这里面包含的45个2维样本点进行PAC和白化处理，数据中每一列代表一个样本点。

第一步 画出原始数据：



第二步：执行PCA，找到数据变化最大的方向：



第三步：将原始数据投射到上面找的两个方向上：



第四步：降维，此例中把数据由2维降维到1维，画出降维后的数据：



第五步：PCA白化处理：



第六步：ZCA白化处理：



下面是程序matlab源代码：

close all;clear all;clc;

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

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
xHat = zeros(size(x)); % You need to compute this
z = u(:, 1:k)' * x;
xHat = u(:,1:k) * z;

% --------------------------------------------------------
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)) * xRot;

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