【转】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 thissigma = x * x'/ size(x, 2);[u,S,V] = svd(sigma);% -------------------------------------------------------- hold onplot([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 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 thisz = 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 thisxPCAWhite = 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 thisxZCAWhite = 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. :)

出处：http://www.cnblogs.com/90zeng/ 作者：博客园-太白路上的小混混