数据清洗和特征选择→PCA→2.实战

《数据清洗和特征选择→PCA→2.实战
》

程序目的

对2*45的数据进行读取、旋转(投影)(基为协方差矩阵的特征向量)、降维、白化操作

并展现可视化过程

总结

求解原数据协方差矩阵特征向量时，用的是奇异值分解，且左奇异向量为要投影的特征向量 1.17
在进行数据的旋转时，要从投影的角度看 2.6
3.9 的含义是：从线性表示的角度看，在数据的第一主元方向的坐标是xRot,xHat是在标准基下的坐标参考见线性代数→矩阵→矩阵的秩→矩阵的乘法
在PCA白化时，有时一些特征值(奇异值)在数值上接近于0，这样在缩放步骤时我们除以将导致除

以一个接近0的值。因而在实践中，我们使用少量的正则化实现这个缩放过程，即给特征

值加上一个很小的常数ε 4.7

使用plot画出向量(x,y): plot([0,x],[0,y]) 1.20

错误赵

pacData.txt下载参考见七牛云→picturetemp→机器学习工程师→pca实战→pcaData.txt

%% 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（困惑已久的自然函数） 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');%未处理的数据
%%================================================================
%% 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
sigm = x*x'./(size(x, 2));%covariance matrix
[u,s] = svd(sigm);
% -------------------------------------------------------- 
hold on
plot([0 u(1,1)], [0 u(2,1)]);%The eigenvector corresponding to the largest eigenvalue
plot([0 u(1,2)], [0 u(2,2)]);%The eigenvector corresponding to the second largest eigenvalue
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)); 
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
xRot = u(:,1:k)'*x;
xHat = u(:,1:k)*xRot;
% -------------------------------------------------------- 
figure(3);
scatter(xHat(1, :), xHat(2, :));
title('xHat');
%%================================================================

%% Step 3: PCA Whitening
%  Complute xPCAWhite and plot the results.
epsilon = 1e-5; % 1*10^(-5)
% -------------------- YOUR CODE HERE -------------------- 
xPCAWhite = zeros(size(x)); % You need to compute this
xRot = u'*x;%2x45 double
xPCAWhite = bsxfun(@rdivide,xRot,sqrt(diag(s)+epsilon));
% -------------------------------------------------------- 
figure(4);
scatter(xPCAWhite(1, :), xPCAWhite(2, :));
title('xPCAWhite');
%%================================================================

数据清洗和特征选择→PCA→2.实战

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