编程练习:PCA in 2D部分
<span style="font-size:14px;">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)); % 获得特征向量U;
[n m] = size(x);
%x = x-repmat(mean(x,2),1,m);%预处理,均值为0
sigma = (1.0/m)*x*x';
[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)); % 旋转操作
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)); % 降维操作;
xHat = u*([u(:,1),zeros(n,1)]'*x);
% --------------------------------------------------------
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)); % 初始化,然后求xPCAWhite;
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)); % 初始化,然后求xZCAWhite;
xZCAWhite = u*diag(1./sqrt(diag(s)+epsilon))*u'*x;
% --------------------------------------------------------
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. :)
</span>
编程练习:PCA and Whitening部分
%%================================================================
%% Step 0a: Load data
% Here we provide the code to load natural image data into x.
% x will be a 144 * 10000 matrix, where the kth column x(:, k) corresponds to
% the raw image data from the kth 12x12 image patch sampled.
% You do not need to change the code below.
x = sampleIMAGESRAW();
figure('name','Raw images');
randsel = randi(size(x,2),200,1); % A random selection of samples for visualization
display_network(x(:,randsel));
%%================================================================
%% Step 0b: Zero-mean the data (by row)
% You can make use of the mean and repmat/bsxfun functions.
% -------------------- YOUR CODE HERE --------------------
x = x-repmat(mean(x,1),size(x,1),1);%求的是每一列的均值
%x = x-repmat(mean(x,2),1,size(x,2));
%%================================================================
%% Step 1a: Implement PCA to obtain xRot
% Implement PCA to obtain xRot, the matrix in which the data is expressed
% with respect to the eigenbasis of sigma, which is the matrix U.
% -------------------- YOUR CODE HERE --------------------
xRot = zeros(size(x)); % 旋转操作;
[n m]=size(x);
sigma=(1.0/m)*x*x';
[u s v]=svd(sigma);
xRot=u'*x;
%%================================================================
%% Step 1b: Check your implementation of PCA
% The covariance matrix for the data expressed with respect to the basis U
% should be a diagonal matrix with non-zero entries only along the main
% diagonal. We will verify this here.
% Write code to compute the covariance matrix, covar.
% When visualised as an image, you should see a straight line across the
% diagonal (non-zero entries) against a blue background (zero entries).
% -------------------- YOUR CODE HERE --------------------
covar = zeros(size(x, 1)); % 检查PCA,为0,则为直线,否则为蓝色背景;
covar=(1./m)*xRot*xRot';
% Visualise the covariance matrix. You should see a line across the
% diagonal against a blue background.
figure('name','Visualisation of covariance matrix');
imagesc(covar);
%%================================================================
%% Step 2: Find k, the number of components to retain
% Write code to determine k, the number of components to retain in order
% to retain at least 99% of the variance.
% -------------------- YOUR CODE HERE --------------------
k = 0; % 计算最合适的k值;
ss=diag(s);
k=length(ss((cumsum(ss)/sum(ss))<=0.99));
%%================================================================
%% Step 3: Implement PCA with dimension reduction
% Now that you have found k, you can reduce the dimension of the data by
% discarding the remaining dimensions. In this way, you can represent the
% data in k dimensions instead of the original 144, which will save you
% computational time when running learning algorithms on the reduced
% representation.
%
% Following the dimension reduction, invert the PCA transformation to produce
% the matrix xHat, the dimension-reduced data with respect to the original basis.
% Visualise the data and compare it to the raw data. You will observe that
% there is little loss due to throwing away the principal components that
% correspond to dimensions with low variation.
% -------------------- YOUR CODE HERE --------------------
xHat = zeros(size(x)); % 降维操作;
xHat=u*[u(:,1:k)'*x;zeros(n-k,m)];
% Visualise the data, and compare it to the raw data
% You should observe that the raw and processed data are of comparable quality.
% For comparison, you may wish to generate a PCA reduced image which
% retains only 90% of the variance.
figure('name',['PCA processed images ',sprintf('(%d / %d dimensions)', k, size(x, 1)),'']);
display_network(xHat(:,randsel));
figure('name','Raw images');
display_network(x(:,randsel));
%%================================================================
%% Step 4a: Implement PCA with whitening and regularisation
% Implement PCA with whitening and regularisation to produce the matrix
% xPCAWhite.
epsilon = 0.1;
xPCAWhite = zeros(size(x));
% -------------------- YOUR CODE HERE --------------------
xPCAWhite = diag(1./sqrt(diag(s)+epsilon))*u'*x;
figure('name','PCA whitened images');
display_network(xPCAWhite(:,randsel));
%%================================================================
%% Step 4b: Check your implementation of PCA whitening
% Check your implementation of PCA whitening with and without regularisation.
% PCA whitening without regularisation results a covariance matrix
% that is equal to the identity matrix. PCA whitening with regularisation
% results in a covariance matrix with diagonal entries starting close to
% 1 and gradually becoming smaller. We will verify these properties here.
% Write code to compute the covariance matrix, covar.
%
% Without regularisation (set epsilon to 0 or close to 0),
% when visualised as an image, you should see a red line across the
% diagonal (one entries) against a blue background (zero entries).
% With regularisation, you should see a red line that slowly turns
% blue across the diagonal, corresponding to the one entries slowly
% becoming smaller.
% -------------------- YOUR CODE HERE --------------------
covar = (1./m)*xPCAWhite*xPCAWhite';%检验xPCAWhite;
% Visualise the covariance matrix. You should see a red line across the
% diagonal against a blue background.
figure('name','Visualisation of covariance matrix');
imagesc(covar);
%%================================================================
%% Step 5: Implement ZCA whitening
% Now implement ZCA whitening to produce the matrix xZCAWhite.
% Visualise the data and compare it to the raw data. You should observe
% that whitening results in, among other things, enhanced edges.
xZCAWhite = zeros(size(x));
% -------------------- YOUR CODE HERE --------------------
xZCAWhite = u*xPCAWhite;
% Visualise the data, and compare it to the raw data.
% You should observe that the whitened images have enhanced edges.
figure('name','ZCA whitened images');
display_network(xZCAWhite(:,randsel));
figure('name','Raw images');
display_network(x(:,randsel));<span style="color:#ff0000;">
</span>
编程练习
今天的文章PCA and Whitening编程代码整理分享到此就结束了,感谢您的阅读,如果确实帮到您,您可以动动手指转发给其他人。
版权声明:本文内容由互联网用户自发贡献,该文观点仅代表作者本人。本站仅提供信息存储空间服务,不拥有所有权,不承担相关法律责任。如发现本站有涉嫌侵权/违法违规的内容, 请发送邮件至 举报,一经查实,本站将立刻删除。
如需转载请保留出处:http://bianchenghao.cn/28955.html