Principal Component Analysis (PCA)

Principal Component Analysis (PCA)

Principal Component Analysis (PCA)

Principal Component Analysis (PCA)

Principal Component Analysis (PCA)

Principal Component Analysis (PCA)

Principal Component Analysis (PCA)

Principal Component Analysis (PCA)

Principal Component Analysis (PCA)

Principal Component Analysis (PCA)

Alternative Derivation (PCA)

Alternative Derivation (PCA)

Alternative Derivation (PCA)

Alternative Derivation (PCA)

Singular Value Decomposition

Singular Value Decomposition

Singular Value Decomposition

Example 1Use the data set "noisy.mat" available on your CD. The data set consists of 1965, 20-pixel-by-28-pixel grey-scale images distorted by adding Gaussian noises to each pixel with s=25.

Example 1Apply PCA to the noisy data. Suppose the intrinsic dimensionality of the data is 10. Compute reconstructed images using the top d = 10 eigenvectors and plot original and reconstructed images

Example 1If original images are stored in matrix X (it is 560 by 1965 matrix) and reconstructed images are in matrix X_hat , you can type in colormap gray and thenimagesc(reshape(X(:, 10), 20 28))imagesc(reshape(X_hat(:, 10), 20 28))to plot the 10th original image and its reconstruction.

Example 2

Example 2Load the sample data, which includes digits 2 and 3 of64 measurements on a sample of 400. load 2_3.mat

Extract appropriate features by PCA

[u s v]=svd(X','econ');

Create data

Low_dimensional_data=u(:,1:2);Observe low dimensional dataImagesc(Low_dimensional_data)