Principal Component Analysis (PCA)

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Presentation transcript:

Principal Component Analysis (PCA) Dated back to Pearson (1901) A set of data are summarized as a linear combination of an ortonormal set of vectors which define a new coordinate system. Figure from Hastie, et, al 2001

Principal Components Analysis (PCA)

Principal Components Analysis (PCA)

Singular Value Decomposition

Singular Value Decomposition

Singular Value Decomposition

Singular Value Decomposition

Example 1 Use the data set "noisy.mat" available on the course web page. 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 1 Apply 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 1 If 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 then imagesc(reshape(X(:, 10), 20 28)’) imagesc(reshape(X_hat(:, 10), 20 28)’) to plot the 10th original image and its reconstruction.

Example 2

Example 2 Load the sample data, which includes digits 2 and 3 of 64 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 data Imagesc(Low_dimensional_data)

Kernel Methods

Kernel Trick

Observed and Feature