principal components analysis (alternative to princomp).
A simple dimensionality reduction technique. Use this function to create an orthonormal
basis for the space R^N. This basis has the property that the coordinates of a vector x
in R^N are of decreasing importance. Hence instead of using all N coordinates to
specify the location of x, using only the first k<N still gives a vector xhat that is
very close to x with high probability.
Use this function to retrieve the basis U. Use pca_apply.m to retrieve that basis
coefficients for a novel vector x. Also, use pca_visualize(X,...) for visualization of
the approximated X.
This function operates on arrays of arbitrary dimension, by first converting the array
to a vector. That is if X is n dimensional, say of dims d1 x d2 x...x dn-1 x dn. Then
the first n-1 dimensions of X are comined. That is X is flattened to be 2 dimensional:
(d1 * d2 * ...* dn-1) x dn.
Once X is converted to 2 dimensions of size Nxn, each column represents a single
observation, and each row is a different variable. Note that this is the opposite of
many matlab functions such as princomp. So if X is MxNxK, then X(:,:,i) representes the
ith observation. This is useful if X is a stack of images (each image will
automatically get vectorized). Likewise if X is MxNxKxR, then X(:,:,:,i) becomes the
ith observation, which is useful if X is a collection of videos each of size MxNxK.
If X is very large, it is samlped before running PCA, using randomsample.
To calculate residuals:
residuals = variances / sum(variances);
residuals = cumsum(residuals); plot( residuals, '- .' )
INPUTS
X - n-dim array of size (d1 x d2 x...x dn-1) x dn (treated as dn elements)
OUTPUTS
U - 2D array of size (d1 * d2 * ...* dn-1) x r, where each column represents
- a principal component of X (after X is flattened).
mu - Array of size d1 x d2 x...x dn-1 which represents the mean of X.
variances - sorted eigenvalues corresponding to eigenvectors in U
EXAMPLE
load pca_data;
[ U, mu, variances ] = pca( I3D1(:,:,1:12) );
[ Y, Xhat, pe ] = pca_apply( I3D1(:,:,1), U, mu, variances, 5 );
figure(1); im(I3D1(:,:,1)); figure(2); im(Xhat);
pca_visualize( U, mu, variances, I3D1, 13, [0:12], [], 3 );
DATESTAMP
29-Nov-2005 2:00pm
See also PRINCOMP, PCA_APPLY, PCA_VISUALIZE, VISUALIZE_DATA, RANDOMSAMPLE