Principal components analysis (alternative to princomp).


function [U,mu,vars] = pca( X )


 Principal components analysis (alternative to princomp).

 A simple linear dimensionality reduction technique. Use to create an
 orthonormal basis for the points in R^d such that the coordinates of a
 vector x in this basis are of decreasing importance. Instead of using all
 d basis vectors to specify the location of x, using only the first k<d
 still gives a vector xhat that is close to x.

 This function operates on arrays of arbitrary dimension, by first
 converting the arrays to vectors. If X is m+1 dimensional, say of size
 [d1 x d2 x...x dm x n], then the first m dimensions of X are combined. X
 is flattened to be 2 dimensional: [dxn], with d=prod(di). Once X is
 converted to 2 dimensions of size dxn, 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. If X is MxNxn, then
 X(:,:,i) represents the ith observation (useful for stack of n images),
 likewise for n videos X is MxNxKxn. If X is very large, it is sampled
 before running PCA. Use this function to retrieve the basis U. Use
 pcaApply to retrieve that basis coefficients for a novel vector x. Use
 pcaVisualize(X,...) for visualization of approximated X.

 To calculate residuals:
  residuals = cumsum(vars/sum(vars)); plot(residuals,'-.')

  [U,mu,vars] = pca( X )

  X         - [d1 x ... x dm x n], treated as n [d1 x ... x dm] elements

  U         - [d x r], d=prod(di), each column is a principal component
  mu        - [d1 x ... x dm] mean of X
  vars      - sorted eigenvalues corresponding to eigenvectors in U

  load pcaData;
  [U,mu,vars] = pca( I3D1(:,:,1:12) );
  [Y,Xhat,avsq] = pcaApply( I3D1(:,:,1), U, mu, 5 );
  pcaVisualize( U, mu, vars, I3D1, 13, [0:12], [], 1 );
  Xr = pcaRandVec( U, mu, vars, 1, 25, 0, 3 );

 See also princomp, pcaApply, pcaVisualize, pcaRandVec, visualizeData

 Piotr's Computer Vision Matlab Toolbox      Version 3.24
 Copyright 2014 Piotr Dollar.  [pdollar-at-gmail.com]
 Licensed under the Simplified BSD License [see external/bsd.txt]

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