pca

PURPOSE ^

Principal components analysis (alternative to princomp).

SYNOPSIS ^

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

DESCRIPTION ^

 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 vector.  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, using subsampleMatrix. 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, '-.' )

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

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

 OUTPUTS
  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

 EXAMPLE
  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, SUBSAMPLEMATRIX

 Piotr's Image&Video Toolbox      Version 2.0
 Copyright 2008 Piotr Dollar.  [pdollar-at-caltech.edu]
 Please email me if you find bugs, or have suggestions or questions!
 Licensed under the Lesser GPL [see external/lgpl.txt]

CROSS-REFERENCE INFORMATION ^

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