# 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 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,'-.')

     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

     Piotr's Image&Video Toolbox      Version 3.24
     Copyright 2013 Piotr Dollar.  [pdollar-at-caltech.edu]
     Please email me if you find bugs, or have suggestions or questions!
     Licensed under the Simplified BSD License [see external/bsd.txt]