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Statistical Learning Toolbox

from Statistical Learning Toolbox by Dahua Lin
Functions for statistical learning, pattern recognition and computer vision, covering many topics.

Description of slwhiten_from_samples
Home > sltoolbox > stat > slwhiten_from_samples.m

slwhiten_from_samples

PURPOSE ^

SLWHITEN_FROM_SAMPLES Compute the whitening matrix from sample matrix

SYNOPSIS ^

function W = slwhiten_from_samples(X, varargin)

DESCRIPTION ^

SLWHITEN_FROM_SAMPLES Compute the whitening matrix from sample matrix

 $ Syntax $
   - W = slwhiten_from_samples(X)
   - W = slwhiten_from_samples(X, ...)

 $ Arguments $
   - X:            the sample matrix
   - W:            the computed whitening transform matrix

 $ Description $
   - W = slwhiten_from_samples(X) computes the whiten transform matrix
     from samples using the automatic-selection scheme.

   - W = slwhiten_from_samples(X, ...) computes the whiten transform
     matrix from samples according to the user-specified properties.
     \*   
     \t   Table 1. The properties of sample-whitening computation \\
     \h     name     &      description                   \\
           'scheme'  &  The scheme of computation procedure, 
                        default = 'auto' \\
           'evproc'  &  The {method, ...} form of eigenvalue processing. 
                        default = {'std'} \\
                        This will be input to the slinvevals function. \\
           'weights' &  The sample weights.  default = []. \\
     \*      
     The available schemes are listed as follows
     \* 
     \t   Table 2. The schemes of computing whitening matrix       \\
     \h     name   &     description                               \\
           'auto'  &  Automatically select a proper scheme for computing.
                      \\ 
           'std'   &  Standard scheme: first compute the covariance matrix
                      and then derive the whitening transform matrix. \\
           'svd'   &  SVD-based scheme. Using svd for eigen-decomposition.
           'trans' &  Use a transpose-based way. It is more efficient for
                      the case with high-dimensionality and small sample
                      size.                                            \\
     \*
     The methods for processing the eigenvalues, i.e. computing their
     reciprocals can be referred to the slinvevals function.

 $ Remarks $
   - It is a prerequisite that the samples are properly centered. 

 $ History $
   - Created by Lin Dahua on Apr 30, 2006
   - Modified by Dahua Lin on Sep 10, 2006
       - replace slmul by slmulvec to increase efficiency

CROSS-REFERENCE INFORMATION ^

This function calls:
  • slmulvec SLMULVEC multiplies a vector to columns or rows of a matrix
  • slnormalize SLNORMALIZE Normalize the sub-arrays
  • slsymeig SLSYMEIG Compute the eigenvalues and eigenvectors for symmetric matrix
  • slinvevals SLINVEVALS Compute the reciprocals of eigenvalues in a robust way
  • slparseprops SLPARSEPROPS Parses input parameters
This function is called by:
  • slfld SLFLD Performs Fisher Linear Discriminant Analysis

SUBFUNCTIONS ^

SOURCE CODE ^

0001 function W = slwhiten_from_samples(X, varargin)
0002 %SLWHITEN_FROM_SAMPLES Compute the whitening matrix from sample matrix
0003 %
0004 % $ Syntax $
0005 %   - W = slwhiten_from_samples(X)
0006 %   - W = slwhiten_from_samples(X, ...)
0007 %
0008 % $ Arguments $
0009 %   - X:            the sample matrix
0010 %   - W:            the computed whitening transform matrix
0011 %
0012 % $ Description $
0013 %   - W = slwhiten_from_samples(X) computes the whiten transform matrix
0014 %     from samples using the automatic-selection scheme.
0015 %
0016 %   - W = slwhiten_from_samples(X, ...) computes the whiten transform
0017 %     matrix from samples according to the user-specified properties.
0018 %     \*
0019 %     \t   Table 1. The properties of sample-whitening computation \\
0020 %     \h     name     &      description                   \\
0021 %           'scheme'  &  The scheme of computation procedure,
0022 %                        default = 'auto' \\
0023 %           'evproc'  &  The {method, ...} form of eigenvalue processing.
0024 %                        default = {'std'} \\
0025 %                        This will be input to the slinvevals function. \\
0026 %           'weights' &  The sample weights.  default = []. \\
0027 %     \*
0028 %     The available schemes are listed as follows
0029 %     \*
0030 %     \t   Table 2. The schemes of computing whitening matrix       \\
0031 %     \h     name   &     description                               \\
0032 %           'auto'  &  Automatically select a proper scheme for computing.
0033 %                      \\
0034 %           'std'   &  Standard scheme: first compute the covariance matrix
0035 %                      and then derive the whitening transform matrix. \\
0036 %           'svd'   &  SVD-based scheme. Using svd for eigen-decomposition.
0037 %           'trans' &  Use a transpose-based way. It is more efficient for
0038 %                      the case with high-dimensionality and small sample
0039 %                      size.                                            \\
0040 %     \*
0041 %     The methods for processing the eigenvalues, i.e. computing their
0042 %     reciprocals can be referred to the slinvevals function.
0043 %
0044 % $ Remarks $
0045 %   - It is a prerequisite that the samples are properly centered.
0046 %
0047 % $ History $
0048 %   - Created by Lin Dahua on Apr 30, 2006
0049 %   - Modified by Dahua Lin on Sep 10, 2006
0050 %       - replace slmul by slmulvec to increase efficiency
0051 %
0052 
0053 %% parse and verify input arguments
0054 
0055 if ndims(X) ~= 2
0056     error('sltoolbox:invaliddims', ...
0057         'The sample matrix X should be a 2D matrix');
0058 end
0059 n = size(X, 2);
0060 
0061 % check options
0062 
0063 opts.scheme = 'auto';
0064 opts.evproc = {'std'};
0065 opts.weights = [];
0066 opts = slparseprops(opts, varargin{:});
0067 
0068 switch opts.scheme 
0069     case 'auto'
0070         fh_compW = @compute_whiten_auto;
0071     case 'std'
0072         fh_compW = @compute_whiten_std;
0073     case 'svd'
0074         fh_compW = @compute_whiten_svd;
0075     case 'trans'
0076         fh_compW = @compute_whiten_trans;
0077     otherwise
0078         error('sltoolbox:invalidarg', ...
0079             'Invalid whiten matrix computing scheme %s', opts.scheme);
0080 end
0081 
0082 if ~isempty(opts.weights)
0083     if ~isequal(size(opts.weights), [1, n])
0084         error('sltoolbox:sizmismatch', ...
0085             'The weights should be a 1 x n row vector');
0086     end
0087 end
0088 
0089 
0090 %% prepare the samples
0091 
0092 if ~isempty(opts.weights)
0093     X = slmulvec(X, sqrt(max(opts.weights, 0)), 2);
0094 end
0095 
0096 %% compute
0097 
0098 W = fh_compW(X, opts.evproc);
0099 
0100 %% The functions for computing whiten matrix
0101 
0102 function W = compute_whiten_auto(X, evproc)
0103 
0104 if size(X, 1) > size(X, 2)
0105     W = compute_whiten_trans(X, evproc);
0106 else
0107     W = compute_whiten_std(X, evproc);
0108 end
0109 
0110 function W = compute_whiten_std(X, evproc)
0111 
0112 S = X * X';
0113 [evs, U] = slsymeig(S);
0114 [revs, U] = proc_eigs(evs, U, evproc);
0115 W = slmulvec(U, sqrt(revs)', 2);
0116 
0117 function W = compute_whiten_svd(X, evproc)
0118 
0119 [U, D] = svd(X, 0);
0120 evs = diag(D) .^ 2;
0121 clear D;
0122 
0123 [revs, U] = proc_eigs(evs, U, evproc);
0124 W = slmulvec(U, sqrt(revs)', 2);
0125 
0126 function W = compute_whiten_trans(X, evproc)
0127 
0128 S = X' * X;
0129 [evs, V] = slsymeig(S);
0130 U = X * V; 
0131 clear V;
0132 [revs, U] = proc_eigs(evs, U, evproc);
0133 U = slnormalize(U);
0134 
0135 W = slmulvec(U, sqrt(revs)', 2);
0136 
0137 
0138 
0139 %% The auxiliary function for obtaining truncated eigen-reciprocals
0140 
0141 function [revs, U] = proc_eigs(evs, U, evproc)
0142 
0143 revs = slinvevals(evs, evproc{:});
0144 
0145 si = find(revs == 0);
0146 if ~isempty(si)
0147     revs(si) = [];
0148     U(:, si) = [];
0149 end
0150
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