Description of slinvevals

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slinvevals

PURPOSE

SLINVEVALS Compute the reciprocals of eigenvalues in a robust way

SYNOPSIS

function revs = slinvevals(evals, method, r)

DESCRIPTION

SLINVEVALS Compute the reciprocals of eigenvalues in a robust way

 $ Syntax $
   - revs = slinvevals(evals)
   - revs = slinvevals(evals, method, r)

 $ Description $
   - revs = slinvevals(evals) computes the reciprocals of eigenvalues in
     the default way: using the default method and its corresponding
     default r value.
   
   - revs = slinvevals(evals, method, r) computes the reciprocals of 
     eigenvalues in a user-specified way.
     \*
     \t   Table 1. The methods of computing eigenvalue reciprocals
          name      &        revs
          'std'     & For effective eigenvalues, their reciprocals are
                      computed as usual; for the rest ones, their 
                      reciprocals are set to zeros. r values here is 
                      the ratio of minimum allowable effective eigenvalues
                      to the maximum eigenvalue. default r = 1e-7;
          'reg'     & Regularize the eigenvalues before computing their
                      reciprocals. By regularization, a small positive 
                      value is added to all eigenvalues. r value here is
                      the ratio of the addend to the maximum eigenvalue.
                      default r = 1e-6.
          'bound'   & Enforce lower bound to eigenvalues before computing
                      their reciprocals. The eigenvalues below the lower
                      bound is set to the lower bound value. r value here
                      is the ratio of the lower bound to the maximum 
                      eigenvalue. default r = 1e-6.
          'gapprox' & Computing the reciprocals of the eigenvalues in the
                      way of optimal non-singular approximation of 
                      Gaussian distribution. r values here is the ratio
                      of the minimum effective eigenvalues to the
                      maximum eigenvalue. The eigenvalues below are 
                      considered to be corresponding to isometric noises.
                      default r = 1e-6.
     \*

 $ Remarks $
   - The eigenvalues should be a column vector with values arranged in 
     a descending order.

 $ History $
   - Created by Dahua Lin on Apr 30th, 2006

CROSS-REFERENCE INFORMATION

This function calls:

This function is called by:

• slgaussinv SLGAUSSINV Computes the inverse of variance/covariance in Gaussian model
• slinvcov SLINVCOV Compute the inverse of an covariance matrix
• slwhiten_from_cov SLWHITEN_FROM_COV Compute the whitening transform from covariance matrix
• slwhiten_from_samples SLWHITEN_FROM_SAMPLES Compute the whitening matrix from sample matrix

SOURCE CODE

0001 function revs = slinvevals(evals, method, r)
0002 %SLINVEVALS Compute the reciprocals of eigenvalues in a robust way
0003 %
0004 % $ Syntax $
0005 %   - revs = slinvevals(evals)
0006 %   - revs = slinvevals(evals, method, r)
0007 %
0008 % $ Description $
0009 %   - revs = slinvevals(evals) computes the reciprocals of eigenvalues in
0010 %     the default way: using the default method and its corresponding
0011 %     default r value.
0012 %
0013 %   - revs = slinvevals(evals, method, r) computes the reciprocals of
0014 %     eigenvalues in a user-specified way.
0015 %     \*
0016 %     \t   Table 1. The methods of computing eigenvalue reciprocals
0017 %          name      &        revs
0018 %          'std'     & For effective eigenvalues, their reciprocals are
0019 %                      computed as usual; for the rest ones, their
0020 %                      reciprocals are set to zeros. r values here is
0021 %                      the ratio of minimum allowable effective eigenvalues
0022 %                      to the maximum eigenvalue. default r = 1e-7;
0023 %          'reg'     & Regularize the eigenvalues before computing their
0024 %                      reciprocals. By regularization, a small positive
0025 %                      value is added to all eigenvalues. r value here is
0026 %                      the ratio of the addend to the maximum eigenvalue.
0027 %                      default r = 1e-6.
0028 %          'bound'   & Enforce lower bound to eigenvalues before computing
0029 %                      their reciprocals. The eigenvalues below the lower
0030 %                      bound is set to the lower bound value. r value here
0031 %                      is the ratio of the lower bound to the maximum
0032 %                      eigenvalue. default r = 1e-6.
0033 %          'gapprox' & Computing the reciprocals of the eigenvalues in the
0034 %                      way of optimal non-singular approximation of
0035 %                      Gaussian distribution. r values here is the ratio
0036 %                      of the minimum effective eigenvalues to the
0037 %                      maximum eigenvalue. The eigenvalues below are
0038 %                      considered to be corresponding to isometric noises.
0039 %                      default r = 1e-6.
0040 %     \*
0041 %
0042 % $ Remarks $
0043 %   - The eigenvalues should be a column vector with values arranged in
0044 %     a descending order.
0045 %
0046 % $ History $
0047 %   - Created by Dahua Lin on Apr 30th, 2006
0048 %
0049 
0050 %% parse and verify input arguments
0051 
0052 n = length(evals);
0053 if n ~= numel(evals)
0054     error('sltoolbox:invalidarg', 'evals should be a vector');
0055 end
0056 
0057 if nargin < 2 || isempty(method)
0058     method = 'std';
0059 end
0060 
0061 %% compute
0062 
0063 % make strictly non-negative
0064 evals = max(evals, 0);
0065 
0066 switch method    
0067     case 'std'
0068         if nargin < 3 || isempty(r)
0069             r = 1e-7;
0070         end
0071         lb = r * evals(1);
0072         k = sum(evals >= lb);
0073         if k == n
0074             revs = 1 ./ evals;
0075         else
0076             revs = [1 ./ evals(1:k); zeros(n-k, 1)];
0077         end
0078         
0079     case 'reg'
0080         if nargin < 3 || isempty(r)
0081             r = 1e-6;
0082         end
0083         a = r * evals(1);
0084         revs = 1 ./ (evals + a);
0085         
0086     case 'bound'
0087         if nargin < 3 || isempty(r)
0088             r = 1e-6;
0089         end
0090         lb = r * evals(1);
0091         evals = max(evals, lb);
0092         revs = 1 ./ evals;
0093         
0094     case 'gapprox'
0095         if nargin < 3 || isempty(r)
0096             r = 1e-6;
0097         end
0098         lb = r * evals(1);
0099         k = sum(evals >= lb);
0100         if k == n
0101             revs = 1 ./ evals;
0102         else
0103             nv = sum(evals(k+1:n)) / (n-k);
0104             rnv = 1 / nv;
0105             rnvs = rnv(ones(n-k, 1));
0106             revs = [1 ./ evals(1:k); rnvs];
0107         end
0108         
0109 end
0110

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