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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 slcovs
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slcovs

PURPOSE ^

SLCOVS Computes the sample covariance matrix

SYNOPSIS ^

function varargout = slcovs(X, w, nums, M)

DESCRIPTION ^

SLCOVS Computes the sample covariance matrix

 $ Syntax $
   - C = slcovs(X)
   - C = slcovs(X, w)
   - Cs = slcovs(X, [], nums)
   - Cs = slcovs(X, w, nums)
   - C = slcovs(X, w, [], M)
   - Cs = slcovs(X, w, nums, M)
   - [Cs, Cpool] = slcovs(...)

 $ Arguments $
   - X:           the sample matrix with each column representing a sample
   - w:           the sample weights
   - nums:        the number of samples in the groups
   - M:           the means or related information
   - C:           the overall covariance matrix
   - Cs:          the covariance matrices for the groups
   - Cpool:       the pooled covariance matrix

 $ Description $
   - C = slcovs(X) computes the covariance matrix for the samples in X.

   - C = slcovs(X, w) computes the weighted covariance matrix for the 
     samples in X, with weights specified in w.

   - C = slcovs(X, [], nums) computes the covariance matrices for 
     collections of samples, with the number of samples in each 
     collections specified by the row vector nums.

   - C = slcovs(X, w, nums) computes the weighted covariance matrices for
     collections of samples, with the sample weights specified in w,
     and the number of samples in collections specified in nums.

   - C = slcovs(X, w, [], M) also provides mean information via M. M = 0
     indicates that the mean vector is certainly zero vector. M can
     also be a d x 1 row vector specifying the (weighted) mean vector
     of X. If w is [], then all samples are treated equally.
         
   - Cs = slcovs(X, w, nums, M) computes the (weighted) covariance matrices 
     for collections of vectors. When computing normal covariance without 
     weighting you can set w = []. M = 0 indicates that the (weighted) mean 
     vector for all groups of samples are zero vectors. M can be a 
     d x k matrix storing the (weighted) mean vectors for all groups 
     of samples.

   - [Cs, Cpool] = slcovs(...) computes the pool covariance matrices for
     all collections of vectors. If the samples is not groupped, then
     Cpool is equal to Cs.

 $ History $
   - Created by Dahua Lin on Dec 18th, 2005

CROSS-REFERENCE INFORMATION ^

This function calls:
  • slcov SLCOV Compute the covariance matrix
  • slnums2bounds SLNUMS2BOUNDS Compute the index-boundaries from section sizes
This function is called by:

SUBFUNCTIONS ^

SOURCE CODE ^

0001 function varargout = slcovs(X, w, nums, M)
0002 %SLCOVS Computes the sample covariance matrix
0003 %
0004 % $ Syntax $
0005 %   - C = slcovs(X)
0006 %   - C = slcovs(X, w)
0007 %   - Cs = slcovs(X, [], nums)
0008 %   - Cs = slcovs(X, w, nums)
0009 %   - C = slcovs(X, w, [], M)
0010 %   - Cs = slcovs(X, w, nums, M)
0011 %   - [Cs, Cpool] = slcovs(...)
0012 %
0013 % $ Arguments $
0014 %   - X:           the sample matrix with each column representing a sample
0015 %   - w:           the sample weights
0016 %   - nums:        the number of samples in the groups
0017 %   - M:           the means or related information
0018 %   - C:           the overall covariance matrix
0019 %   - Cs:          the covariance matrices for the groups
0020 %   - Cpool:       the pooled covariance matrix
0021 %
0022 % $ Description $
0023 %   - C = slcovs(X) computes the covariance matrix for the samples in X.
0024 %
0025 %   - C = slcovs(X, w) computes the weighted covariance matrix for the
0026 %     samples in X, with weights specified in w.
0027 %
0028 %   - C = slcovs(X, [], nums) computes the covariance matrices for
0029 %     collections of samples, with the number of samples in each
0030 %     collections specified by the row vector nums.
0031 %
0032 %   - C = slcovs(X, w, nums) computes the weighted covariance matrices for
0033 %     collections of samples, with the sample weights specified in w,
0034 %     and the number of samples in collections specified in nums.
0035 %
0036 %   - C = slcovs(X, w, [], M) also provides mean information via M. M = 0
0037 %     indicates that the mean vector is certainly zero vector. M can
0038 %     also be a d x 1 row vector specifying the (weighted) mean vector
0039 %     of X. If w is [], then all samples are treated equally.
0040 %
0041 %   - Cs = slcovs(X, w, nums, M) computes the (weighted) covariance matrices
0042 %     for collections of vectors. When computing normal covariance without
0043 %     weighting you can set w = []. M = 0 indicates that the (weighted) mean
0044 %     vector for all groups of samples are zero vectors. M can be a
0045 %     d x k matrix storing the (weighted) mean vectors for all groups
0046 %     of samples.
0047 %
0048 %   - [Cs, Cpool] = slcovs(...) computes the pool covariance matrices for
0049 %     all collections of vectors. If the samples is not groupped, then
0050 %     Cpool is equal to Cs.
0051 %
0052 % $ History $
0053 %   - Created by Dahua Lin on Dec 18th, 2005
0054 %
0055 
0056 %% parse and verify input arguments
0057 
0058 % basic
0059 if ndims(X) ~= 2
0060     error('sltoolbox:invaliddims', 'X should be a 2D matrix');
0061 end
0062 [d, n] = size(X);
0063 
0064 % for weights
0065 if nargin < 2 || isempty(w)
0066     w = [];
0067 else
0068     if ~isequal(size(w), [1, n])
0069         error('sltoolbox:sizmismatch', ...
0070             'the weight vector should be a 1 x n row vector');
0071     end
0072 end
0073 
0074 % for groupping
0075 if nargin < 3 || isempty(nums)
0076     isgrouped = false;
0077     k = 1;
0078 else
0079     isgrouped = true;
0080     if size(nums, 1) ~= 1 
0081         error('sltoolbox:sizmismatch', ...
0082             'the nums vector should be a row vector');
0083     end
0084     if sum(nums) ~= n
0085         error('sltoolbox:sizmismatch', ...
0086             'the nums vector does not match the total number of vectors');
0087     end    
0088     k = length(nums);
0089 end
0090 
0091 % for mean vectors
0092 if nargin < 4
0093     M = [];
0094 elseif ~isempty(M) && ~isequal(M, 0) 
0095     if ~isequal(size(M), [d, k])        
0096         error('sltoolbox:sizmismatch', ...
0097             'the matrix of means should be a d x k matrix');
0098     end
0099 end
0100 
0101 
0102 %% compute the covariances
0103 
0104 if ~isgrouped      % not groupped
0105     
0106     % compute the covariance
0107     C = slcov(X, w, M);
0108     
0109     % output
0110     if nargout == 1
0111         varargout = {C};
0112     elseif nargout == 2
0113         varargout = {C, C};
0114     end
0115            
0116 else                % groupped
0117     
0118     % compute groupwise covariances
0119     Cs = compute_groupwise_covariances(nums, d, k, X, w, M);
0120     
0121     % output
0122     if nargout == 1        
0123         varargout = {Cs};        
0124     elseif nargout == 2        
0125         % compute pooled covariance
0126         Cpool = compute_pooled_covariance(Cs, d, k, nums, w);                        
0127         varargout = {Cs, Cpool};
0128     end
0129     
0130 end
0131  
0132 
0133 %% Sub functions
0134 
0135 
0136 %% the function for computing group-wise covariance matrices
0137 function Cs = compute_groupwise_covariances(nums, d, k, X, w, M)
0138 
0139 [sp, ep] = slnums2bounds(nums);
0140 
0141 % prepare storage
0142 Cs = zeros(d, d, k);
0143 
0144 if isempty(M)           % no precomputed means
0145 
0146     if isempty(w)
0147         for i = 1 : k
0148             Cs(:, :, i) = slcov(X(:, sp(i):ep(i)), [], []);
0149         end
0150     else
0151         for i = 1 : k
0152             Cs(:, :, i) = slcov(X(:, sp(i):ep(i)), w(sp(i):ep(i)), []);
0153         end
0154     end
0155    
0156 elseif isequal(M, 0)        % pre centralized
0157 
0158     if isempty(w)
0159         for i = 1 : k
0160             Cs(:, :, i) = slcov(X(:, sp(i):ep(i)), [], 0);
0161         end
0162     else
0163         for i = 1 : k
0164             Cs(:, :, i) = slcov(X(:, sp(i):ep(i)), w(sp(i):ep(i)), 0);
0165         end
0166     end
0167       
0168 else                        % precomputed means
0169 
0170     if isempty(w)
0171         for i = 1 : k
0172             Cs(:, :, i) = slcov(X(:, sp(i):ep(i)), [], M(:, i));
0173         end
0174     else
0175         for i = 1 : k
0176             Cs(:, :, i) = slcov(X(:, sp(i):ep(i)), w(sp(i):ep(i)), M(:, i));
0177         end
0178     end
0179 
0180 end
0181 
0182 
0183 %% the function for computing pooled covariance
0184 function Cpool = compute_pooled_covariance(Cs, d, k, nums, w)
0185 
0186 [sp, ep] = slnums2bounds(nums);
0187 
0188 % compute group weights
0189 if isempty(w)
0190     gw = nums;
0191 else
0192     gw = zeros(1, k);
0193     for i = 1 : k
0194         gw(i) = sum(w(sp(i):ep(i)));
0195     end
0196 end
0197 gw = gw(:) / sum(gw);
0198 
0199 % compute with reshape
0200 Cs = reshape(Cs, [d*d, k]);
0201 Cpool = Cs * gw;
0202 Cpool = reshape(Cpool, [d, d]);
0203 
0204 
0205
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