/usr/share/octave/packages/3.2/nan-2.4.4/corrcoef.m is in octave-nan 2.4.4-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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% CORRCOEF calculates the correlation matrix from pairwise correlations.
% The input data can contain missing values encoded with NaN.
% Missing data (NaN's) are handled by pairwise deletion [15].
% In order to avoid possible pitfalls, use case-wise deletion or
% or check the correlation of NaN's with your data (see below).
% A significance test for testing the Hypothesis
% 'correlation coefficient R is significantly different to zero'
% is included.
%
% [...] = CORRCOEF(X);
% calculates the (auto-)correlation matrix of X
% [...] = CORRCOEF(X,Y);
% calculates the crosscorrelation between X and Y
%
% [...] = CORRCOEF(..., Mode);
% Mode='Pearson' or 'parametric' [default]
% gives the correlation coefficient
% also known as the 'product-moment coefficient of correlation'
% or 'Pearson''s correlation' [1]
% Mode='Spearman' gives 'Spearman''s Rank Correlation Coefficient'
% This replaces SPEARMAN.M
% Mode='Rank' gives a nonparametric Rank Correlation Coefficient
% This is the "Spearman rank correlation with proper handling of ties"
% This replaces RANKCORR.M
%
% [...] = CORRCOEF(..., param1, value1, param2, value2, ... );
% param value
% 'Mode' type of correlation
% 'Pearson','parametric'
% 'Spearman'
% 'rank'
% 'rows' how do deal with missing values encoded as NaN's.
% 'complete': remove all rows with at least one NaN
% 'pairwise': [default]
% 'alpha' 0.01 : significance level to compute confidence interval
%
% [R,p,ci1,ci2,nansig] = CORRCOEF(...);
% R is the correlation matrix
% R(i,j) is the correlation coefficient r between X(:,i) and Y(:,j)
% p gives the significance of R
% It tests the null hypothesis that the product moment correlation coefficient is zero
% using Student's t-test on the statistic t = r*sqrt(N-2)/sqrt(1-r^2)
% where N is the number of samples (Statistics, M. Spiegel, Schaum series).
% p > alpha: do not reject the Null hypothesis: 'R is zero'.
% p < alpha: The alternative hypothesis 'R is larger than zero' is true with probability (1-alpha).
% ci1 lower (1-alpha) confidence interval
% ci2 upper (1-alpha) confidence interval
% If no alpha is provided, the default alpha is 0.01. This can be changed with function flag_implicit_significance.
% nan_sig p-value whether H0: 'NaN''s are not correlated' could be correct
% if nan_sig < alpha, H1 ('NaNs are correlated') is very likely.
%
% The result is only valid if the occurence of NaN's is uncorrelated. In
% order to avoid this pitfall, the correlation of NaN's should be checked
% or case-wise deletion should be applied.
% Case-Wise deletion can be implemented
% ix = ~any(isnan([X,Y]),2);
% [...] = CORRCOEF(X(ix,:),Y(ix,:),...);
%
% Correlation (non-random distribution) of NaN's can be checked with
% [nan_R,nan_sig]=corrcoef(X,isnan(X))
% or [nan_R,nan_sig]=corrcoef([X,Y],isnan([X,Y]))
% or [R,p,ci1,ci2] = CORRCOEF(...);
%
% Further recommandation related to the correlation coefficient:
% + LOOK AT THE SCATTERPLOTS to make sure that the relationship is linear
% + Correlation is not causation because
% it is not clear which parameter is 'cause' and which is 'effect' and
% the observed correlation between two variables might be due to the action of other, unobserved variables.
%
% see also: SUMSKIPNAN, COVM, COV, COR, SPEARMAN, RANKCORR, RANKS,
% PARTCORRCOEF, flag_implicit_significance
%
% REFERENCES:
% on the correlation coefficient
% [ 1] http://mathworld.wolfram.com/CorrelationCoefficient.html
% [ 2] http://www.geography.btinternet.co.uk/spearman.htm
% [ 3] Hogg, R. V. and Craig, A. T. Introduction to Mathematical Statistics, 5th ed. New York: Macmillan, pp. 338 and 400, 1995.
% [ 4] Lehmann, E. L. and D'Abrera, H. J. M. Nonparametrics: Statistical Methods Based on Ranks, rev. ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 292, 300, and 323, 1998.
% [ 5] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 634-637, 1992
% [ 6] http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html
% on the significance test of the correlation coefficient
% [11] http://www.met.rdg.ac.uk/cag/STATS/corr.html
% [12] http://www.janda.org/c10/Lectures/topic06/L24-significanceR.htm
% [13] http://faculty.vassar.edu/lowry/ch4apx.html
% [14] http://davidmlane.com/hyperstat/B134689.html
% [15] http://www.statsoft.com/textbook/stbasic.html%Correlations
% others
% [20] http://www.tufts.edu/~gdallal/corr.htm
% $Id: corrcoef.m 8223 2011-04-20 09:16:06Z schloegl $
% Copyright (C) 2000-2004,2008,2009 by Alois Schloegl <alois.schloegl@gmail.com>
% This function is part of the NaN-toolbox
% http://pub.ist.ac.at/~schloegl/matlab/NaN/
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
% Features:
% + handles missing values (encoded as NaN's)
% + pairwise deletion of missing data
% + checks independence of missing values (NaNs)
% + parametric and non-parametric (rank) correlation
% + Pearson's correlation
% + Spearman's rank correlation
% + Rank correlation (non-parametric, Spearman rank correlation with proper handling of ties)
% + is fast, using an efficient algorithm O(n.log(n)) for calculating the ranks
% + significance test for null-hypthesis: r=0
% + confidence interval included
% - rank correlation works for cell arrays, too (no check for missing values).
% + compatible with Octave and Matlab
global FLAG_NANS_OCCURED;
NARG = nargout; % needed because nargout is not reentrant in Octave, and corrcoef is recursive
mode = [];
if nargin==1
Y = [];
Mode='Pearson';
elseif nargin==0
fprintf(2,'Error CORRCOEF: Missing argument(s)\n');
elseif nargin>1
if ischar(Y)
varg = [Y,varargin];
Y=[];
else
varg = varargin;
end;
if length(varg)<1,
Mode = 'Pearson';
elseif length(varg)==1,
Mode = varg{1};
else
for k = 2:2:length(varg),
mode = setfield(mode,lower(varg{k-1}),varg{k});
end;
if isfield(mode,'mode')
Mode = mode.mode;
end;
end;
end;
if isempty(Mode) Mode='pearson'; end;
Mode=[Mode,' '];
FLAG_WARNING = warning; % save warning status
warning('off');
[r1,c1]=size(X);
if ~isempty(Y)
[r2,c2]=size(Y);
if r1~=r2,
fprintf(2,'Error CORRCOEF: X and Y must have the same number of observations (rows).\n');
return;
end;
NN = real(~isnan(X)')*real(~isnan(Y));
else
[r2,c2]=size(X);
NN = real(~isnan(X)')*real(~isnan(X));
end;
%%%%% generate combinations using indices for pairwise calculation of the correlation
YESNAN = any(isnan(X(:))) | any(isnan(Y(:)));
if YESNAN,
FLAG_NANS_OCCURED=(1==1);
if isfield(mode,'rows')
if strcmp(mode.rows,'complete')
ix = ~any([X,Y],2);
X = X(ix,:);
if ~isempty(Y)
Y = Y(ix,:);
end;
YESNAN = 0;
NN = size(X,1);
elseif strcmp(mode.rows,'all')
fprintf(1,'Warning: data contains NaNs, rows=pairwise is used.');
%%NN(NN < size(X,1)) = NaN;
elseif strcmp(mode.rows,'pairwise')
%%% default
end;
end;
end;
if isempty(Y),
IX = ones(c1)-diag(ones(c1,1));
[jx, jy ] = find(IX);
[jxo,jyo] = find(IX);
R = eye(c1);
else
IX = sparse([],[],[],c1+c2,c1+c2,c1*c2);
IX(1:c1,c1+(1:c2)) = 1;
[jx,jy] = find(IX);
IX = ones(c1,c2);
[jxo,jyo] = find(IX);
R = zeros(c1,c2);
end;
if strcmp(lower(Mode(1:7)),'pearson');
% see http://mathworld.wolfram.com/CorrelationCoefficient.html
if ~YESNAN,
[S,N,SSQ] = sumskipnan(X,1);
if ~isempty(Y),
[S2,N2,SSQ2] = sumskipnan(Y,1);
CC = X'*Y;
M1 = S./N;
M2 = S2./N2;
cc = CC./NN - M1'*M2;
R = cc./sqrt((SSQ./N-M1.*M1)'*(SSQ2./N2-M2.*M2));
else
CC = X'*X;
M = S./N;
cc = CC./NN - M'*M;
v = SSQ./N - M.*M; %max(N-1,0);
R = cc./sqrt(v'*v);
end;
else
if ~isempty(Y),
X = [X,Y];
end;
for k = 1:length(jx),
%ik = ~any(isnan(X(:,[jx(k),jy(k)])),2);
ik = ~isnan(X(:,jx(k))) & ~isnan(X(:,jy(k)));
[s,n,s2] = sumskipnan(X(ik,[jx(k),jy(k)]),1);
v = (s2-s.*s./n)./n;
cc = X(ik,jx(k))'*X(ik,jy(k));
cc = cc/n(1) - prod(s./n);
%r(k) = cc./sqrt(prod(v));
R(jxo(k),jyo(k)) = cc./sqrt(prod(v));
end;
end
elseif strcmp(lower(Mode(1:4)),'rank');
% see [ 6] http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html
if ~YESNAN,
if isempty(Y)
R = corrcoef(ranks(X));
else
R = corrcoef(ranks(X),ranks(Y));
end;
else
if ~isempty(Y),
X = [X,Y];
end;
for k = 1:length(jx),
%ik = ~any(isnan(X(:,[jx(k),jy(k)])),2);
ik = ~isnan(X(:,jx(k))) & ~isnan(X(:,jy(k)));
il = ranks(X(ik,[jx(k),jy(k)]));
R(jxo(k),jyo(k)) = corrcoef(il(:,1),il(:,2));
end;
X = ranks(X);
end;
elseif strcmp(lower(Mode(1:8)),'spearman');
% see [ 6] http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html
if ~isempty(Y),
X = [X,Y];
end;
n = repmat(nan,c1,c2);
if ~YESNAN,
iy = ranks(X); % calculates ranks;
for k = 1:length(jx),
[R(jxo(k),jyo(k)),n(jxo(k),jyo(k))] = sumskipnan((iy(:,jx(k)) - iy(:,jy(k))).^2); % NN is the number of non-missing values
end;
else
for k = 1:length(jx),
%ik = ~any(isnan(X(:,[jx(k),jy(k)])),2);
ik = ~isnan(X(:,jx(k))) & ~isnan(X(:,jy(k)));
il = ranks(X(ik,[jx(k),jy(k)]));
% NN is the number of non-missing values
[R(jxo(k),jyo(k)),n(jxo(k),jyo(k))] = sumskipnan((il(:,1) - il(:,2)).^2);
end;
X = ranks(X);
end;
R = 1 - 6 * R ./ (n.*(n.*n-1));
elseif strcmp(lower(Mode(1:7)),'partial');
fprintf(2,'Error CORRCOEF: use PARTCORRCOEF \n',Mode);
return;
elseif strcmp(lower(Mode(1:7)),'kendall');
fprintf(2,'Error CORRCOEF: mode ''%s'' not implemented yet.\n',Mode);
return;
else
fprintf(2,'Error CORRCOEF: unknown mode ''%s''\n',Mode);
end;
if (NARG<2),
warning(FLAG_WARNING); % restore warning status
return;
end;
% CONFIDENCE INTERVAL
if isfield(mode,'alpha')
alpha = mode.alpha;
elseif exist('flag_implicit_significance','file'),
alpha = flag_implicit_significance;
else
alpha = 0.01;
end;
% fprintf(1,'CORRCOEF: confidence interval is based on alpha=%f\n',alpha);
% SIGNIFICANCE TEST
R(isnan(R))=0;
tmp = 1 - R.*R;
tmp(tmp<0) = 0; % prevent tmp<0 i.e. imag(t)~=0
t = R.*sqrt(max(NN-2,0)./tmp);
if exist('t_cdf','file');
sig = t_cdf(t,NN-2);
elseif exist('tcdf','file')>1;
sig = tcdf(t,NN-2);
else
fprintf('CORRCOEF: significance test not completed because of missing TCDF-function\n')
sig = repmat(nan,size(R));
end;
sig = 2 * min(sig,1 - sig);
if NARG<3,
warning(FLAG_WARNING); % restore warning status
return;
end;
tmp = R;
%tmp(ix1 | ix2) = nan; % avoid division-by-zero warning
z = log((1+tmp)./(1-tmp))/2; % Fisher's z-transform;
%sz = 1./sqrt(NN-3); % standard error of z
sz = sqrt(2)*erfinv(1-alpha)./sqrt(NN-3); % confidence interval for alpha of z
ci1 = tanh(z-sz);
ci2 = tanh(z+sz);
%ci1(isnan(ci1))=R(isnan(ci1)); % in case of isnan(ci), the interval limits are exactly the R value
%ci2(isnan(ci2))=R(isnan(ci2));
if (NARG<5) || ~YESNAN,
nan_sig = repmat(NaN,size(R));
warning(FLAG_WARNING); % restore warning status
return;
end;
%%%%% ----- check independence of NaNs (missing values) -----
[nan_R, nan_sig] = corrcoef(X,double(isnan(X)));
% remove diagonal elements, because these have not any meaning %
nan_sig(isnan(nan_R)) = nan;
% remove diagonal elements, because these have not any meaning %
nan_R(isnan(nan_R)) = 0;
if 0, any(nan_sig(:) < alpha),
tmp = nan_sig(:); % Hack to skip NaN's in MIN(X)
min_sig = min(tmp(~isnan(tmp))); % Necessary, because Octave returns NaN rather than min(X) for min(NaN,X)
fprintf(1,'CORRCOFF Warning: Missing Values (i.e. NaNs) are not independent of data (p-value=%f)\n', min_sig);
fprintf(1,' Its recommended to remove all samples (i.e. rows) with any missing value (NaN).\n');
fprintf(1,' The null-hypotheses (NaNs are uncorrelated) is rejected for the following parameter pair(s).\n');
[ix,iy] = find(nan_sig < alpha);
disp([ix,iy])
end;
%%%%% ----- end of independence check ------
warning(FLAG_WARNING); % restore warning status
return;
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