386 lines
14 KiB
Matlab
386 lines
14 KiB
Matlab
function [R,sig,ci1,ci2,nan_sig] = corrcoef(X,Y,varargin)
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% CORRCOEF calculates the correlation matrix from pairwise correlations.
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% The input data can contain missing values encoded with NaN.
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% Missing data (NaN's) are handled by pairwise deletion [15].
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% In order to avoid possible pitfalls, use case-wise deletion or
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% or check the correlation of NaN's with your data (see below).
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% A significance test for testing the Hypothesis
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% 'correlation coefficient R is significantly different to zero'
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% is included.
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%
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% [...] = CORRCOEF(X);
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% calculates the (auto-)correlation matrix of X
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% [...] = CORRCOEF(X,Y);
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% calculates the crosscorrelation between X and Y
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%
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% [...] = CORRCOEF(..., Mode);
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% Mode='Pearson' or 'parametric' [default]
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% gives the correlation coefficient
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% also known as the 'product-moment coefficient of correlation'
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% or 'Pearson''s correlation' [1]
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% Mode='Spearman' gives 'Spearman''s Rank Correlation Coefficient'
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% This replaces SPEARMAN.M
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% Mode='Rank' gives a nonparametric Rank Correlation Coefficient
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% This is the "Spearman rank correlation with proper handling of ties"
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% This replaces RANKCORR.M
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%
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% [...] = CORRCOEF(..., param1, value1, param2, value2, ... );
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% param value
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% 'Mode' type of correlation
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% 'Pearson','parametric'
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% 'Spearman'
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% 'rank'
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% 'rows' how do deal with missing values encoded as NaN's.
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% 'complete': remove all rows with at least one NaN
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% 'pairwise': [default]
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% 'alpha' 0.01 : significance level to compute confidence interval
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%
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% [R,p,ci1,ci2,nansig] = CORRCOEF(...);
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% R is the correlation matrix
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% R(i,j) is the correlation coefficient r between X(:,i) and Y(:,j)
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% p gives the significance of R
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% It tests the null hypothesis that the product moment correlation coefficient is zero
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% using Student's t-test on the statistic t = r*sqrt(N-2)/sqrt(1-r^2)
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% where N is the number of samples (Statistics, M. Spiegel, Schaum series).
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% p > alpha: do not reject the Null hypothesis: 'R is zero'.
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% p < alpha: The alternative hypothesis 'R is larger than zero' is true with probability (1-alpha).
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% ci1 lower (1-alpha) confidence interval
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% ci2 upper (1-alpha) confidence interval
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% If no alpha is provided, the default alpha is 0.01. This can be changed with function flag_implicit_significance.
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% nan_sig p-value whether H0: 'NaN''s are not correlated' could be correct
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% if nan_sig < alpha, H1 ('NaNs are correlated') is very likely.
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%
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% The result is only valid if the occurence of NaN's is uncorrelated. In
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% order to avoid this pitfall, the correlation of NaN's should be checked
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% or case-wise deletion should be applied.
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% Case-Wise deletion can be implemented
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% ix = ~any(isnan([X,Y]),2);
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% [...] = CORRCOEF(X(ix,:),Y(ix,:),...);
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%
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% Correlation (non-random distribution) of NaN's can be checked with
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% [nan_R,nan_sig]=corrcoef(X,isnan(X))
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% or [nan_R,nan_sig]=corrcoef([X,Y],isnan([X,Y]))
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% or [R,p,ci1,ci2] = CORRCOEF(...);
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%
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% Further recommandation related to the correlation coefficient:
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% + LOOK AT THE SCATTERPLOTS to make sure that the relationship is linear
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% + Correlation is not causation because
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% it is not clear which parameter is 'cause' and which is 'effect' and
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% the observed correlation between two variables might be due to the action of other, unobserved variables.
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%
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% see also: SUMSKIPNAN, COVM, COV, COR, SPEARMAN, RANKCORR, RANKS,
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% PARTCORRCOEF, flag_implicit_significance
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%
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% REFERENCES:
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% on the correlation coefficient
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% [ 1] http://mathworld.wolfram.com/CorrelationCoefficient.html
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% [ 2] http://www.geography.btinternet.co.uk/spearman.htm
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% [ 3] Hogg, R. V. and Craig, A. T. Introduction to Mathematical Statistics, 5th ed. New York: Macmillan, pp. 338 and 400, 1995.
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% [ 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.
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% [ 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
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% [ 6] http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html
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% on the significance test of the correlation coefficient
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% [11] http://www.met.rdg.ac.uk/cag/STATS/corr.html
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% [12] http://www.janda.org/c10/Lectures/topic06/L24-significanceR.htm
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% [13] http://faculty.vassar.edu/lowry/ch4apx.html
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% [14] http://davidmlane.com/hyperstat/B134689.html
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% [15] http://www.statsoft.com/textbook/stbasic.html%Correlations
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% others
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% [20] http://www.tufts.edu/~gdallal/corr.htm
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% [21] Fisher transformation http://en.wikipedia.org/wiki/Fisher_transformation
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% $Id: corrcoef.m 9387 2011-12-15 10:42:14Z schloegl $
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% Copyright (C) 2000-2004,2008,2009,2011 by Alois Schloegl <alois.schloegl@gmail.com>
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% Copyright (C) 2014-2017 Dynare Team
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% This function is part of the NaN-toolbox
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% http://pub.ist.ac.at/~schloegl/matlab/NaN/
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% This program is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with this program. If not, see <http://www.gnu.org/licenses/>.
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% Features:
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% + handles missing values (encoded as NaN's)
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% + pairwise deletion of missing data
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% + checks independence of missing values (NaNs)
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% + parametric and non-parametric (rank) correlation
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% + Pearson's correlation
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% + Spearman's rank correlation
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% + Rank correlation (non-parametric, Spearman rank correlation with proper handling of ties)
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% + is fast, using an efficient algorithm O(n.log(n)) for calculating the ranks
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% + significance test for null-hypthesis: r=0
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% + confidence interval included
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% - rank correlation works for cell arrays, too (no check for missing values).
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% + compatible with Octave and Matlab
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global FLAG_NANS_OCCURED;
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NARG = nargout; % needed because nargout is not reentrant in Octave, and corrcoef is recursive
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mode = [];
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if nargin==1
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Y = [];
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Mode='Pearson';
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elseif nargin==0
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fprintf(2,'Error CORRCOEF: Missing argument(s)\n');
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elseif nargin>1
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if ischar(Y)
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varg = [Y,varargin];
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Y=[];
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else
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varg = varargin;
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end
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if length(varg)<1
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Mode = 'Pearson';
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elseif length(varg)==1
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Mode = varg{1};
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else
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for k = 2:2:length(varg)
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mode = setfield(mode,lower(varg{k-1}),varg{k});
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end
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if isfield(mode,'mode')
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Mode = mode.mode;
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end
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end
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end
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if isempty(Mode), Mode='pearson'; end
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Mode=[Mode,' '];
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FLAG_WARNING = warning; % save warning status
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warning('off');
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[r1,c1]=size(X);
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if ~isempty(Y)
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[r2,c2]=size(Y);
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if r1~=r2
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fprintf(2,'Error CORRCOEF: X and Y must have the same number of observations (rows).\n');
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return
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end
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NN = real(~isnan(X)')*real(~isnan(Y));
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else
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[r2,c2]=size(X);
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NN = real(~isnan(X)')*real(~isnan(X));
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end
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%%%%% generate combinations using indices for pairwise calculation of the correlation
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YESNAN = any(isnan(X(:))) | any(isnan(Y(:)));
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if YESNAN
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FLAG_NANS_OCCURED=(1==1);
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if isfield(mode,'rows')
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if strcmp(mode.rows,'complete')
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ix = ~any([X,Y],2);
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X = X(ix,:);
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if ~isempty(Y)
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Y = Y(ix,:);
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end
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YESNAN = 0;
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NN = size(X,1);
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elseif strcmp(mode.rows,'all')
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fprintf(1,'Warning: data contains NaNs, rows=pairwise is used.');
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%%NN(NN < size(X,1)) = NaN;
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elseif strcmp(mode.rows,'pairwise')
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%%% default
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end
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end
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end
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if isempty(Y)
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IX = ones(c1)-diag(ones(c1,1));
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[jx, jy ] = find(IX);
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[jxo,jyo] = find(IX);
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R = eye(c1);
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else
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IX = sparse([],[],[],c1+c2,c1+c2,c1*c2);
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IX(1:c1,c1+(1:c2)) = 1;
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[jx,jy] = find(IX);
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IX = ones(c1,c2);
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[jxo,jyo] = find(IX);
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R = zeros(c1,c2);
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end
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if strcmp(lower(Mode(1:7)),'pearson')
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% see http://mathworld.wolfram.com/CorrelationCoefficient.html
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if ~YESNAN
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[S,N,SSQ] = sumskipnan(X,1);
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if ~isempty(Y)
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[S2,N2,SSQ2] = sumskipnan(Y,1);
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CC = X'*Y;
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M1 = S./N;
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M2 = S2./N2;
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cc = CC./NN - M1'*M2;
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R = cc./sqrt((SSQ./N-M1.*M1)'*(SSQ2./N2-M2.*M2));
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else
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CC = X'*X;
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M = S./N;
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cc = CC./NN - M'*M;
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v = SSQ./N - M.*M; %max(N-1,0);
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R = cc./sqrt(v'*v);
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end
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else
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if ~isempty(Y)
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X = [X,Y];
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end
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for k = 1:length(jx)
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%ik = ~any(isnan(X(:,[jx(k),jy(k)])),2);
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ik = ~isnan(X(:,jx(k))) & ~isnan(X(:,jy(k)));
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[s,n,s2] = sumskipnan(X(ik,[jx(k),jy(k)]),1);
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v = (s2-s.*s./n)./n;
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cc = X(ik,jx(k))'*X(ik,jy(k));
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cc = cc/n(1) - prod(s./n);
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%r(k) = cc./sqrt(prod(v));
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R(jxo(k),jyo(k)) = cc./sqrt(prod(v));
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end
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end
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elseif strcmp(lower(Mode(1:4)),'rank')
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% see [ 6] http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html
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if ~YESNAN
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if isempty(Y)
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R = corrcoef(ranks(X));
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else
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R = corrcoef(ranks(X),ranks(Y));
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end
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else
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if ~isempty(Y)
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X = [X,Y];
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end
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for k = 1:length(jx)
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%ik = ~any(isnan(X(:,[jx(k),jy(k)])),2);
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ik = ~isnan(X(:,jx(k))) & ~isnan(X(:,jy(k)));
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il = ranks(X(ik,[jx(k),jy(k)]));
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R(jxo(k),jyo(k)) = corrcoef(il(:,1),il(:,2));
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end
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X = ranks(X);
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end
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elseif strcmp(lower(Mode(1:8)),'spearman')
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% see [ 6] http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html
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if ~isempty(Y)
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X = [X,Y];
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end
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n = repmat(nan,c1,c2);
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if ~YESNAN
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iy = ranks(X); % calculates ranks;
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for k = 1:length(jx)
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[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
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end
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else
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for k = 1:length(jx)
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%ik = ~any(isnan(X(:,[jx(k),jy(k)])),2);
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ik = ~isnan(X(:,jx(k))) & ~isnan(X(:,jy(k)));
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il = ranks(X(ik,[jx(k),jy(k)]));
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% NN is the number of non-missing values
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[R(jxo(k),jyo(k)),n(jxo(k),jyo(k))] = sumskipnan((il(:,1) - il(:,2)).^2);
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end
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X = ranks(X);
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end
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R = 1 - 6 * R ./ (n.*(n.*n-1));
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elseif strcmp(lower(Mode(1:7)),'partial')
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fprintf(2,'Error CORRCOEF: use PARTCORRCOEF \n',Mode);
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return
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elseif strcmp(lower(Mode(1:7)),'kendall')
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fprintf(2,'Error CORRCOEF: mode ''%s'' not implemented yet.\n',Mode);
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return
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else
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fprintf(2,'Error CORRCOEF: unknown mode ''%s''\n',Mode);
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end
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if (NARG<2)
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warning(FLAG_WARNING); % restore warning status
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return
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end
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% CONFIDENCE INTERVAL
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if isfield(mode,'alpha')
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alpha = mode.alpha;
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elseif exist('flag_implicit_significance','file')
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alpha = flag_implicit_significance;
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else
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alpha = 0.01;
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end
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% fprintf(1,'CORRCOEF: confidence interval is based on alpha=%f\n',alpha);
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% SIGNIFICANCE TEST
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R(isnan(R))=0;
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tmp = 1 - R.*R;
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tmp(tmp<0) = 0; % prevent tmp<0 i.e. imag(t)~=0
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t = R.*sqrt(max(NN-2,0)./tmp);
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if exist('t_cdf','file')
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sig = t_cdf(t,NN-2);
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elseif exist('tcdf','file')>1
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sig = tcdf(t,NN-2);
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else
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fprintf('CORRCOEF: significance test not completed because of missing TCDF-function\n')
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sig = repmat(nan,size(R));
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end
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sig = 2 * min(sig,1 - sig);
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if NARG<3
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warning(FLAG_WARNING); % restore warning status
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return
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end
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tmp = R;
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%tmp(ix1 | ix2) = nan; % avoid division-by-zero warning
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z = log((1+tmp)./(1-tmp))/2; % Fisher transformation [21]
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%sz = 1./sqrt(NN-3); % standard error of z
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sz = sqrt(2)*erfinv(1-alpha)./sqrt(NN-3); % confidence interval for alpha of z
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ci1 = tanh(z-sz);
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ci2 = tanh(z+sz);
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%ci1(isnan(ci1))=R(isnan(ci1)); % in case of isnan(ci), the interval limits are exactly the R value
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%ci2(isnan(ci2))=R(isnan(ci2));
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if (NARG<5) || ~YESNAN
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nan_sig = repmat(NaN,size(R));
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warning(FLAG_WARNING); % restore warning status
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return
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end
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%%%%% ----- check independence of NaNs (missing values) -----
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[nan_R, nan_sig] = corrcoef(X,double(isnan(X)));
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% remove diagonal elements, because these have not any meaning %
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nan_sig(isnan(nan_R)) = nan;
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% remove diagonal elements, because these have not any meaning %
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nan_R(isnan(nan_R)) = 0;
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if 0, any(nan_sig(:) < alpha)
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tmp = nan_sig(:); % Hack to skip NaN's in MIN(X)
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min_sig = min(tmp(~isnan(tmp))); % Necessary, because Octave returns NaN rather than min(X) for min(NaN,X)
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fprintf(1,'CORRCOFF Warning: Missing Values (i.e. NaNs) are not independent of data (p-value=%f)\n', min_sig);
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fprintf(1,' Its recommended to remove all samples (i.e. rows) with any missing value (NaN).\n');
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fprintf(1,' The null-hypotheses (NaNs are uncorrelated) is rejected for the following parameter pair(s).\n');
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[ix,iy] = find(nan_sig < alpha);
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disp([ix,iy])
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end
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%%%%% ----- end of independence check ------
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warning(FLAG_WARNING); % restore warning status
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