function r = ranks(X,DIM,Mode); % RANKS gives the rank of each element in a vector. % This program uses an advanced algorithm with averge effort O(m.n.log(n)) % NaN in the input yields NaN in the output. % % r = ranks(X[,DIM]) % if X is a vector, return the vector of ranks of X adjusted for ties. % if X is matrix, the rank is calculated along dimension DIM. % if DIM is zero or empty, the lowest dimension with more then 1 element is used. % r = ranks(X,DIM,'traditional') % implements the traditional algorithm with O(n^2) computational % and O(n^2) memory effort % r = ranks(X,DIM,'mtraditional') % implements the traditional algorithm with O(n^2) computational % and O(n) memory effort % r = ranks(X,DIM,'advanced ') % implements an advanced algorithm with O(n*log(n)) computational % and O(n.log(n)) memory effort % % see also: CORRCOEF, SPEARMAN, RANKCORR % % REFERENCES: % -- % $Id: ranks.m 4585 2008-02-04 13:47:45Z adb014 $ % Copyright (C) 2000-2002,2005 by Alois Schloegl <a.schloegl@ieee.org> % This script is part of the NaN-toolbox % http://www.dpmi.tu-graz.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 2 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: % + is fast, uses an efficient algorithm for the rank correlation % + computational effort is O(n.log(n)) instead of O(n^2) % + memory effort is O(n.log(n)), instead of O(n^2). % Now, the ranks of 8000 elements can be easily calculated % + NaN's in the input yield NaN in the output % + compatible with this software and Matlab % + traditional method is also implemented for comparison. if nargin<2, DIM = 0; end; if ischar(DIM), Mode= DIM; DIM = 0; elseif (nargin<3), Mode = ''; end; if isempty(Mode), Mode='advanced '; end; sz = size(X); if (~DIM) [tmp,DIM] = min(find(sz>1)); end; [N,M] = size(X); if (DIM==2), X = X'; [N,M] = size(X); end; if strcmp(Mode(1:min(11,length(Mode))),'traditional'), % traditional, needs O(m.n^2) % this method was originally implemented by: KH <Kurt.Hornik@ci.tuwien.ac.at> % Comment of KH: This code is rather ugly, but is there an easy way to get the ranks adjusted for ties from sort? r = zeros(size(X)); for i = 1:M; p = X(:, i(ones(1,N))); r(:,i) = [(sum (p < p') + (sum (p == p') + 1) / 2)']; end; % r(r<1)=NaN; elseif strcmp(Mode(1:min(12,length(Mode))),'mtraditional'), % advanced % + memory effort is lower r = zeros(size(X)); for k = 1:N; for i = 1:M; r(k,i) = [(sum (X(:,i) < X(k,i)) + (sum (X(:,i) == X(k,i)) + 1) / 2)]; end; end; % r(r<1)=NaN; elseif strcmp(Mode(1:min(11,length(Mode))),'advanced '), % advanced % + uses sorting, hence needs only O(m.n.log(n)) computations % [tmp,ix] = sort([X,Y]); % [tmp,r] = sort(ix); % r yields rank. % but because sort does not work accordingly for cell arrays, % and DIM argument not supported by Octave % and DIM argument does not work for cell-arrays in Matlab % we sort each column separately: r = zeros(size(X)); n = N; for k = 1:M, [sX,ix] = sort(X(:,k)); [tmp,r(:,k)] = sort(ix); % r yields the rank of each element % identify multiple occurences (not sure if this important, but implemented to be compatible with traditional version) if isnumeric(X) n=sum(~isnan(X(:,k))); end; x = [0;find(sX~=[sX(2:N);n])]; % for this reason, cells are not implemented yet. d = find(diff(x)>1); % correct rank of multiple occurring elements for l = 1:length(d), t = (x(d(l))+1:x(d(l)+1))'; r(ix(t),k) = mean(t); end; end; tmp = version; if str2num(tmp(1))*1000+str2num(tmp(3))*100+str2num(tmp(5))*10<2020, for k1=1:size(X,1), for k2=1:size(X,2), % needed for 2.0.17 if isnan(X(k1,k2)), r(k1,k2) = nan; end; end; end; else r(isnan(X)) = nan; end; elseif strcmp(Mode,'=='), % the results of both algorithms are compared for testing. % % if the Mode-argument is omitted, both methods are applied and % the results are compared. Once the advanced algorithm is confirmed, % it will become the default Mode. r = ranks(X,'advanced '); r(isnan(r)) = 1/2; if N>100, r1 = ranks(X,'mtraditional'); % Memory effort is lower else r1 = ranks(X,'traditional'); end; if ~all(all(r==r1)), fprintf(2,'WARNING RANKS: advanced algorithm does not agree with traditional one\n Please report to <a.schloegl@ieee.org>\n'); r = r1; end; r(isnan(X)) = nan; end; if (DIM==2) r=r'; end;

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