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kstest2d.m
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kstest2d.m
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% kstest2d Two-dimensional, 2-sample Kolmorogov-Smirnov test
%
% [p,D] = kstest2d(s1,s2);
%
% Compare two, 2-dimensional distributions using Fasano & Franceschini's
% generalization of the KS-test.
%
% The analytic distribution of the statistic is unknown, and p-values
% are estimated using an approximation (Press et al., 1992) to FF's Monte
% Carlo simulations.
%
% INPUTS
% s1 - [n1 x 2] matrix
% s2 - [n2 x 2] matrix
%
% OUTPUTS
% p - approximate p-value
% D - K-S statistic
%
% REFERENCE
% Fasano, G, Franceschini, A (1987) A multidimensional version of the
% Kolmorogov-Smirnov test. Mon Not R astr Soc 225: 155-170
% Press et al (1992). Numerical Recipes in C, section 14.7
%
% SEE ALSO
% minentest, hotell2
% $ Copyright (C) 2014 Brian Lau http://www.subcortex.net/ $
% The full license and most recent version of the code can be found on GitHub:
% https://github.com/brian-lau/multdist
%
% 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/>.
%
% REVISION HISTORY:
% brian 03.14.06 written
% brian 08.23.11 added flag to assign point to quadrant that maximizes D
% http://www.nr.com/forum/showthread.php?t=576
function [p,D] = kstest2d(s1,s2)
assign_point = false; % Set true to assign center point to maximizing quadrant
% Leave this false if you want FF's original procedure
[n1,m1] = size(s1);
[n2,m2] = size(s2);
if ~all([m1,m2]==2)
error('# of columns in X and Y must equal 2');
end
D = zeros(n1+n2,4);
count = 0;
for i = 1:n1
count = count + 1;
[a1,b1,c1,d1] = quadcnt(s1(i,1),s1(i,2),s1,n1-1);
[a2,b2,c2,d2] = quadcnt(s1(i,1),s1(i,2),s2,n2);
temp = abs([a1-a2 , b1-b2 , c1-c2 , d1-d2]);
if assign_point
% Assign point to quadrant where it maximizes difference
ind = find(max(temp));
if length(ind) >= 1
ind = ind(1); % take first maximum
temp(ind) = temp(ind) + 1/length(s1);
end
end
D(count,:) = temp;
end
for i = 1:n2
count = count + 1;
[a1,b1,c1,d1] = quadcnt(s2(i,1),s2(i,2),s1,n1);
[a2,b2,c2,d2] = quadcnt(s2(i,1),s2(i,2),s2,n2-1);
temp = abs([a1-a2 , b1-b2 , c1-c2 , d1-d2]);
if assign_point
% Assign point to quadrant where it maximizes difference
ind = find(max(temp));
if length(ind) >= 1
ind = ind(1); % take first maximum
temp(ind) = temp(ind) + 1/length(s2);
end
end
D(count,:) = temp;
end
D = max(max(D));
% Average correlation coefficients
r1 = corrcoef(s1); r1 = r1(1,2);
r2 = corrcoef(s2); r2 = r2(1,2);
rr = 0.5*(r1*r1 + r2*r2);
p = probks(n1,n2,D,rr);
%----- Count fractions of points in s in quadrants defined around point (x,y).
% s is a nx2 matrix
%
% a|b
%-----
% c|d
%
% Currently, the point x,y is not counted in any fraction
function [a,b,c,d] = quadcnt(x,y,s,d)
slx = s(:,1)<x;
sgx = s(:,1)>x;
sly = s(:,2)<y;
sgy = s(:,2)>y;
inda = slx & sgy;
indb = sgx & sgy;
indc = slx & sly;
indd = sgx & sly;
a = sum(inda)/d;
b = sum(indb)/d;
c = sum(indc)/d;
d = sum(indd)/d;
%----- Asymptotic Q-function to approximate the 2-sided P-value
function p = probks(n1,n2,D,rr)
% Numerical Recipes in C, section 14.7
N = (n1*n2)/(n1+n2);
lambda = (sqrt(N)*D) / (1 + sqrt(1 - rr)*(.25 - .75/sqrt(N)));
j = (1:101)';
p = 2 * sum((-1).^(j-1).*exp(-2*lambda*lambda*j.^2));
p = min(max(p,0),1);