ourMELONS/matlab/general/kstest2.m

function [H, pValue, KSstatistic] = kstest2(x1, x2, alpha, tail)
%KSTEST2 Two-sample Kolmogorov-Smirnov goodness-of-fit hypothesis test.
%   H = KSTEST2(X1,X2,ALPHA,TAIL) performs a Kolmogorov-Smirnov (K-S) test 
%   to determine if independent random samples, X1 and X2, are drawn from 
%   the same underlying continuous population. ALPHA and TAIL are optional
%   scalar inputs: ALPHA is the desired significance level (default = 0.05); 
%   TAIL indicates the type of test (default = 0). H indicates the result of
%   the hypothesis test:
%      H = 0 => Do not reject the null hypothesis at significance level ALPHA.
%      H = 1 => Reject the null hypothesis at significance level ALPHA.
% 
%   Let S1(x) and S2(x) be the empirical distribution functions from the
%   sample vectors X1 and X2, respectively, and F1(x) and F2(x) be the
%   corresponding true (but unknown) population CDFs. The two-sample K-S
%   test tests the null hypothesis that F1(x) = F2(x) for all x, against the
%   alternative specified by TAIL:
%       'unequal' -- "F1(x) not equal to F2(x)" (two-sided test)
%       'larger'  -- "F1(x) > F2(x)" (one-sided test)
%       'smaller' -- "F1(x) < F2(x)" (one-sided test)
%
%   For TAIL = 'unequal', 'larger', and 'smaller', the test statistics are
%   max|S1(x) - S2(x)|, max[S1(x) - S2(x)], and max[S2(x) - S1(x)],
%   respectively.
%
%   The decision to reject the null hypothesis occurs when the significance 
%   level, ALPHA, equals or exceeds the P-value.
%
%   X1 and X2 are vectors of lengths N1 and N2, respectively, and represent
%   random samples from some underlying distribution(s). Missing
%   observations, indicated by NaNs (Not-a-Number), are ignored.
%
%   [H,P] = KSTEST2(...) also returns the asymptotic P-value P.
%
%   [H,P,KSSTAT] = KSTEST2(...) also returns the K-S test statistic KSSTAT
%   defined above for the test type indicated by TAIL.
%
%   The asymptotic P-value becomes very accurate for large sample sizes, and
%   is believed to be reasonably accurate for sample sizes N1 and N2 such 
%   that (N1*N2)/(N1 + N2) >= 4.
%
%   See also KSTEST, LILLIETEST, CDFPLOT.
%

% Copyright 1993-2007 The MathWorks, Inc.
% $Revision: 1.5.2.5 $   $ Date: 1998/01/30 13:45:34 $

% References:
%   Massey, F.J., (1951) "The Kolmogorov-Smirnov Test for Goodness of Fit",
%         Journal of the American Statistical Association, 46(253):68-78.
%   Miller, L.H., (1956) "Table of Percentage Points of Kolmogorov Statistics",
%         Journal of the American Statistical Association, 51(273):111-121.
%   Stephens, M.A., (1970) "Use of the Kolmogorov-Smirnov, Cramer-Von Mises and
%         Related Statistics Without Extensive Tables", Journal of the Royal
%         Statistical Society. Series B, 32(1):115-122.
%   Conover, W.J., (1980) Practical Nonparametric Statistics, Wiley.
%   Press, W.H., et. al., (1992) Numerical Recipes in C, Cambridge Univ. Press.
 
if nargin < 2
    error('stats:kstest2:TooFewInputs','At least 2 inputs are required.');
end

%
% Ensure each sample is a VECTOR.
%

if ~isvector(x1) || ~isvector(x2) 
    error('stats:kstest2:VectorRequired','The samples X1 and X2 must be vectors.');
end

%
% Remove missing observations indicated by NaN's, and 
% ensure that valid observations remain.
%

x1  =  x1(~isnan(x1));
x2  =  x2(~isnan(x2));
x1  =  x1(:);
x2  =  x2(:);

if isempty(x1)
   error('stats:kstest2:NotEnoughData', 'Sample vector X1 contains no data.');
end

if isempty(x2)
   error('stats:kstest2:NotEnoughData', 'Sample vector X2 contains no data.');
end

%
% Ensure the significance level, ALPHA, is a scalar 
% between 0 and 1 and set default if necessary.
%

if (nargin >= 3) && ~isempty(alpha)
   if ~isscalar(alpha) || (alpha <= 0 || alpha >= 1)
      error('stats:kstest2:BadAlpha',...
            'Significance level ALPHA must be a scalar between 0 and 1.'); 
   end
else
   alpha  =  0.05;
end

%
% Ensure the type-of-test indicator, TAIL, is a scalar integer from 
% the allowable set, and set default if necessary.
%

if (nargin >= 4) && ~isempty(tail)
   if ischar(tail)
      tail = strmatch(lower(tail), {'smaller','unequal','larger'}) - 2;
      if isempty(tail)
         error('stats:kstest2:BadTail',...
               'Type-of-test indicator TAIL must be ''unequal'', ''smaller'', or ''larger''.');
      end
   elseif ~isscalar(tail) || ~((tail==-1) || (tail==0) || (tail==1))
      error('stats:kstest2:BadTail',...
            'Type-of-test indicator TAIL must be ''unequal'', ''smaller'', or ''larger''.');
   end
else
   tail  =  0;
end

%
% Calculate F1(x) and F2(x), the empirical (i.e., sample) CDFs.
%

binEdges    =  [-inf ; sort([x1;x2]) ; inf];

binCounts1  =  histc (x1 , binEdges, 1);
binCounts2  =  histc (x2 , binEdges, 1);

sumCounts1  =  cumsum(binCounts1)./sum(binCounts1);
sumCounts2  =  cumsum(binCounts2)./sum(binCounts2);

sampleCDF1  =  sumCounts1(1:end-1);
sampleCDF2  =  sumCounts2(1:end-1);

%
% Compute the test statistic of interest.
%

switch tail
   case  0      %  2-sided test: T = max|F1(x) - F2(x)|.
      deltaCDF  =  abs(sampleCDF1 - sampleCDF2);

   case -1      %  1-sided test: T = max[F2(x) - F1(x)].
      deltaCDF  =  sampleCDF2 - sampleCDF1;

   case  1      %  1-sided test: T = max[F1(x) - F2(x)].
      deltaCDF  =  sampleCDF1 - sampleCDF2;
end

KSstatistic   =  max(deltaCDF);

%
% Compute the asymptotic P-value approximation and accept or
% reject the null hypothesis on the basis of the P-value.
%

n1     =  length(x1);
n2     =  length(x2);
n      =  n1 * n2 /(n1 + n2);
lambda =  max((sqrt(n) + 0.12 + 0.11/sqrt(n)) * KSstatistic , 0);

if tail ~= 0        % 1-sided test.

   pValue  =  exp(-2 * lambda * lambda);

else                % 2-sided test (default).
%
%  Use the asymptotic Q-function to approximate the 2-sided P-value.
%
   j       =  (1:101)';
   pValue  =  2 * sum((-1).^(j-1).*exp(-2*lambda*lambda*j.^2));
   pValue  =  min(max(pValue, 0), 1);

end

H  =  (alpha >= pValue);