/MatlabToolbox/CircStat2010e/circ_otest.m

function [pval m] = circ_otest(alpha, sz, w)

%

% [pval, m] = circ_otest(alpha,sz,w)

%   Computes Omnibus or Hodges-Ajne test for non-uniformity of circular data.

%   H0: the population is uniformly distributed around the circle

%   HA: the population is not distributed uniformly around the circle

%

%   Alternative to the Rayleigh and Rao's test. Works well for unimodal,

%   bimodal or multimodal data. If requirements of the Rayleigh test are 

%   met, the latter is more powerful.

%

%   Input:

%     alpha	sample of angles in radians

%     [sz   step size for evaluating distribution, default 1 degree

%     [w		number of incidences in case of binned angle data]



%   Output:

%     pval  p-value 

%     m     minimum number of samples falling in one half of the circle

%

% PHB 3/16/2009

%

% References:

%   Biostatistical Analysis, J. H. Zar

%   A bivariate sign test, J. L. Hodges et al., 1955

%   A simple test for uniformity of a circular distribution, B. Ajne, 1968

%

% Circular Statistics Toolbox for Matlab



% By Philipp Berens, 2009

% berens@tuebingen.mpg.de - www.kyb.mpg.de/~berens/circStat.html



if size(alpha,2) > size(alpha,1)

	alpha = alpha';

end



if nargin < 2 || isempty(sz)

  sz = circ_ang2rad(1);

end



if nargin < 3

  w = ones(size(alpha));

else

  if length(alpha)~=length(w)

    error('Input length does not match.')

  end

  w =w(:);  

end



alpha = mod(alpha,2*pi);

n = sum(w);

dg = 0:sz:pi;



m1 = zeros(size(dg));

m2 = zeros(size(dg));

for i=1:length(dg)

  m1(i) = sum((alpha > dg(i) & alpha < pi + dg(i)).*w);    

  m2(i) = n - m1(i);

end

m = min(min([m1;m2]));



if n > 50

  % approximation by Ajne (1968)

  A = pi*sqrt(n) / 2 / (n-2*m);

  pval = sqrt(2*pi) / A * exp(-pi^2/8/A^2);

else

  % exact formula by Hodges (1955)

  pval = 2^(1-n) * (n-2*m) * nchoosek(n,m);  

end