function [pval z] = circ_rtest(alpha, w, d)
%
% [pval, z] = circ_rtest(alpha,w)
%   Computes Rayleigh test for non-uniformity of circular data.
%   H0: the population is uniformly distributed around the circle
%   HA: the populatoin is not distributed uniformly around the circle
%   Assumption: the distribution has maximally one mode and the data is 
%   sampled from a von Mises distribution!
%
%   Input:
%     alpha	sample of angles in radians
%     [w		number of incidences in case of binned angle data]
%     [d    spacing of bin centers for binned data, if supplied 
%           correction factor is used to correct for bias in 
%           estimation of r, in radians (!)]
%
%   Output:
%     pval  p-value of Rayleigh's test
%     z     value of the z-statistic
%
% PHB 7/6/2008
%
% References:
%   Statistical analysis of circular data, N. I. Fisher
%   Topics in circular statistics, S. R. Jammalamadaka et al. 
%   Biostatistical Analysis, J. H. Zar
%
% 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
	r =  circ_r(alpha);
  n = length(alpha);
else
  if length(alpha)~=length(w)
    error('Input dimensions do not match.')
  end
  if nargin < 3
    d = 0;
  end
  r =  circ_r(alpha,w(:),d);
  n = sum(w);
end

% compute Rayleigh's R (equ. 27.1)
R = n*r;

% compute Rayleigh's z (equ. 27.2)
z = R^2 / n;

% compute p value using approxation in Zar, p. 617
pval = exp(sqrt(1+4*n+4*(n^2-R^2))-(1+2*n));

% outdated version:
% compute the p value using an approximation from Fisher, p. 70
% pval = exp(-z);
% if n < 50
%   pval = pval * (1 + (2*z - z^2) / (4*n) - ...
%    (24*z - 132*z^2 + 76*z^3 - 9*z^4) / (288*n^2));
% end