Circular Statistics Toolbox (Directional Statistics)

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08 Apr 2006 (Updated )

Compute descriptive and inferential statistics for circular or directional data.

Editor's Notes:

This file was selected as MATLAB Central Pick of the Week

circ_vmrnd(theta, kappa, n)
function alpha = circ_vmrnd(theta, kappa, n)

% alpha = circ_vmrnd(theta, kappa, n)
%   Simulates n random angles from a von Mises distribution, with preferred 
%   direction thetahat and concentration parameter kappa.
%
%   Input:
%     [theta    preferred direction, default is 0]
%     [kappa    width, default is 1]
%     [n        number of samples, default is 10]
%
%     If n is a vector with two entries (e.g. [2 10]), the function creates
%     a matrix output with the respective dimensionality.
%
%   Output:
%     alpha     samples from von Mises distribution
%
%
%   References:
%     Statistical analysis of circular data, Fisher, sec. 3.3.6, p. 49
%
% Circular Statistics Toolbox for Matlab

% By Philipp Berens and Marc J. Velasco, 2009
% velasco@ccs.fau.edu


% default parameter
if nargin < 3
    n = 10;
end

if nargin < 2
    kappa = 1;
end

if nargin < 1
    theta = 0;
end

if numel(n) > 2
  error('n must be a scalar or two-entry vector!')
elseif numel(n) == 2
  m = n;
  n = n(1) * n(2);
end  

% if kappa is small, treat as uniform distribution
if kappa < 1e-6
    alpha = 2*pi*rand(n,1);
    return
end

% other cases
a = 1 + sqrt((1+4*kappa.^2));
b = (a - sqrt(2*a))/(2*kappa);
r = (1 + b^2)/(2*b);

alpha = zeros(n,1);
for j = 1:n
  while true
      u = rand(3,1);

      z = cos(pi*u(1));
      f = (1+r*z)/(r+z);
      c = kappa*(r-f);

      if u(2) < c * (2-c) || ~(log(c)-log(u(2)) + 1 -c < 0)
         break
      end

      
  end

  alpha(j) = theta +  sign(u(3) - 0.5) * acos(f);
  alpha(j) = angle(exp(1i*alpha(j)));
end

if exist('m','var')
  alpha = reshape(alpha,m(1),m(2));
end






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