Concentration

Concentration - Estimate the concentration parameter for circular data.

 Estimate the concentration parameter for circular data assuming a Von Mises distribution.
 Uses the approximation described in "Statistical Analysis of Circular Data" (Fisher, p. 88).

  USAGE

    kappa = Concentration(angles)

    angles         angles in radians

  SEE

    See also ConcentrationTest, CircularMean, CircularVariance, CircularConfidenceIntervals.

0001 function kappa = Concentration(angles)
0002 
0003 %Concentration - Estimate the concentration parameter for circular data.
0004 %
0005 % Estimate the concentration parameter for circular data assuming a Von Mises distribution.
0006 % Uses the approximation described in "Statistical Analysis of Circular Data" (Fisher, p. 88).
0007 %
0008 %  USAGE
0009 %
0010 %    kappa = Concentration(angles)
0011 %
0012 %    angles         angles in radians
0013 %
0014 %  SEE
0015 %
0016 %    See also ConcentrationTest, CircularMean, CircularVariance, CircularConfidenceIntervals.
0017 
0018 % Copyright (C) 2004-2011 by Michaël Zugaro
0019 %
0020 % This program is free software; you can redistribute it and/or modify
0021 % it under the terms of the GNU General Public License as published by
0022 % the Free Software Foundation; either version 3 of the License, or
0023 % (at your option) any later version.
0024 
0025 isradians(angles);
0026 
0027 n = length(angles);
0028 angles = exp(i*angles);
0029 r_bar = abs(mean(angles));
0030 
0031 if r_bar < 0.53,
0032     kappa = 2*r_bar+r_bar^3+5*r_bar^5/6;
0033 elseif r_bar < 0.85,
0034     kappa = -0.4+1.39*r_bar+0.43/(1-r_bar);
0035 else
0036     kappa = 1/(r_bar^3-4*r_bar^2+3*r_bar);
0037 end
0038 
0039 % Correction for small samples
0040 if n <= 15,
0041     if kappa < 2,
0042         kappa = max([kappa-2/(n*kappa) 0]);
0043     else
0044         kappa = (n-1)^3*kappa/(n^3+n);
0045     end
0046 end