Code covered by the BSD License

# Circular Statistics Toolbox (Directional Statistics)

### Philipp Berens (view profile)

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_kappa(alpha,w)
```function kappa = circ_kappa(alpha,w)
%
% kappa = circ_kappa(alpha,[w])
%   Computes an approximation to the ML estimate of the concentration
%   parameter kappa of the von Mises distribution.
%
%   Input:
%     alpha   angles in radians OR alpha is length resultant
%     [w      number of incidences in case of binned angle data]
%
%   Output:
%     kappa   estimated value of kappa
%
%   References:
%     Statistical analysis of circular data, Fisher, equation p. 88
%
% Circular Statistics Toolbox for Matlab

% By Philipp Berens, 2009
% berens@tuebingen.mpg.de - www.kyb.mpg.de/~berens/circStat.html

alpha = alpha(:);

if nargin<2
% if no specific weighting has been specified
% assume no binning has taken place
w = ones(size(alpha));
else
if size(w,2) > size(w,1)
w = w';
end
end

N = length(alpha);

if N>1
R = circ_r(alpha,w);
else
R = alpha;
end

if R < 0.53
kappa = 2*R + R^3 + 5*R^5/6;
elseif R>=0.53 && R<0.85
kappa = -.4 + 1.39*R + 0.43/(1-R);
else
kappa = 1/(R^3 - 4*R^2 + 3*R);
end

if N<15 && N>1
if kappa < 2
kappa = max(kappa-2*(N*kappa)^-1,0);
else
kappa = (N-1)^3*kappa/(N^3+N);
end
end
```