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_r(alpha, w, d, dim)
```function r = circ_r(alpha, w, d, dim)
% r = circ_r(alpha, w, d)
%   Computes mean resultant vector length for circular data.
%
%   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 (!)]
%     [dim  compute along this dimension, default is 1]
%
%     If dim argument is specified, all other optional arguments can be
%     left empty: circ_r(alpha, [], [], dim)
%
%   Output:
%     r		mean resultant length
%
% 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 nargin < 4
dim = 1;
end

if nargin < 2 || isempty(w)
% if no specific weighting has been specified
% assume no binning has taken place
w = ones(size(alpha));
else
if size(w,2) ~= size(alpha,2) || size(w,1) ~= size(alpha,1)
error('Input dimensions do not match');
end
end

if nargin < 3 || isempty(d)
% per default do not apply correct for binned data
d = 0;
end

% compute weighted sum of cos and sin of angles
r = sum(w.*exp(1i*alpha),dim);

% obtain length
r = abs(r)./sum(w,dim);

% for data with known spacing, apply correction factor to correct for bias
% in the estimation of r (see Zar, p. 601, equ. 26.16)
if d ~= 0
c = d/2/sin(d/2);
r = c*r;
end

```