circ_r

 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

0001 function r = circ_r(alpha, w, d, dim)
0002 % r = circ_r(alpha, w, d)
0003 %   Computes mean resultant vector length for circular data.
0004 %
0005 %   Input:
0006 %     alpha    sample of angles in radians
0007 %     [w        number of incidences in case of binned angle data]
0008 %     [d    spacing of bin centers for binned data, if supplied
0009 %           correction factor is used to correct for bias in
0010 %           estimation of r, in radians (!)]
0011 %     [dim  compute along this dimension, default is 1]
0012 %
0013 %     If dim argument is specified, all other optional arguments can be
0014 %     left empty: circ_r(alpha, [], [], dim)
0015 %
0016 %   Output:
0017 %     r        mean resultant length
0018 %
0019 % PHB 7/6/2008
0020 %
0021 % References:
0022 %   Statistical analysis of circular data, N.I. Fisher
0023 %   Topics in circular statistics, S.R. Jammalamadaka et al.
0024 %   Biostatistical Analysis, J. H. Zar
0025 %
0026 % Circular Statistics Toolbox for Matlab
0027 
0028 % By Philipp Berens, 2009
0029 % berens@tuebingen.mpg.de - www.kyb.mpg.de/~berens/circStat.html
0030 
0031 if nargin < 4
0032   dim = 1;
0033 end
0034 
0035 if nargin < 2 || isempty(w) 
0036   % if no specific weighting has been specified
0037   % assume no binning has taken place
0038     w = ones(size(alpha));
0039 else
0040   if size(w,2) ~= size(alpha,2) || size(w,1) ~= size(alpha,1) 
0041     error('Input dimensions do not match');
0042   end 
0043 end
0044 
0045 if nargin < 3 || isempty(d)
0046   % per default do not apply correct for binned data
0047   d = 0;
0048 end
0049 
0050 % compute weighted sum of cos and sin of angles
0051 r = sum(w.*exp(1i*alpha),dim);
0052 
0053 % obtain length
0054 r = abs(r)./sum(w,dim);
0055 
0056 % for data with known spacing, apply correction factor to correct for bias
0057 % in the estimation of r (see Zar, p. 601, equ. 26.16)
0058 if d ~= 0
0059   c = d/2/sin(d/2);
0060   r = c*r;
0061 end
0062