How do you calculate the average of a set of circu

Question

I want to calculate the average of a set of circular data. For example, I might have several samples from the reading of a compass. The problem of course is how to deal with the wraparound. The same algorithm might be useful for a clockface.

The actual question is more complicated - what do statistics mean on a sphere or in an algebraic space which "wraps around", e.g. the additive group mod n. The answer may not be unique, e.g. the average of 359 degrees and 1 degree could be 0 degrees or 180, but statistically 0 looks better.

This is a real programming problem for me and I'm trying to make it not look like just a Math problem.

Answer 1

Like all averages, the answer depends upon the choice of metric. For a given metric M, the average of some angles a_k in [-pi,pi] for k in [1,N] is that angle a_M which minimizes the sum of squared distances d^2_M(a_M,a_k). For a weighted mean, one simply includes in the sum the weights w_k (such that sum_k w_k = 1). That is,

a_M = arg min_x sum_k w_k d^2_M(x,a_k)

Two common choices of metric are the Frobenius and the Riemann metrics. For the Frobenius metric, a direct formula exists that corresponds to the usual notion of average bearing in circular statistics. See "Means and Averaging in the Group of Rotations", Maher Moakher, SIAM Journal on Matrix Analysis and Applications, Volume 24, Issue 1, 2002, for details.
http://link.aip.org/link/?SJMAEL/24/1/1

Here's a function for GNU Octave 3.2.4 that does the computation:

function ma=meanangleoct(a,w,hp,ntype)
%   ma=meanangleoct(a,w,hp,ntype) returns the average of angles a
%   given weights w and half-period hp using norm type ntype
%   Ref: "Means and Averaging in the Group of Rotations",
%   Maher Moakher, SIAM Journal on Matrix Analysis and Applications,
%   Volume 24, Issue 1, 2002.

if (nargin<1) | (nargin>4), help meanangleoct, return, end 
if isempty(a), error('no measurement angles'), end
la=length(a); sa=size(a); 
if prod(sa)~=la, error('a must be a vector'); end
if (nargin<4) || isempty(ntype), ntype='F'; end
if ~sum(ntype==['F' 'R']), error('ntype must be F or R'), end
if (nargin<3) || isempty(hp), hp=pi; end
if (nargin<2) || isempty(w), w=1/la+0*a; end
lw=length(w); sw=size(w); 
if prod(sw)~=lw, error('w must be a vector'); end
if lw~=la, error('length of w must equal length of a'), end
if sum(w)~=1, warning('resumming weights to unity'), w=w/sum(w); end

a=a(:);     % make column vector
w=w(:);     % make column vector
a=mod(a+hp,2*hp)-hp;    % reduce to central period
a=a/hp*pi;              % scale to half period pi
z=exp(i*a); % U(1) elements

% % NOTA BENE:
% % fminbnd can get hung up near the boundaries.
% % If that happens, shift the input angles a
% % forward by one half period, then shift the
% % resulting mean ma back by one half period.
% X=fminbnd(@meritfcn,-pi,pi,[],z,w,ntype);

% % seems to work better
x0=imag(log(sum(w.*z)));
X=fminbnd(@meritfcn,x0-pi,x0+pi,[],z,w,ntype);

% X=real(X);              % truncate some roundoff
X=mod(X+pi,2*pi)-pi;    % reduce to central period
ma=X*hp/pi;             % scale to half period hp

return
%%%%%%

function d2=meritfcn(x,z,w,ntype)
x=exp(i*x);
if ntype=='F'
    y=x-z;
else % ntype=='R'
    y=log(x'*z);
end
d2=y'*diag(w)*y;
return
%%%%%%

% %   test script
% % 
% % NOTA BENE: meanangleoct(a,[],[],'R') will equal mean(a) 
% % when all abs(a-b) < pi/2 for some value b
% % 
% na=3, a=sort(mod(randn(1,na)+1,2)-1)*pi;
% da=diff([a a(1)+2*pi]); [mda,ndx]=min(da);
% a=circshift(a,[0 2-ndx])    % so that diff(a(2:3)) is smallest
% A=exp(i*a), B1=expm(a(1)*[0 -1; 1 0]), 
% B2=expm(a(2)*[0 -1; 1 0]), B3=expm(a(3)*[0 -1; 1 0]),
% masimpl=[angle(mean(exp(i*a))) mean(a)]
% Bsum=B1+B2+B3; BmeanF=Bsum/sqrt(det(Bsum)); 
% % this expression for BmeanR should be correct for ordering of a above
% BmeanR=B1*(B1'*B2*(B2'*B3)^(1/2))^(2/3);
% mamtrx=real([[0 1]*logm(BmeanF)*[1 0]' [0 1]*logm(BmeanR)*[1 0]'])
% manorm=[meanangleoct(a,[],[],'F') meanangleoct(a,[],[],'R')]
% polar(a,1+0*a,'b*'), axis square, hold on
% polar(manorm(1),1,'rs'), polar(manorm(2),1,'gd'), hold off

%     Meanangleoct Version 1.0
%     Copyright (C) 2011 Alphawave Research, [e-mail]
%     Released under GNU GPLv3 -- see file COPYING for more info.
%
%     Meanangle is free software: you can redistribute it and/or modify
%     it under the terms of the GNU General Public License as published by
%     the Free Software Foundation, either version 3 of the License, or (at
%     your option) any later version.
%
%     Meanangle is distributed in the hope that it will be useful, but
%     WITHOUT ANY WARRANTY; without even the implied warranty of
%     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
%     General Public License for more details.
%
%     You should have received a copy of the GNU General Public License
%     along with this program.  If not, see `http://www.gnu.org/licenses/'.

Answer 2

There's no single "right answer". I recommend reading the book, K. V. Mardia and P. E. Jupp, "Directional Statistics", (Wiley, 1999), for a thorough analysis.

Answer 3

The average angle phi_avg should have the property that sum_i|phi_avg-phi_i|^2 becomes minimal, where the difference has to be in [-Pi, Pi) (because it might be shorter to go the other way around!). This is easily achieved by normalizing all input values to [0, 2Pi), keeping a running average phi_run and choosing normalizing |phi_i-phi_run| to [-Pi,Pi) (by adding or subtractin 2Pi). Most suggestions above do something else that does not have that minimal property, i.e., they average something, but not angles.

Answer 4

Here is some java code to average angles, I think it's reasonably robust.

public static double getAverageAngle(List<Double> angles)
{
    // r = right (0 to 180 degrees)

    // l = left (180 to 360 degrees)

    double rTotal = 0;
    double lTotal = 0;
    double rCtr = 0;
    double lCtr = 0;

    for (Double angle : angles)
    {
        double norm = normalize(angle);
        if (norm >= 180)
        {
            lTotal += norm;
            lCtr++;
        } else
        {
            rTotal += norm;
            rCtr++;
        }
    }

    double rAvg = rTotal / Math.max(rCtr, 1.0);
    double lAvg = lTotal / Math.max(lCtr, 1.0);

    if (rAvg > lAvg + 180)
    {
        lAvg += 360;
    }
    if (lAvg > rAvg + 180)
    {
        rAvg += 360;
    }

    double rPortion = rAvg * (rCtr / (rCtr + lCtr));
    double lPortion = lAvg * (lCtr / (lCtr + rCtr));
    return normalize(rPortion + lPortion);
}

public static double normalize(double angle)
{
    double result = angle;
    if (angle >= 360)
    {
        result = angle % 360;
    }
    if (angle < 0)
    {
        result = 360 + (angle % 360);
    }
    return result;
}

Answer 5

Here is the full solution: (the input is an array of bearing in degrees (0-360)

public static int getAvarageBearing(int[] arr)
{
    double sunSin = 0;
    double sunCos = 0;
    int counter = 0;

    for (double bearing : arr)
    {
        bearing *= Math.PI/180;

        sunSin += Math.sin(bearing);
        sunCos += Math.cos(bearing);
        counter++; 
    }

    int avBearing = INVALID_ANGLE_VALUE;
    if (counter > 0)
    {
        double bearingInRad = Math.atan2(sunSin/counter, sunCos/counter);
        avBearing = (int) (bearingInRad*180f/Math.PI);
        if (avBearing<0)
            avBearing += 360;
    }

    return avBearing;
}

Answer 6

Here's an idea: build the average iteratively by always calculating the average of the angles that are closest together, keeping a weight.

Another idea: find the largest gap between the given angles. Find the point that bisects it, and then pick the opposite point on the circle as the reference zero to calculate the average from.