The Spherical Bingham Distribution edit page 
plottingConvention.default('y↑→x');

The Bingham distribution on the sphere is an antipodal symmetric distribution (Bingham, 1974) with a probability density function given by

\[p_{b}(\hat{x}\vert AKA^T) = \frac{1}{F(\kappa_{1},\kappa_{2},\kappa_{3})}\exp (\hat{x}^T AZA^T \hat{x})\]

where \(A\) is an orthogonal covariance matrix, and \(Z\) a concentration matrix with \(\mathrm{diag}(\kappa_{1},\kappa_{2},\kappa_{3})\) with \(\kappa_{1} < \kappa_{2} < \kappa_{3}\).

In MTEX \(Z\) is given by Z = [k1,k2,k3] with k3 = 0 and \(A\) is given by three orthogonal vectors.

Bingham, C., An Antipodally Symmetric Distribution on the Sphere, The Annals of Statistics Vol. 2, No. 6 (Nov., 1974), pp. 1201-1225 

% A simple example:
Z = [-10 -4 0];
a = rotation.rand(1) .* vector3d([xvector yvector zvector]);
bingFun = S2FunBingham(Z,a);
plot(bingFun)

Meaning of \(Z\)

\(k1 = k2\) defines a rotational symmetric point maximum and \(k2 = 0\) defines a girdle distribution.

close
kappa = [0 4 8 12 24];
mtexFig = newMtexFigure('layout',[length(kappa) length(kappa)]);
for k2 = kappa
  for k1 = kappa
    if k1 >= k2
      bFun = S2FunBingham([-k1 -k2 0]);
      plot(bFun,'colorRange',[0,25],'noLabel')
       mtexTitle(['\(\kappa_1=\)' num2str(k1)  '  ' '\(\kappa_2=\)' num2str(k2)],'FontSize',12)
      nextAxis
    else
      nextAxis
    end
  end
end
setColorRange('equal')
mtexFig.drawNow;

Drawing a random sample of the Bingham distribution

close
v = bingFun.discreteSample(50)
plot(bingFun)
hold on
plot(v,'MarkerEdgeColor','k','MarkerFaceColor','gray','MarkerFaceAlpha',0.5)
hold off
v = vector3d (y↑→x)
 size: 50 x 1
 antipodal: true

Estimating a spherical Bingham distribution from discrete data

Given arbitrarily scattered data v on the sphere we can estimate the best fitting Bingham distribution by

% estimate a Bingham distribution
[bingFunEst,ab,rot] = S2FunBingham.fit(v)
bingFunEst = S2FunBingham (y↑→x)
ab =
    0.1541    0.1716
 
rot = rotation
 
  Bunge Euler angles in degree
     phi1     Phi    phi2    Inv.
  149.397 154.782 48.1841       1

Lets plot the fitted distribution with the data

plot(bingFunEst)
hold on
plot(v,'MarkerEdgeColor','k','MarkerFaceColor','gray','MarkerFaceAlpha',0.5)
hold off

The function S2FunBingham.fit.html provides two additional output arguments ab and rot. Those describe the half axes \(a\) and \(b\) and the orientation rot of the confidence ellipse of the mean mean direction bingFunEst.a(3) of the estimated Bingham distribution at the confidence level \(p=0.95\). We may visualize this confidence ellipse by the commands

% mark the mean direction
annotate(bingFunEst.a(3),'MarkerFaceColor','red','MarkerSize',10)

% annotate the p=0.95 confidence ellipse
ellipse(rot,ab(1),ab(2), 'linewidth',3,'lineColor','k')