CDF of VOn Mises distribution

Hello all,

How can I calculate cdf of Von Mises distribution with MATLAB?

An explanation about Von Mises distribution is provided in this Wikipedia page.

Thanks

[KSSV]

If you have the data, substitute them in the formula and analyse.

[Aep]

The CDF formula itself is a little complicated and it is not analytic. I was wondering whether there are any tools that can help.

[KSSV]

https://in.mathworks.com/matlabcentral/fileexchange/37241-vmrand-fmu-fkappa-varargin

https://in.mathworks.com/matlabcentral/fileexchange/10676-circular-statistics-toolbox-directional-statistics

[Aep]

Thanks @KSSV. I had searched in the circular statistics toolbox, however it seems that it does not calculate the CDF of Von Mises distribution. The other code that you sent also, as far as I understood, generates random Von Mises distribution, but it does not calculate CDF.

Answer 1

[David Goodmanson]

HI Aep,

This is straight numerical integration and provides the cdf at 1e6 equally spaced points in theta, from -pi to pi.

kappa = 3;
mu = 1;
theta = linspace(-pi,pi,1e6);
f = exp(kappa*cos(theta-mu))/(2*pi*besseli(0,kappa));
c = cumtrapz(theta,f);  % cdf
plot(theta,c)
c(end)-1       % check to see how close the last point of the cdf is to 1            
  ans = 2.0650e-14

% assume a set of angles theta1  with -pi <= theta1 <= pi;
theta1 = [.7  1.4  2.1];
c1 = interp1(theta,c,theta1,'spline')  
  c1 = 0.3148    0.7455    0.9578
 
  

For any reasonable kappa the last point of the cdf will be very close to 1, but the look of the plot depends on how far the maximum of f (at theta = mu) is from the starting point. Here the starting point is -pi, but it could be anywhere on the circle.

For an array of input angles of your choosing, interp1 provides the result.

Answer 2

[Paul]

This code seems to recreate one of the CDF plots on the linked wikipedia page. It doesn't run very fast.

syms x mu kappa x0 real
syms j integer
phi(x,mu,kappa) = 1/2/sym(pi)*(x + 2/besseli(0,kappa)*symsum(besseli(j,kappa)*sin(j*(x-mu))/j,j,1,inf));
F(x,mu,kappa,x0) = phi(x,mu,kappa) - phi(x0,mu,kappa);
cdf = F(-pi:.1:pi,0,1,-pi); % mu = 0, kappa = 1, support from -pi to pi
plot(-pi:.1:pi,double(cdf))
xlim([-pi pi]);
set(gca,'XTick',(-1:.5:1)*pi);
set(gca,'XMinorTick','on');
set(gca,'XMinorGrid','on');
grid

Answer 3

[Jeff Miller]

The Von Mises distribution is included in Cupid. You could use it like this to calculate CDF values:

>> location = 10;
>> concentration = 1.5;
>> vm = VonMises(location,concentration);
>> vm.CDF(11)
ans =
      0.84863