Statistics: vmcdf

### Function Reference: `vmcdf`

statistics: p = vmcdf (x, mu, k)
statistics: p = vmcdf (x, mu, k, `"upper"`)

Von Mises probability density function (PDF).

For each element of x, compute the cumulative distribution function (CDF) of the von Mises distribution with location parameter mu and concentration parameter k on the interval [-pi,pi]. The size of p is the common size of x, mu, and k. A scalar input functions as a constant matrix of the same same size as the other inputs.

`p = vmcdf (x, mu, k, "upper")` computes the upper tail probability of the von Mises distribution with parameters mu and k, at the values in x.

Note: the CDF of the von Mises distribution is not analytic. Hence, it is calculated by integrating its probability density which is expressed as a series of Bessel functions. Balancing between performance and accuracy, the integration uses a step of `1e-5` on the interval [-pi,pi], which results to an accuracy of about 10 significant digits.

Further information about the von Mises distribution can be found at https://en.wikipedia.org/wiki/Von_Mises_distribution

See also: vminv, vmpdf, vmrnd

Source Code: vmcdf

### Example: 1

     ## Plot various CDFs from the von Mises distribution
     x1 = [-pi:0.1:pi];
     p1 = vmcdf (x1, 0, 0.5);
     p2 = vmcdf (x1, 0, 1);
     p3 = vmcdf (x1, 0, 2);
     p4 = vmcdf (x1, 0, 4);
     plot (x1, p1, "-r", x1, p2, "-g", x1, p3, "-b", x1, p4, "-c")
     grid on
     xlim ([-pi, pi])
     legend ({"μ = 0, k = 0.5", "μ = 0, k = 1", ...
              "μ = 0, k = 2", "μ = 0, k = 4"}, "location", "northwest")
     title ("Von Mises CDF")
     xlabel ("values in x")
     ylabel ("probability")