Sampling the Fisher Distribution
A preliminary plot
sphere = ParametricPlot3D@
8Cos@thetaD Cos@phiD, Cos@thetaD Sin@phiD, Sin@thetaD<,
8phi, 0, 2 Pi<, 8theta, -Pi ê 2, Pi ê 2<, Axes Ø False,
PlotPoints Ø 23D
Sampling the Homogeneous Distribution
SeedRandom@1D
set = 8<;
Do@8z = RandomReal@8-1, 1<D, phi = RandomReal@80, 2 p<D, theta = ArcCos@zD,
AppendTo@set, Point@8Sin@thetaD Cos@phiD, Sin@thetaD Sin@phiD, z<DD<, 85000<D
2
SamplingFisher.nb
Show@Graphics3D@setD, sphereD
The Fisher (2D) volumetric probability
Fisher@t_, t0_, f_, f0_, k_D :=
Hk ê H4 p Sinh@kDLL Exp@k HCos@tD Cos@t0D + Sin@tD Sin@t0D Cos@f - f0DLD
Numerical check that it is normalized (total probability equal to
one)
NIntegrate@Sin@tD Fisher@t, Pi ê 3, f, Pi ê 4, 3D, 8t, 0, Pi<, 8f, -Pi, Pi<D
1.
Our Fisher distribution
t0 = Pi ê 6;
f0 = -Pi ê 4;
k = 20;
F@t_, f_D := Fisher@t, t0, f, f0, kD
SamplingFisher.nb
Maximum value?
N@F@t0, f0DD
3.1831
But we only need a value that is larger of equal than the actual maximum value:
Fmax = 4.;
SeedRandom@1D
set = 8<;
Do@8
z = RandomReal@8-1, 1<D,
phi = RandomReal@80, 2 p<D,
theta = ArcCos@zD,
P = F@theta, phiD ê Fmax,
chance = RandomReal@80, 1<D,
If@chance < P, AppendTo@set, Point@8Sin@thetaD Cos@phiD, Sin@thetaD Sin@phiD, z<DDD
<, 850 000<D
Show@Graphics3D@setD, sphereD
3