January – Poisson Process on a Sphere

Poisson process has a generalization in a large category of manifolds. Particularly Poisson point process on a sphere is useful. Nicely enough, Poisson process on a sphere is equivalent to process in a two dimensional area $A = [-\pi,\pi] \times [-1,1]$ through the area preserving mapping from $A$ to surface of a sphere
  $\displaystyle (x,y) \mapsto (r,x,\sin(y)) \nonumber.

Resulting process interpreted in geographical coordinates $(r,\theta,\varphi)$ is a Poisson point process on a sphere of radius $r$. Following codes returns a scatter plot of Poisson points on the unit sphere.

GNU/Octave or Matlab:

%Plot random points on a unit sphere. Returns the points in a vector ref in cartesian coordinates
function refc = poissononsphere(density)
  yMin = -1; yMax = 1;
  xMin=-pi; xMax = pi;
  xDelta=xMax-xMin;yDelta=yMax-yMin; %Rectangle dimensions
  numbPoints=poissrnd(density);    %Number of points in the area is a Poisson variable of intensity given as density
  x=xDelta*(rand(numbPoints,1))+xMin;    %Pick points from uniform distribution
  y=yDelta*(rand(numbPoints,1))+yMin;    %Map referencepoints to geographical coordinates
  ref = [x y]';

  refs = [x'; asin(y)'];%Map geographical coordinates to Cartesian coordinates on a unit circle
  r = 1;
  refc = [r*sin(refs(2,:)+pi/2).*cos(refs(1,:)+pi);...

  figure(1)    %Plot
  [X, Y, Z] = sphere;
  hold on
  axis equal


import numpy as np
import scipy.stats
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import axes3d

#Rectangle dimension
xDelta=xMax-xMin;yDelta=yMax-yMin; #rectangle dimensions

#Density parameter of the Poisson point process. Mean number of points on the sphere

#Simulate Poisson point process

#Number of point in the area is a Poisson variable of intensity lambda0
numbPoints = scipy.stats.poisson( lambda0 ).rvs()
x = xDelta*scipy.stats.uniform.rvs(0,1,((numbPoints,1)))+xMin
y = yDelta*scipy.stats.uniform.rvs(0,1,((numbPoints,1)))+yMin

#Transform to geographical coordinates
x = x
y = np.arcsin(y)
fig = plt.figure()
ax = plt.axes(projection="3d")
ax.scatter(np.sin(y+np.pi/2)*np.cos(x+np.pi),np.sin(y+np.pi/2)*np.sin(x+np.pi),np.cos(y+np.pi/2), color='r' )

Wolfram Language:

(*lambda is the mean number of points on the unit sphere*) 
  poissononsphere[lambda_] := 
  Module[{nrofpoints, phi, theta, radius, refc, polarp}, 
   nrofpoints = RandomVariate[PoissonDistribution[lambda]];
   polarp = 
    Table[{RandomVariate[UniformDistribution[{-Pi, Pi}]], 
      ArcSin[RandomVariate[UniformDistribution[{-1, 1}]]]}, 
   radius = 1;
   refc = 
    Table[{radius*Sin[polarp[[i]][[2]] + Pi/2]*
       Cos[polarp[[i]][[1]] + Pi],
      radius*Sin[polarp[[i]][[2]] + Pi/2]*Sin[polarp[[i]][[1]] + Pi],
      radius*Cos[polarp[[i]][[2]] + Pi/2]}, {i, nrofpoints}];
   ListPointPlot3D[poissononsphere[500], BoxRatios -> {1, 1, 1}]

Figure: Are the stars Poisson distributed in the sky?
Image poissononsphere

Figure: Poisson distributed transmitters and the aggregate signal strength in a satellite by location. Poisson assumption is reasonable in varying situations, but, well, here the sea areas are a bit overrepresented.
Image interferenceenvironment


D. J. Daley and D. Vere-Jones, The General Poisson Process in “An introduction to the theory of point processes”. New York: Springer, 2003, pp. 39.
Stoyan, Dietrich. et al. “Stochastic Geometry and Its Applications”. 3rd ed. Chichester: Wiley, 2013. Print.