In this post, I’ll take a break from the more theoretical posts. Instead I’ll describe how to simulate or sample a homogeneous Poisson point process on a circle. I have already simulated this point process on a rectangle, triangle and disk. In some sense, I should have done this simulation method before the disk one, as it’s easier to simulate. I recommend reading that post first, as the material presented here builds off it.
Sampling a homogeneous Poisson point process on a circle is rather straightforward. It just requires using a fixed radius and uniformly choose angles from interval \((0, 2\pi)\). But the circle setting gives an opportunity to employ a different method for positioning points uniformly on circles and, more generally, spheres. This approach uses Gaussian random variables, and it becomes much more efficient when the points are placed on high dimensional spheres.
Steps
Simulating a Poisson point process requires two steps: simulating the random number of points and then randomly positioning each point.
Number of points
The number of points of a Poisson point process on circle of radius \(r>0\) is a Poisson random variable with mean \(\lambda C\), where \(C=2\pi r\) is the circumference of the circle. You just need to be able to need to produce (pseudo-)random numbers according to a Poisson distribution.
To generate Poisson variables in MATLAB, use the poissrnd function with the argument \(\lambda C\). In Python, use either the scipy.stats.poisson or numpy.random.poisson function from the SciPy or NumPy libraries. (If you’re curious how Poisson simulation works, I suggest seeing this post for details on sampling Poisson random variables or, more accurately, variates.)
Locations of points
For a homogeneous Poisson point process, we need to uniformly position points on the underlying space, which is this case is a circle. We will look at two different ways to position points uniformly on a circle. The first is arguably the most natural approach.
Method 1: Polar coordinates
We use polar coordinates due to the nature of the problem. To position all the points uniformly on a circle, we simple generate uniform numbers on the unit interval \((0,1)\). We then multiply these random numbers by \(2\pi\).
We have generated polar coordinates for points uniformly located on the circle. To plot the points, we have to convert the coordinates back to Cartesian form by using the change of coordinates: \(x=\rho\cos(\theta)\) and \(y=\rho\sin(\theta)\).
Method 2: Normal random variables
For each point, we generate two standard normal or Gaussian random variables, say, \(W_x\) and \(W_y\), which are independent of each other. (The term standard here means the normal random variables have mean \(\mu =0\) and standard deviation \(\sigma=1\).) These two random variables are the Cartesian components of a random point. We then rescale the two values by the Euclidean norm, giving
$$X=\frac{W_x}{(W_x^2+W_y^2)^{1/2}},$$
$$Y=\frac{W_y}{(W_x^2+W_y^2)^{1/2}}.$$
These are the Cartesian coordinates of points uniformly scattered around a unit circle with centre at the origin. We multiply the two random values \(X\) and \(Y\) by the \(r>0\) for a circle with radius \(r\).
How does it work?
The procedure is somewhat like the Box-Muller transform in reverse. I’ll give an outline of the proof. The joint density of the random variables \(W_x\) and \(W_y\) is that of the bivariate normal distribution with zero correlation, meaning it has the joint density
$$ f(x,y)=\frac{1}{2\pi}e^{[-(x^2+y^2)/2]}.$$
We see that the function \(f\) is a constant when we trace around any line for which \((x^2+y^2)\) is a constant, which is simply the Cartesian equation for a circle (where the radius is the square root of the aforementioned constant). This means that the angle of the point \((W_x, W_y)\) will be uniformly distributed.
Now we just need to look at the distance of the random point. The vector formed from the pair of normal variables \((W_x, W_y)\) is a Rayleigh random variable. We can see that the vector from the origin to the point \((X,Y)\) has length one, because we rescaled it with the Euclidean norm.
Results
I have presented some results produced by code written in MATLAB and Python. The blue points are the Poisson points on the sphere. I have used a surface plot (with clear faces) in black to illustrate the underling sphere.
MATLAB
Python
Code
The code for all my posts is located online here. For this post, the code in MATLAB and Python is here.
Further reading
I recommend this blog post, which discusses different methods for randomly placing points on spheres and inside spheres (or, rather, balls) in a uniform manner. (Embedded in two dimensions, a sphere is a circle and a ball is a disk.) A key paper on using normal variables is the following:
1959, Muller, A note on a method for generating points uniformly on n-dimensional spheres.
As I mentioned in the post on the disk, the third edition of the classic book Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke details on page 54 how to uniformly place points on a disk. It just requires a small modification for the circle.
Sometimes with just a little tweaking of a point process, you can get a new point process. An example of this is the Thomas point process, which is a type of cluster point process, meaning that its randomly located points tend to form random clusters. This point process is an example of a family of cluster point processes known as Neyman-Scott point processes, which have been used as models in spatial statistics and telecommunications. If that sounds familiar, that is because this point process is very similar to the Matérn point cluster process, which I covered in the previous post.
The only difference between the two point processes is how the points are randomly located. In each cluster of a Thomas point process, each individual point is located according to two independent zero-mean normal variables with variance \(\sigma^2\), describing the \(x\) and \(y\) coordinates relative to the cluster centre, whereas each point of a Matérn point process is located uniformly in a disk.
Working in polar coordinates, an equivalent way to simulate a Thomas point process is to use independent and identically-distributed Rayleigh random variables for the radial (or \(\rho\)) coordinates, instead of using random variables with a triangular distribution, which are used to simulate the Matérn point process. This method works because in polar coordinates a uniform random variable for the angular (or \(\theta\) ) coordinate and a Rayleigh random variable for the angular (or \(\rho\)) is equivalent to in Cartesian coordinates two independent zero-mean normal variables. This is exactly the trick behind the Box-Muller transform for generating normal random variables using just uniform random variables.
If you’re familiar with simulating the Matérn point process, the most difference is what size to make the simulation window for the parents points. I cover that in the next section.
Overview
Simulating a Thomas cluster point process requires first simulating a homogeneous Poisson point process with intensity \(\lambda>0\) on some simulation window, such as a rectangle, which is the simulation window I will use here. Then for each point of this underlying point process, simulate a Poisson number of points with mean \(\mu>0\), and for each point simulate two independent zero-mean normal variables with variance \(\sigma^2\), corresponding to the (relative) Cartesian coordinates .
The underlying point process is sometimes called the parent (point) process, and its points are centres of the cluster disks. The subsequent point process on all the disks is called daughter (point) process and it forms the clusters. I have already written about simulating the homogeneous Poisson point processes on a rectangle and a disk, so those posts are good starting points, and I will not focus too much on details for these steps steps.
Importantly, like the Matérn point process, it’s possible for daughter points to appear in the simulation window that come from parents points outside the simulation window. To handle these edge effects, the point processes must be first simulated on an extended version of the simulation window. Then only the daughter points within the simulation window are kept and the rest are removed.
We can add a strip of some width \(d\) all around the simulation window. But what value does \(d\) take? Well, in theory, it is possible that a daughter point comes from a parent point that is very far from the simulation window. But that probability becomes vanishingly small as the distance increases, due to the daughter points being located according to zero-mean normal random variables.
For example, if a single parent is at a distance \(d=6 \sigma\) from the simulation, then there is about a $1/1 000 000 000$ chance that a single daughter point will land in the simulation window. The probability is simply \(1-\Phi(6 \sigma)\), where \(\Phi\) is the cumulative distribution function of a normal variable with zero mean and standard deviation \(\sigma>0\). This is what they call a six sigma event. In my code, I set \(d=6 \sigma\), but \(d=4 \sigma\) is good enough, which is the value that the R library spatstat uses by default.
Due to this approximation, this simulation cannot be called a perfect simulation, despite the approximation being highly accurate. In practice, it will not have no measurable effect on simulation results, as the number of simulations will rarely be high enough for (hypothetical) daughter points to come from (hypothetical) parent points outside the window.
Steps
Number of points
Simulate the underlying or parent Poisson point process on the rectangle with \(N_P\) points. Then for each point, simulate a Poisson number of points, where each disk now has \(D_i\) number of points. Then the total number of points is simply \(N=D_1+\dots +D_{P}=\sum_{i=1}^{N_P}D_i \). The random variables \(P\) and \(D_i\) are Poisson random variables with respective means \(\lambda A\) and \(\mu\), where \(A\) is the area of the rectangular simulation window. To simulate these random variables in MATLAB, use the poissrnd function. To do this in R, use the standard function rpois. In Python, we can use either functions scipy.stats.poisson or numpy.random.poisson from the SciPy or NumPy libraries.
Locations of points
The points of the parent point process are randomly positioned by using Cartesian coordinates. For a homogeneous Poisson point process, the \(x\) and \(y\) coordinates of each point are independent uniform points, which is also the case for the binomial point process, covered in an earlier post.
As mentioned in the introduction of this post, the points of all the daughter point process are randomly positioned by either using polar coordinates or Cartesian coordinates, due to the Box-Muller transform. But because we ultimately convert back to Cartesian coordinates (for example, to plot the points), we will work entirely in this coordinate system. Each point is then simply positioned with two independent zero-mean normal random variables, representing the \(x\) and \(y\) coordinates relative to the original parent point.
Shifting all the points in each cluster disk
In practice (that is, in the code), all the daughter points are simulated relative to the origin. Then for each cluster disk, all the points need to be shifted, so the origin coincides with the parent point, which completes the simulation step.
To use vectorization in the code, the coordinates of each cluster point are repeated by the number of daughters in the corresponding cluster by using the functions repelem in MATLAB, rep in R, and repeat in Python.
Code
I have implemented the simulation procedure in MATLAB, R and Python, which as usual are all very similar. The code can be downloaded here.
MATLAB
%Simulation window parameters xMin=-.5; xMax=.5; yMin=-.5; yMax=.5; %Parameters for the parent and daughter point processes lambdaParent=10;%density of parent Poisson point process lambdaDautgher=100;%mean number of points in each cluster sigma=0.05;%sigma for normal variables (ie random locations) of daughters %Extended simulation windows parameters rExt=7*sigma; %extension parameter -- use factor of deviation %for rExt, use factor of deviation sigma eg 6 or 7 xMinExt=xMin-rExt; xMaxExt=xMax+rExt; yMinExt=yMin-rExt; yMaxExt=yMax+rExt; %rectangle dimensions xDeltaExt=xMaxExt-xMinExt; yDeltaExt=yMaxExt-yMinExt; areaTotalExt=xDeltaExt*yDeltaExt; %area of extended rectangle %Simulate Poisson point process for the parents numbPointsParent=poissrnd(areaTotalExt*lambdaParent,1,1);%Poisson number %x and y coordinates of Poisson points for the parent xxParent=xMinExt+xDeltaExt*rand(numbPointsParent,1); yyParent=yMinExt+yDeltaExt*rand(numbPointsParent,1); %Simulate Poisson point process for the daughters (ie final poiint process) numbPointsDaughter=poissrnd(lambdaDautgher,numbPointsParent,1); numbPoints=sum(numbPointsDaughter); %total number of points %Generate the (relative) locations in Cartesian coordinates by %simulating independent normal variables xx0=normrnd(0,sigma,numbPoints,1); yy0=normrnd(0,sigma,numbPoints,1); %replicate parent points (ie centres of disks/clusters) xx=repelem(xxParent,numbPointsDaughter); yy=repelem(yyParent,numbPointsDaughter); %translate points (ie parents points are the centres of cluster disks) xx=xx(:)+xx0; yy=yy(:)+yy0; %thin points if outside the simulation window booleInside=((xx>=xMin)&(xx<=xMax)&(yy>=yMin)&(yy<=yMax)); %retain points inside simulation window xx=xx(booleInside); yy=yy(booleInside); %Plotting scatter(xx,yy);
R
Note: it is a bit tricky to write “<-” in the R code (as it automatically changes to the html equivalent in the HTML editor I am using), so I have usually used “=” instead of the usual “<-”.
#Simulation window parameters xMin=-.5; xMax=.5; yMin=-.5; yMax=.5; #Parameters for the parent and daughter point processes lambdaParent=10;#density of parent Poisson point process lambdaDaughter=100;#mean number of points in each cluster sigma=0.05; #sigma for normal variables (ie random locations) of daughters #Extended simulation windows parameters rExt=7*sigma; #extension parameter #for rExt, use factor of deviation sigma eg 6 or 7 xMinExt=xMin-rExt; xMaxExt=xMax+rExt; yMinExt=yMin-rExt; yMaxExt=yMax+rExt; #rectangle dimensions xDeltaExt=xMaxExt-xMinExt; yDeltaExt=yMaxExt-yMinExt; areaTotalExt=xDeltaExt*yDeltaExt; #area of extended rectangle #Simulate Poisson point process for the parents numbPointsParent=rpois(1,areaTotalExt*lambdaParent);#Poisson number of points #x and y coordinates of Poisson points for the parent xxParent=xMinExt+xDeltaExt*runif(numbPointsParent); yyParent=yMinExt+yDeltaExt*runif(numbPointsParent); #Simulate Poisson point process for the daughters (ie final poiint process) numbPointsDaughter=rpois(numbPointsParent,lambdaDaughter); numbPoints=sum(numbPointsDaughter); #total number of points #Generate the (relative) locations in Cartesian coordinates by #simulating independent normal variables xx0=rnorm(numbPoints,0,sigma); yy0=rnorm(numbPoints,0,sigma); #replicate parent points (ie centres of disks/clusters) xx=rep(xxParent,numbPointsDaughter); yy=rep(yyParent,numbPointsDaughter); #translate points (ie parents points are the centres of cluster disks) xx=xx+xx0; yy=yy+yy0; #thin points if outside the simulation window booleInside=((xx&amp;amp;amp;amp;gt;=xMin)&amp;amp;amp;amp;amp;(xx&amp;amp;amp;amp;lt;=xMax)&amp;amp;amp;amp;amp;(yy&amp;amp;amp;amp;gt;=yMin)&amp;amp;amp;amp;amp;(yy&amp;amp;amp;amp;lt;=yMax)); #retain points inside simulation window xx=xx[booleInside]; yy=yy[booleInside]; #Plotting par(pty="s") plot(xx,yy,'p',xlab='x',ylab='y',col='blue');
Of course, as I have mentioned before, simulating a spatial point processes in R is even easier with the powerful spatial statistics library spatstat. The Thomas cluster point process is simulated by using the function rThomas, but other cluster point processes, including Neyman-Scott types, are possible.
Python
Note: in previous posts I used the SciPy functions for random number generation, but now use the NumPy ones, but there is little difference, as SciPy builds off NumPy.
import numpy as np;&amp;amp;amp;amp;nbsp; # NumPy package for arrays, random number generation, etc import matplotlib.pyplot as plt&amp;amp;amp;amp;nbsp; # For plotting # Simulation window parameters xMin = -.5; xMax = .5; yMin = -.5; yMax = .5; # Parameters for the parent and daughter point processes lambdaParent = 10; # density of parent Poisson point process lambdaDaughter = 100; # mean number of points in each cluster sigma = 0.05; # sigma for normal variables (ie random locations) of daughters #Extended simulation windows parameters rExt=7*sigma; #extension parameter #for rExt, use factor of deviation sigma eg 6 or 7 xMinExt = xMin - rExt; xMaxExt = xMax + rExt; yMinExt = yMin - rExt; yMaxExt = yMax + rExt; # rectangle dimensions xDeltaExt = xMaxExt - xMinExt; yDeltaExt = yMaxExt - yMinExt; areaTotalExt = xDeltaExt * yDeltaExt; # area of extended rectangle # Simulate Poisson point process for the parents numbPointsParent = np.random.poisson(areaTotalExt * lambdaParent);# Poisson number of points # x and y coordinates of Poisson points for the parent xxParent = xMinExt + xDeltaExt * np.random.uniform(0, 1, numbPointsParent); yyParent = yMinExt + yDeltaExt * np.random.uniform(0, 1, numbPointsParent); # Simulate Poisson point process for the daughters (ie final poiint process) numbPointsDaughter = np.random.poisson(lambdaDaughter, numbPointsParent); numbPoints = sum(numbPointsDaughter); # total number of points # Generate the (relative) locations in Cartesian coordinates by # simulating independent normal variables xx0 = np.random.normal(0, sigma, numbPoints); # (relative) x coordinaets yy0 = np.random.normal(0, sigma, numbPoints); # (relative) y coordinates # replicate parent points (ie centres of disks/clusters) xx = np.repeat(xxParent, numbPointsDaughter); yy = np.repeat(yyParent, numbPointsDaughter); # translate points (ie parents points are the centres of cluster disks) xx = xx + xx0; yy = yy + yy0; # thin points if outside the simulation window booleInside=((xx&amp;amp;amp;amp;gt;=xMin)&amp;amp;amp;amp;amp;(xx&amp;amp;amp;amp;lt;=xMax)&amp;amp;amp;amp;amp;(yy&amp;amp;amp;amp;gt;=yMin)&amp;amp;amp;amp;amp;(yy&amp;amp;amp;amp;lt;=yMax)); # retain points inside simulation window xx = xx[booleInside]; yy = yy[booleInside]; # Plotting plt.scatter(xx, yy, edgecolor='b', facecolor='none', alpha=0.5); plt.xlabel("x"); plt.ylabel("y"); plt.axis('equal');
Julia
After writing this post, I later wrote the code in Julia. The code is here and my thoughts about Julia are here.
Results
The results show that the clusters of Thomas point process tend to be more blurred than those of Matérn point process, which has cluster edges clearly defined by the disks. The points of of a Thomas point process can be far away from the centre of each cluster, depending on the variance of the normal random variables used in the simulation.
The Thomas point process has also appeared in models of wireless networks, which we covered in the book Stochastic Geometry Analysis of Cellular Networks by Błaszczyszyn, Haenggi, Keeler, and Mukherjee, Section 8.1.8.
More generally, Neyman-Scott point processes belong to a family of point processes called shot noise Cox point processes; see the paper by Møller.
In my travels on the web, I found this post where the writer also simulates the Thomas and Matérn point processes in Python, independent of my code. That code is a bit different to mine because I use the repeat function and simulate all the Poisson variables at once, instead of using a for-loop and simulating a Poisson variable for each iteration. I also think it’s not quite correct because I do not see how they account for edge effects.
