Binomial point process

The binomial point process is arguably the simplest point process. It consists of a non-random number of points scattered randomly and independently over some bounded region of space. In this post I will describe the binomial point process, how it leads to the Poisson point process, and its historical role as stars in the sky.

The binomial point process is an important stepping stone in the theory of point process. But I stress that for mathematical models, I would always use a Poisson point process instead of a binomial one. The only exception would be if you were developing a model for a small, non-random number of points.

Uniform binomial point process

We start with the simplest binomial point process, which has uniformly located points. (I described simulating this point process in an early post. The code is here.)

Definition

Consider some bounded (or more precisely, compact) region, say, \(W\), of the plane plane \(\mathbb{R}^2\), but the space can be more general.  The uniform binomial point process is created by scattering \(n\) points uniformly and independently across the set \(W\).

A single realization of a binomial point process with n=30 points. The points are uniformly and independently scattered across a unit square.

Distribution

Consider a single point uniformly scattered in the region \(W\), giving a binomial point process with \(n=1\).  We look at some region \(B\), which is a subset of \(W\), implying \(B\subseteq W\).  What is the probability that the single point \(X\) falls in region \(B\)?

First we write \(\nu(W)\) and \(\nu(B)\)  to denote the respective areas (or more precisely, Lebesgue measures) of the regions \(W\) and \(B\), hence \(\nu(B)\leq \nu(W)\). Then this probability, say, \(p\), is simply the ratio of the two areas, giving

$$p= P(X\in B)=\frac{\nu(B)}{\nu(W)}.$$

The event of a single point being found in the set \(B\) is a single Bernoulli trial, like flipping a single coin. But if there are \(n\) points, then there are \(n\) Bernoulli trials, which bring us to the binomial distribution.

For a uniform binomial point process \(N_W\), the number of randomly located points being found in a region \(B\) is a binomial random variable, say, \(N_W(B)\), with probability parameter \(p=\nu(B)/ \nu(W)\). The probability mass function of \(N_W(B)\) is

$$ P(N_W(B)=k)={n\choose k} p^k(1-p)^{n-k}. $$

We can write the expression more explicitly

$$ P(N_W(B)=k)={n\choose k} \left[\frac{\nu(B)}{ \nu(W)}\right]^k\left[1-\frac{\nu(B)}{\nu(W)}\right]^{n-k}. $$

Poisson limit

Poisson random variable

A standard exercise in introductory probability is deriving the Poisson distribution by taking the limit of the binomial distribution. This is done by sending \(n\) (the total number of Bernoulli trials) to infinity while keeping the binomial mean \(\mu:=p n\) fixed, which sends the probability \(p=\mu/n\) to zero.

More precisely, for \(\mu\geq0\), setting \(p_n=\mu/n \) and keeping \(\mu :=p_n n\) fixed, we have the limit result

$$\lim_{n\to \infty} {n \choose k} p_n^k (1-p_n)^{n-k} = \frac{\mu^k}{k!}\, e^{-\mu}.$$

We can use, for example, Stirling’s approximation to prove this limit result.

We can make the same limit argument with the binomial point process.

Homogeneous Poisson point process

We consider the intensity of the uniform binomial point process, which is the average number of points in a unit area. For a binomial point process, this is simply

$$\lambda := \frac{n}{\nu(W)}.$$

For the Poisson limit, we expand the region \(W\) so it covers the whole plane \(\mathbb{R}^2\), while keeping the intensity \(\lambda = n/\nu(W)\) fixed. This means that the area \(\nu(W)\) approaches infinity while the probability \(p=\nu(B)/\nu(W)\) goes to zero. Then in the limit we arrive at the homogeneous Poisson point process \(N\) with intensity \(\lambda\).

The number of points of \(N\) falling in the set \(B\) is a random variable \(N(B)\) with the probability mass function

$$ P(N(B)=k)=\frac{[\lambda \nu(B)]^k}{k!}\,e^{-\lambda \nu(B)}. $$

General binomial point process

Typically in point process literature, one first encounters the uniform binomial point process. But we can generalize it so the points are distributed according to some general distribution.

Definition

We write \(\Lambda\) to denote a non-negative Radon measure on \(W\), meaning \(\Lambda(W)< \infty\) and \(\Lambda(B)\geq 0\) for all (measurable) sets \(B\subseteq W\).  We can also assume a more general space for the underlying space such as a compact metric space, which is (Borel) measurable. But the intuition still works for compact region of the plane \(\mathbb{R}^2\).

For the \(n\) points, we assume each point is distributed according to the probability measure

$$\bar{\Lambda}= \frac{\Lambda}{\Lambda(W)}.$$

The resulting point process is a general binomial point process. The proofs for this point process remain essentially the same, replacing the Lebesgue measure \(\nu\), such as area or volume, with the non-negative measure \(\Lambda\).

Example

A typical example of the intensity measure \(\Lambda\) has the form

$$\Lambda(B)= \int_B f(x) dx\,,$$

where \(f\) is a non-negative density function on \(W\). Then the probability density of a single point is

$$ p(x_1,x_2) = \frac{f}{c},$$

where \(c\) is a normalization constant

$$c= \int_W f(x) dx\,.$$

On the set \(\subseteq \mathbb{R}^2\), a specific example is

$$ f(x_1,x_2) = \lambda e^{-(x_1^2+x_2^2)}.$$

Distribution

Assuming a general binomial point process \(N_W\) on \(W\), we can use the previous arguments to obtain the binomial distribution

$$ P(N_W(B)=k)={n\choose k} \left[\frac{\Lambda(B)}{\Lambda(W)}\right]^k\left[1-\frac{\Lambda(B)}{\Lambda(W)}\right]^{n-k}. $$

General Poisson point process

We can easily adapt the Poisson limit arguments for the general binomial Poisson point process, which results in the general Poisson point process \(N\) with intensity measure \(\Lambda\). The number of points of \(N\) falling in the set \(B\) is a random variable \(N(B)\) with the probability mass function

$$ P(N(B)=k)=\frac{[\Lambda(B)]^k}{k!}\, e^{-\Lambda(B)}. $$

History: Stars in the sky

The uniform binomial point process is an example of a spatial point process. With points being scattered uniformly and independently, its sheer simplicity makes it a natural choice for an early spatial model. But which scattered objects?

Perhaps not surprisingly, it is trying to understand star locations where we find the earliest known example of somebody describing something like a random point process. In 1767 in England John Michell wrote:

what it is probable would have been the least apparent distance of any two or more stars, any where in the whole heavens, upon the supposition that they had been scattered by mere chance, as it might happen

As an example, Michelle studied the six brightest stars in the Pleiades star cluster. He concluded the stars were not scattered by mere chance. Of course “scattered by mere chance” is not very precise in today’s probability language, but we can make the reasonable assumption that Michell meant the points were uniformly and independently scattered.

Years later in 1860 Simon Newcomb  examined Michell’s problem, motivating him to derive the Poisson distribution as the limit of the binomial distribution. Newcomb also studied star locations. Stephen Stigler considers this as the first example of applying the Poisson distribution to real data, pre-dating the famous work by Ladislaus Bortkiewicz who studied rare events such as deaths from horse kicks. We also owe Bortkiewicz the terms Poisson distribution and stochastic (in the sense of random).

Code

Here, on my repository, are some pieces of code that simulate a uniform binomial point process on a rectangle.

Further reading

For an introduction to spatial statistics, I suggest the lectures notes by Baddeley, which form Chapter 1 of these published lectures, edited by Baddeley, Bárány, Schneider, and Weil. The binomial point process is covered in Section 1.3.

The binomial point process is also covered briefly in the classic text Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke; see Section 2.2. Similar material is covered in the book’s previous edition by Stoyan, Kendall and Mecke.

Haenggi also wrote a readable introductory book called Stochastic Geometry for Wireless networks, where he gives the basics of point process theory. The binomial point process is covered in Section 2.4.4.

For some history on point processes and the Poisson distribution, I suggest starting with the respective papers:

  • Guttorp and Thorarinsdottir, What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property?;
  • Stigler, Poisson on the Poisson distribution.

Histories of the Poisson distribution and the Poisson point process are found in the following books:

Point process simulation

Point processes are mathematical objects that can represent a collection of randomly located points on some underlying space. There is a rich range of these random objects, and, depending on the type of points process, there are various steps and methods used to sample, simulate or generate them on computers. (I use these three terms somewhat interchangeably.) This is the first of a series of posts in which I will describe how to simulate some of the more tractable point processes defined on bounded regions of two-dimensional Euclidean space.

Finite point processes

A general family of point processes are the finite point processes, which are, as the name suggests, simply point processes with the property that the total number of points is finite with probability one. Usually it is assumed that both the number of points and the positions of these points are random. These two properties give a natural way to describe and to study point processes mathematically, while also giving an intuitive way to simulate them on computers.

If the number and locations of points can be simulated sequentially, such that the number of points is naturally generated first followed by the placing of the points, then the simulation process is usually easier. Such a point process is a good place to start for an example.

Simple example — Binomial point process

The binomial point process is arguably the simplest point process. It is created by scattering \(n\) points uniformly and independently located in some bounded region, say, \(R\) with area \(|R|\). The number of points located in a region \(B\subset R\) is a binomial random variable, say, \(N\), where its probability parameter \(p=\frac{|B|}{|R|}\) is simply the ratio of the  two areas. This implies that

$$ P(N=k)={n\choose k} p^k(1-p)^{n-k}, $$

The total number of points is non-random number \(n\), so we do not need to generate it as it is given. The independence of point locations means that all the points can be positioned in parallel.

For example, to simulate \(n\) points of a binomial point process on the unit square \([0,1]\times[0,1]\), we just need to independently sample the \(x\) and \(y\) coordinates of the points from a uniform distribution on the unit interval \([0,1]\). In other words, we randomly generate or sample two sets of \(n\) uniform random variables corresponding to the \(x\) and \(y\) of the \(n\) points.

To sample on a general \(w \times h\) rectangle, we just need need to multiple the random \(x\) and \(y\) values by the respective dimensions of the rectangle \(w\) and \(h\). Of course the rectangle can also be shifted up or down by adding or subtracting \(x\) and \(y\) values appropriately.

Essentially every programming language has a function for generating uniform random numbers because it is the default random number generator, which all the other random number generators build off, by employing various techniques like applying transformations. In MATLAB, R and Python, it is respectively rand, runif and scipy.stats.uniform, which all generate uniform points on the open interval \((0,1)\).

Code

Here are some pieces of code that illustrates how to simulate a binomial point process on the unit square. I suggest downloading the code directly here from my repository. (Sometimes my webpage editor mangles simulation code.)

MATLAB

n=10; %number of points

%Simulate binomial point process
xx=rand(n,1);%x coordinates of binomial points
yy=rand(n,1);%y coordinates of binomial points

%Plotting
scatter(xx,yy);
xlabel('x');ylabel('y');

R

n=10; #number of points
#Simulate Binomial point process
xx=runif(n);#x coordinates of binomial points
yy=runif(n);#y coordinates of binomial points

#Plotting
plot(xx,yy,'p',xlab='x',ylab='y',col='blue');

Python

&lt;/pre&gt;
import numpy as np
import matplotlib.pyplot as plt

plt.close('all'); # close all figures

numbPoints = 10; # number of points

# Simulate binomial point process
xx = np.random.uniform(0, 1, numbPoints) # x coordinates of binomial points
yy = np.random.uniform(0, 1, numbPoints) # y coordinates of binomial points

# Plotting
plt.scatter(xx, yy, edgecolor='b', facecolor='none', alpha=0.5)
plt.xlabel('x')
plt.ylabel('y')
plt.show()
&lt;pre&gt;

More complicated point processes

The binomial point process is very simple. One way to have a more complicated point process is to have a random number of points. The binomial point process is obtained from conditioning on the number of points of a homogeneous Poisson point process, implying its simulation builds directly off the binomial point process. The homogeneous Poisson point process is arguably the most used point process, and it is then in turned used to construct even more complicated points processes. This suggests that the homogeneous Poisson point process is the next point process we should try to simulate.

Further reading

For simulation of point processes, see, for example, the books Statistical Inference and Simulation for Spatial Point Processes by Møller and Waagepetersen, or Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke. There are books written by spatial statistics experts such as Stochastic Simulation by Ripley and Spatial Point Patterns: Methodology and Applications with R by Baddeley, Rubak and Turner, where the second book covers the spatial statistics R-package spatstat. Kroese and Botev also have a good introduction in the edited collection Stochastic Geometry, Spatial Statistics and Random Fields : Models and Algorithms by Schmidt, where the relevant chapter (number 12) is also freely available online. More general stochastic simulation books that cover relevant material include Uniform Random Variate Generation by Devroye and Stochastic Simulation: Algorithms and Analysis by Asmussen and Glynn.

For a mathematical introduction to finite point processes, see the standard reference An Introduction to the Theory of Point Processes by Daley and Vere-Jones (Chapter 5 in either the single-volume edition or the first volume of the second two-volume edition). Finite point processes are also covered in a recent tutorial on Palm calculus and Gibbs point processes.