2 Point Processes 2.1 Introduction 2.3 Point Process Intensities

2.2 The Poisson Process

Definition 2.4.

Poisson Process. Let $\lambda(x)$ be a non-negative valued function, called the intensity function of a spatial Poisson process. Then:

1.

the number of events, $N(A)$ , in any planar region $A$ follows a Poisson distribution with mean

$\mu(A)=\int_{A}\lambda(x)dx$
2.

given $N(A)=n$ , the locations of the $n$ events in $A$ form an independent random sample from the distribution on $A$ with pdf proportional to $\lambda(x)$

2.2.1 Homogeneous and inhomogeneous Poisson processes

If $\lambda(x)=\lambda$ , for all $x$ , the process is a homogeneous Poisson process, sometimes referred to as complete spatial randomness. Otherwise the process is inhomogeneous. Informally, complete spatial randomness means that the (expected) density of points is constant across any region, and there is no interaction between the points. See Figure 2.1.

Figure 2.1: Link, Caption: Left: a realisation of a homogeneous Poisson process. Right: a realisation of an inhomogeneous Poisson process. The two point patterns look very similar to the eye, but one process is homogeneous and the other is inhomogeneous. The grey background shows

\lambda(x)

on the same scale for each graph, with lighter areas denoting a lower intensity. The expected number of points for each process within the square

A

\mu(A)=100

2.2.2 Properties of the Homogeneous Poisson Process

Proposition 2.1.

Let $X$ be a homogeneous Poisson process.

1.

for any two disjoint planar regions $A$ and $B$ , the random variables $N(A)$ and $N(B)$ are independent
2.

${\rm Var}\{N(A)\}/{\rm E}[N(A)]=1$ , for all $A$
3.

distribution function of the distance from an arbitrary point to the nearest event is

$F(x)=1-\exp(-\pi\lambda x^{2}):x>0$

Proofs of 2 and 3 will be given in the lecture. A proof of 1 can be found in Rényi (1967).
$\Box$

2.2.3 Simulation of a Poisson Process

There are various approaches to simulating a realisation of a Poisson process. We outline one for a Poisson process with intensity function $\lambda(x)$ , which we wish to simulate on a region $A$ .

Recall from Definition 2.4:

•

The number of points in $A$ has a Poisson distribution with mean $\int_{A}\lambda(x)\mathrm{d}x$ .
•

The location of each point is iid from the density proportional to $\lambda(x)$ .

Therefore we (i) simulate the number of points; then (ii) simulate the position of each point. This is straightforward for homogeneous processes, for which $\lambda(x)=\lambda$ .

For inhomogeneous processes, if the simulation in (ii) is not straightforward then we can resort to rejection sampling (assuming $\lambda(x)<M$ for some $M$ and all $x\in A$ ). This proceeds by simulating points uniformly on $A$ , and accepting each with probability $\lambda(x)/M$ . We repeat this until we have simulated the correct number of points (the number simulated in part (i)).

Examples are given in the lab sessions.

2.2.4 A Simple Test of Complete Spatial Randomness

An early, simple, but now rarely-used method used to test complete spatial randomness (CSR) was based on quadrats. This method relies on the fact that, under CSR, the expected number of observations within any region of equal size is the same.

In Figure 2.2, there are $n=90$ observed points, and $m=9$ regions of equal size, so the expected number in each region is $n^{*}=10$ . Let $n_{i}$ be the number of points in region $i$ .

It can be shown that under CSR, the quantity

X^{2}=\sum_{i=1}^{m}\frac{(n_{i}-n^{*})^{2}}{n^{*}}

has a $\chi^{2}_{m-1}$ distribution. The chi-squared distribution with $d$ degrees of freedom ( $\chi^{2}_{d}$ ) is a special case of the gamma distribution, specifically a $\Gamma(d/2,1/2)$ distribution. It has mean $d$ and variance $2d$ .

Figure 2.2: Link, Caption: Realisation of a Poisson Process with 90 events. The observation window has been split into 9 equally-sized sections in order to conduct a

\chi^{2}

test for complete spatial randomness, see text for further detail.

The observed counts for the point process in Figure 2.2 are

11	5	16
8	12	7
13	4	14

This gives rise to the test statistic $(1+25+36+4+4+9+9+36+16)/10=14$ . The critical point of the $\chi^{2}_{8}$ distribution (at the 5% level of significance) is 15.5, so there is no strong evidence against CSR (the $p$ -value is 0.08). Indeed, the points in the figure were generated from a homogeneous Poisson process. The test described here is an example of a chi-squared goodness-of-fit test.