3  The Poisson Process

3.1 Motivation: Complete Spatial Randomness

In many applications, it is useful to have a baseline model representing no spatial structure.

Intuitively, this corresponds to a situation where:

  • events occur independently of one another
  • there is no clustering or inhibition
  • the spatial distribution is entirely random

This notation is formalized by the Poisson process, often referred to as a model of complete spatial randomness (CSR).

3.2 Definition (Informal)

A Poisson process is a point process with two key properties:

  1. Counts in disjoint regions are independent

  2. The number of points in a region depends only on its size (or intensity).

3.3 Homogeneous Poisson Process

We first consider the simplest case.

A point process \(N(\cdot)\) on \(W \subset \mathbb{R}^d\) is called a homogeneous Poisson process (HPPP) with intensity \(\lambda > 0\) if it satisfies:

(i) Poisson counts

For any region \(B \subset W\):

\[ N(B) \sim \text{Poisson}(\lambda |B|) \]

where \(|B|\) is the volume (area in 2D) of \(B\).

(ii) Independent increments

If \(B_1, \dots, B_k\) are disjoint regions, then:

\[ N(B_1), \dots, N(B_k) \ \text{are independent} \]

3.4 Interpretation

The parameter \(\lambda\) has a clear meaning:

  • it represents the average number of points per area
  • and for any region \(B\), we have:

\[ \mathbb{E}[N(B)] = \lambda |B| \]

3.5 Equivalent construction

The Poisson process can be constructed by the following two steps:

  1. Draw the number of points:

\[ N \sim \text{Poisson}(\lambda |W|) \]

  1. Given \(N = n\), sample locations:

\[ x_1, \dots, x_n \stackrel{\text{iid}}{\sim} \text{Uniform}(W) \] ## Inhomogeneous Poisson process

In many applications, the density of events varies across space.

A point process is called an inhomogeneous Poisson point process (IPPP) with intensity function \(\lambda(u)\) if:

(i) Poisson counts

For any region \(B \subset W\):

\[ N(B) \sim \text{Poisson}(\int_B \lambda(u)\ du) \]

where \(|B|\) is the volume (area in 2D) of \(B\).

(ii) Independent increments

If \(B_1, \dots, B_k\) are disjoint regions, then:

\[ N(B_1), \dots, N(B_k) \ \text{are independent} \]

3.6 Interpretation of \(\lambda(u)\)

The function \(\lambda(u)\) represents:

The expected number of points per unit area at location \(u\)

More precisely,

\[ \mathbb{E}[N(B)] = \int_B \lambda(u) \ du \]

3.7 Why the Poisson process matters

The Poisson process plays a central role because:

  • it represents complete spatial randomness
  • it serves as a baseline for detecting structure
  • many summary statistics are defined relative to it