2 Point Process Definition
2.1 From Point patterns to counting
In the previous section, we described a point pattern as a random set of locations:
\[ X = \{x_1, x_2, \dots, x_n\} \subset W. \]
While this representation is intuitive, it is not mathematically convenient. In particular, it does not easily allow us to:
- count the number of points in a region
- define expectations
- describe spatial structure
To address this, we introduce an equivalent but more useful representation.
2.2 The counting measure
Instead of working directly with the set of points, we define a function that counts how many points fall inside any region.
For a set \(B \subset W\), define:
\[ N(B) = \text{number of points of X that lie in B}. \]
This function \(N(\cdot)\) is called the counting measure associated with the point pattern.
Example:
- \(N(B) = 0\): no points in \(B\)
- \(N(B) = 5\): 5 points in \(B\)
- \(N(W) = n\): total number of points
2.3 Definition of a point process
A point process is a random mechanism that assigns a count to every region \(B \subset W\).
Formally, a point process is a collection of random variables:
\[ \{N(B): B \subset W\} \]
where \(N(B)\) gives the number of points in a region \(B\).
OK so a point process is just an infinite collection of random variables, indexed by every possible subset of the observation window \(W\)?
Can it be a finite collection?
If \(W = \{1\}\), then is \(\{N(1)\}\) a point process.
A: “…infinite collection of random variables…”
It is, but also with the requirement that the random variables are counting variables (?).
Technically, the example \(W = \{1\}\), \(\{N(1)\}\), is a point process, but it is a trivial one.
“A point process generalizes integer-valued random variables to spatial domains.”
On that last quote there, is there a generalization of real-valued variables to spatial domains?
A: Right so this is literally how we arrive at random fields.
A point process models where events occur, while a random field models how an underlying quantity varies across space
2.4 Consistency properties
The counting variables \(N(B)\) must satisfy some natural properties:
- Non-negativity
\[ N(B) \in \{0,1,2,\dots\} \]
- Additivity
If \(B_1, B_2\) are disjoint, then:
\[ N(B_1 \cup B_2) = N(B_1) + N(B_2) \]
- Finite counts (locally finite)
For bounded regions \(B\), \(N(B) < \infty\)
Are these axioms or properties?
Why not define point process as:
Collection of random variables \(\{N(B): B \subset W\}\)
where \(N(B)\) are counting variables (measures?) which satisfy (1, 2, 3) above.
A: Yes this would be better if asked for the exact definition of a point process.
2.5 Equivalence with point patterns
The counting measure representation and the set representation are equivalent.
- Given a set of points \(\{x_1, \dots, x_n \}\), we can define:
\[ N(B) = \sum_{i=1}^n\mathbb{1}\{x_i \in B\} \]
- Conversely, given \(N(\cdot)\), we can recover the point locations.
This allows us to switch between:
- a geometric view (points in space)
- a measure-theoretic view (counts over regions)
So this is just explaining that, we could view the point process as a “random set of locations”, but it is not as mathematically convenient, so instead we construct the counting-variable version?
A: Yes this is correct.
How would you explicitly construct the random set of locations version?
Choose a random number of locations, and then choose each location independently? Or we would need a joint distribution? Possible over the count too?
A: Basically yes.
2.6 Simple point process
In most applications, we assume the process is simple, meaning:
- no two points occur at exactly the same location
Formally:
\[ N(\{u\}) \in \{0, 1\}\quad \forall \ u \in W \]
This rules out multiple points stacking on top of each other.