Point process notation

One tricky part about learning point processes is the use of different notation.  In this post I cover some basic notation used in the theory of point processes.

The reason for different notation is due to the different interpretations of a point process, which is where we will start. For those unfamiliar with them, I suggest the previous post for more details on the definition of a point process.

Interpretation

Historically, there’s a couple main interpretations of a point process, which is also called a random point field. The different interpretations partly explain the various terminology and notation used in the theory of point processes, but now a standard mathematical approach is used, as covered in more detail a previous post.

There are different ways to interpret a point process, which is often denoted by a single letter, for example, \(N\) or \(\Phi\).  If the point process is defined on a space like the real like, where the points can be ordered, then additional interpretations exist, but mathematicians assume the order of the points does not matter, limiting the possible interpretations.

Random measures

The now standard definition of a point process is given in terms of random measures.

A point process can be interpreted a random counting measure.

More specifically, a point process is defined as a mapping from a sample space \(\Omega\) to the space of counting measures \(\mathbb{M}\), meaning that each realization of a point process is a counting measure \(\phi\in \mathbb{M}\).

Notation

The standard interpretation of a point process as a random (counting) measure means that point process theory borrows heavily notation from measure theory and calculus. For example, in measure theory we can write a (non-random) counting measure as \(\#\), so \(\#(B)=n\) is how we write that the set \(B\) contains \(n\) points. We can then write the the number of points of a point process \(\Phi\) located in some (Borel) set \(B\) as \(N(B) =\#( B \cap \Phi)\), where \(N(B)\) is a random variable. In this expression, the point process is denoted by \(\Phi\), while\(N(B)\) is the number of points of \(\Phi\) in \(B\), meaning \(N\) is a random counting measure.

The main interpretations of point processes as random sets and counting measures is captured with the notation:

  • \(\Phi\) is a set of random points.
  • \(\Phi(B)\) is a random variable that gives the number of points of \(\Phi\) located in the (Borel) set \(B\).

This is the notation often used in point process theory. It implies
$$
\Phi(B) =\#(B \cap \Phi).
$$

We now look at how this notation is used in point process theory.

Sums

If \(f\) is some (measurable) function on the underlying space \(\mathbb{S}\), such as Euclidean space \(\mathbb{R}^d\), then we can write the sum of \(f(x)\) over all the points of a simple point process \(\Phi\) as
$$
\sum_{x\in \Phi}f(x)\,,
$$
where we are using the random set interpretation.

For any point process \(\Phi\), we can also write the sum as
$$
\int_{\mathbb{S}} f(x) \,\Phi(dx) \,,
$$
which highlights the interpretation of the point process \(\Phi\) as a random counting measure. Of course, we can use different integral notation, giving, for example, the expression
$$
\int_{\mathbb{S}} f \,d\Phi \,,
$$
which denotes the same sum.

We can illustrate the dual interpretation of a point process by writing the number of point of a simple point process \(\Phi\) existing in a set \(B\) as
$$
\Phi(B)= \sum_{x\in \Phi}1_B(x)\,,
$$
where the indicator function \(1_B(x) =1\) if the point \(x\) is exists in the set \(B\), and \(1_B(x) =0\) otherwise. In this setting, \(1_B(x)\) is also known as a Dirac measure, as it gives a measure of the set \(B\). We can see in this expression that the random measure interpretation is on the left-hand side, while the random set notation is on the right-hand side.

Expectations

We can write the average or expected value of a sum of functions over a simple point process \(\Phi\) as
$$
\mathbb{E}\left[\sum_{x\in \Phi}f(x)\right] \,,
$$
or for any point process \(\Phi\) as
$$
\int_{\textbf{N}}\sum_{x\in \Phi}f(x) \mathbb{P}(d\Phi)\,,
$$
where \(\mathbb{P}\) is an appropriate probability measure defined on the space of counting functions \(\textbf{N}\), thus illustrating the random measure interpretation.

We can write the expected value of \(\Phi(B)\), which is the definition of the intensity measure of a point process \(\Phi\), as
$$
\mathbb{E}[\Phi(B)]=\mathbb{E}\left( \sum_{x\in \Phi}1_B(x)\right) \qquad \text{or} \qquad \mathbb{E}[\Phi(B)]=\int_{\textbf{N}}\sum_{x\in \Phi}1_B(x) P(d\Phi) \,,
$$
which is also known as the mean measure or first moment measure of \(\Phi\).

Events

In probability we want to describe the behaviour of certain events, such as flipping at last three heads across ten coin flips. For point processes, events are simply configurations with a certain (geometric) property, such as no points existing in a certain region or all the points being a fixed minium distance from each other.

Typically, when being mathematically abstract, we denote an event with a single letter, such as \(\Gamma\). Then to denote that a point process satisfies this condition we write \(\Phi\in \Gamma\). In other words, the point process \(\Phi\) has the property \(\Gamma\). We can then write the probability of the event (or configuration) \(\Gamma\) of occurring as
$$
\mathbb{P}(\Gamma)= \mathbb{P}(\Phi\in \Gamma ) \,.
$$

Uppercase and subscript notation

The convention in probability is usually to denote random objects, such as random variables and point processes, with uppercase (or capital) letters. Conversely, a non-random object, such as the realization of a random variable or point process, is denoted by a lowercase letter. For example, \(\Phi\) is a point processes, while \(\phi\) is a point pattern, which may be a realization of the point process \(\Phi\).

With this convention, we can denote an arbitrary point process of a point process \(\Phi\) by \(X\), meaning \(X\in \Phi\). (But such a point is also a point on the underlying non-random space \(\mathbb{S}\) on which the point process \(\Phi\) is defined.) We also see lowercase used for the point, giving \(x\in \Phi\).

Sometimes subscripts are used to emphasize some type of numbering of points, giving, for example, two points \(X_1\in \Phi\) and \(X_2\in \Phi\). Sometimes authors will write something like

$$
\sum_{X_i\in \Phi}f(X_i)\,,
$$

but this redundant notation as \(X_i\) is a dummy variable, so you can omit the subscript in such an expression.

Some authors use a notation where the letter with and without a subscript denotes, respectively, the point process and a point belonging to the point process. Using this convention, we write, for example, \(X=\{ X_i\}_i\) and \(X_i\in X\).

Further reading

For point process theory, Wikipedia is a good start place to start, particularly the articles on point processes and point process notation, though the former is too mathematical for a Wikipedia article. The standard reference on point processes was the An Introduction to the Theory of Point Processes by Daley and Vere-Jones, which now spread across volume one and two, but I would not learn the subject with these books.

The classic text Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke covers point processes and the varying notation in Chapters 2 and 4. The similar material is covered in the previous edition by Stoyan, Kendall and Mecke. A more mathematical book that covers point processes and random sets is Stochastic and integral geometry by Schneider and Weil.

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Author: Paul Keeler

I am a researcher with interests in mathematical models involving randomness, particularly models with some element of geometry. Much of my work studies wireless networks with a focus on using tools from probability theory such as point processes. I come from Australia, where I call Melbourne home, but I have lived several years in Europe. I grew up in country Queensland and New South Wales.

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