A non-random collection of points located on some space is called a *point pattern *in spatial statistics. Informally, you can interpret a *point process* as a collection of random points scattered over some underlying mathematical space, meaning each outcome or realization of a point process forms a point pattern. Using such intuition gets you pretty far. But if we want to be more mathematically formal, we need to use precise mathematical objects.

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.

In this post I will cover the main definitions, terminology and notation used in the theory and application of point processes. I won’t delve too much into the precise details, giving just an outline with references at the end.

## Underlying mathematical space

We consider a point process defined on some underlying mathematical space \(\mathbb{S}\), which is sometimes called the *carrier space* or *state space. * We further assume that the space is measurable by having a Borel \(\sigma\)*-algebra* \(\mathcal{S}\).

In practice, the underlying space is usually the real line \(\mathbb{R}\), the plane \(\mathbb{R}^2\), or some other familiar mathematical space like a square lattice. More generally, a point process can be defined on any metric space, allowing for the notion of distance. Mathematicians study point processes in even more general settings by defining them on, for example, a locally compact second countable Hausdorff space, but such generality is not needed for most people and their applications.

## Modern probability approach

In modern probability theory, if we want to define a random mathematical object, we start with a random experiment in the context of a *probability space* or *triple* \((\Omega,\mathcal{A},\mathbb{P})\), where:

- \(\Omega\) is a
*sample space*, which is the set of all possible outcomes;
- \(\mathcal{A}\) is a \(\sigma\)
*-algebra* or \(\sigma\)*-field*, which is a family of events (subsets of \(\Omega\));
- \(\mathbb{P}\) is a
*probability measure*, which assigns probability to each event in \(\mathcal{A}\).

To gain some intuition, David Williams says to imagine that Tyche, Goddess of Chance, chooses a point \(\omega\in\Omega\) at random according to the *law* \(\mathbb{P}\) such that an event \(A\in \mathcal{A}\) has a probability given by \(\mathbb{P}(A)\), where we understand probability with our own intuition. (We can also choose \(\omega\in\Omega\) by using some physical experiment, as long as it is truly random.)

Now we can define random objects by using a certain *measurable function* or *mapping* that maps to a suitable space of objects. For example, a real-valued random variable is a measurable function from \(\Omega\) to the real line; a random matrix is a measurable function from \(\Omega\) to some space of matrices; and, as John Kingman quips in his classic book Poisson Processes, a random elephant is just a measurable function from \(\Omega\) to some suitable space of elephants.

But what space should we use for a point process? To answer that, we need to interpret a point process as a suitable mathematical object.

## Interpretation

There are different ways to interpret a point process, which is often denoted by a single letter, for example, \(N\) or \(\Phi\). (The convention of using the Greek letter \(\Phi\) comes from German mathematicians, but some prefer not to use \(\Phi\), as it’s often used for the normal cumulative distribution function.) 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 closed set

In mathematics a collection of distinct things is formalized by a mathematical object called a set. We say that a set contains *elements* or *members, *and a set never contains more than one of the same element. Sets are fundamental objects with set concepts and notation being found everywhere in mathematics.

We now define a common type of point process, which we can formalize with the concept of a set.

A point process is *simple *if the probability of all points of the point process being distinct is one.

In other words, for a simple point process, there is zero probability of two or more of its points being found in the same location of the underlying state space \(\mathbb{S}\), which brings us to our first interpretation.

A simple point process can be interpreted as a *random **closed** set*.

More specifically, we can interpret a simple point process as a (measurable) mapping from a sample space \(\Omega\) to the space of closed sets \(\mathbb{F}\), meaning that each realization of a simple point process is a closed set \(\phi\in \mathbb{F}\).

Point process theory has adopted the notation from set theory. For example, if we want to say some point, which we denote by \(x\), of the underlying space \(\mathbb{S}\) *belongs to* or *is a member of* a simple point process \(\Phi\), then we can simply write \(x\in \Phi\). We can also write a point process as \(\{x\}_i\) to highlight its interpretation as a random closed set of points.

The theory of random sets, which is a field of study in its own right, can be applied to point processes owing to this interpretation. But for non-simple point processes, we need another point process interpretation.

### Random measures

Modern integration theory is based on measure theory, which revolves around the concept of a set function known as a *measure*. In addition to a couple of other properties, when you apply this function to a set, it gives a number, such as a integer or real number. For example, a counting measure gives you the number elements in a set, which could be a subspace \(B\), such as a region of the plane \(B \subset \mathbb{R}^2\). (The letter \(B\) is often used for sets in measure and probability theory as it’s typically assumed that the sets are Borel sets, which form a very large family of well-behaved sets in terms of measurability.) The concept of a counting measure gives us the second and now standard interpretation of a point process.

A point process can be interpreted a *random counting measure*.

More specifically, we define a point process 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 \(\phi\in \mathbb{M}\). Some mathematicians even say a point process is just another name for a random counting measure. The techniques of random measure theory provide alternative (and arguably main) approach to study point processes.

This standard interpretation of a point process 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 .

## Notation

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 \(X_i\in X\).

## Important concepts

In the theory point processes, like any other field of mathematics, there are various important concepts for understanding and proving various results. Without going in the details, these include *shot noise*, *Campbell’s theorem*, *Laplace functional*, *Palm calculus*, *void probability*, and *factorial moment measures*. In future posts, I’ll detail some of these concepts.

## Further reading

There are many, many books covering the fundamentals of modern probability theory, including those (in roughly order of difficulty) by Grimmett and Stirzaker, Karr, Rosenthal, Shiryaev, Durrett, and Billingsley. A very quick introduction is given in this web article.

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. Point processes are also covered in a recent readable tutorial on Palm calculus and Gibbs point processes, which will be the subject of another post.

Spatial statistics builds of point process theory, giving good texts for learning the basics of point processes. I suggest the lectures notes by Baddeley, which form Chapter 1 of these published lectures, edited by Baddeley, Bárány, Schneider, and Weil. I always recommend the book *Spatial Point Patterns: Methodology and Applications with R* written by spatial statistics experts Baddeley, Rubak and Turner, which covers the spatial statistics (and point process simulation) R-package spatstat. Another good book is *Statistical Inference and Simulation for Spatial Point Processes* by Møller and Waagepetersen, but there are many more.

In recent years, point process theory (under the guise of *stochastic geometry*) has been used to model wireless networks. An early treatment of the subject is the two-volume textbook *Stochastic Geometry and Wireless Networks* by Baccelli and Błaszczyszyn, where the first volume is on theory and the second volume is on applications. Haenggi wrote a very readable introductory book called *Stochastic Geometry for Wireless networks*.

For the more mathematically brave, there’s the recent book *Random Measures, Theory and Applications* by Kallenberg who is the authority of the subject, having written earlier books, but these have now become obsolete with this recent publication. Another mathematically challenging book is *The Theory of Random Sets* by Molchanov, but it has less emphasis on point processes.

A very recent book (manuscript) is *Random Measures, Point Processes, and Stochastic Geometry* by Baccelli, Błaszczyszyn, and Karray, which contains much material, including new results and proofs. Finally, Last and Penrose wrote a mathematical monograph *Lectures on the Poisson process*, which is freely available online here.