Point process notation

In probability and statistics, point process notation is the varying mathematical notation used to represent stochastic objects known as point processes, which are used in related fields of stochastic geometry, spatial statistics and continuum percolation and frequently serve as mathematical models of random phenomena, representable as points, in time, space or both.

The notation varies due to the intertwining history of certain mathematical fields and the different interpretations of point processes , and borrows notation from mathematical areas of study such as measure theory and set theory.

Interpretation of point processes

The notation, as well as the terminology, of point processes depends on their setting and interpretation as mathematical objects which under certain mathematical assumptions can be interpreted as a random sequences of points, random sets of points or random counting measures.

Random sequences of points

In some mathematical frameworks, a given point process may be considered as a sequence of points with each point randomly positioned in ${\textbf {R}}^{d}$ as well as some other mathematical spaces. In general, whether or not a random sequence is equivalent to the other interpretations of a point process depends on the underling mathematical space, but this holds true for the setting of finite-dimensional Euclidean space ${\textbf {R}}^{d}$.

Random set of points

Given that often point processes are simple and the order of the points does not matter, a collection of random points can be considered as a random set of points. The theory of random sets was independently developed by David Kendall and Georges Matheron. In terms of being considered as a random set, a sequence of random points is a random closed set if the sequence has no accumulation points with probability one.

A point process is often denoted by a single letter , for example $\Phi$, and if $\Phi$ is considered as a random set, then the corresponding notation:

$x\in \Phi ,$

is used to denote that a random point $x$ belongs to the point process $\Phi$. The theory of random sets can be applied to point processes owing to this interpretation. These two interpretations have resulted in a point process being written as $\{x_{1},x_{2},\dots \}=\{x\}_{i}$ to highlight its interpretation as either a sequence or random closed set of points.

Random measures

To denote the number of points of $\Phi$ located in some set Borel $B$, it is sometimes written

$N(B)=\#(B\cap \Phi ),$

where $N(B)$ is a random variable and $\#$ is a counting measure, which gives the number of points in some set. In this expression point process is denoted by $\Phi$ while $N$ represents the number of points of $\Phi$ in $B$. In the context of random measures, one can write $N(B)=n$ to denote that there is the set $B$ that contains $n$ points of $\Phi$. In other words, $N$ can be considered as a random measure that assigns some non-negative integer-valued measure to sets. This interpretation has motivated a point process being considered just another name for a random counting measure and the techniques of random measure theory offering another way to study point processes, also which leads to the various notations used in integration and measure theory. \footnote{As discussed in Chapter 1 of Stoyan, Kendall and Mechke, varying integral notation in general applies to all integrals here and elsewhere.}

Dual notation

The different interpretations of point processes as random sets and counting measures is captured with the popular notation :\ -- $\Phi$ is a set of random points.\ -- $\Phi (B)$ is a random variable that gives the number of points of $\Phi$ in $B$. \ Denoting the counting measure again with $\#$, this new notation would imply:

$\Phi (B)=\#(B\cap \Phi ).$

Sums

If $f$ is some (measurable) function on ${\textbf {R}}^{d}$, then the sum of $f(x)$ over all the points $x$ in $\Phi$ can be written as:

$f(x_{1})+f(x_{2})\dots$

which has the random sequence appearance, or more compactly with set notation as:

$\sum _{x\in \Phi }f(x)$

or equivalently as:

$\int _{\textbf {N}}f(x)\Phi (dx)$

where ${\textbf {N}}$ is the space of all counting measures, hence putting an emphasis on the interpretation of $\Phi$ as a random counting measure. An alternative integration notation may be used to write this integral as:

$\int _{\textbf {N}}fd\Phi$

The dual interpretation of point processes is illustrated when writing the number of $\Phi$ points 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 $B$ and zero otherwise, which is also known as a Dirac measure. In this expression the random measure interpretation is on the left-hand side while the random set notation is used is on the right-hand side.

Expectations

The average or expected value of a sum of functions over a point process is written as:

$E\left[\sum _{x\in \Phi }f(x)\right]\qquad {\text{or}}\qquad \int _{\textbf {N}}\sum _{x\in \Phi }f(x)P(d\Phi ),$

where (in the random measure sense) $P$ is an appropriate probability measure defined on the space of counting functions ${\textbf {N}}$. The expected value of $\Phi (B)$, which is the definition of $\Lambda (B)$, can be written as:

$E[\Phi (B)]=E\left(\sum _{x\in \Phi }1_{B}(x)\right)\qquad {\text{or}}\qquad \int _{\textbf {N}}\sum _{x\in \Phi }1_{B}(x)P(d\Phi ).$

which is also known as the first moment measure of $\Phi$.

Uses in other fields

Point processes serve as cornerstones in other mathematical and statistical disciplines, hence the notation may be used in fields such stochastic geometry, spatial statistics or continuum percolation, and areas which use the methods and theory from these fields.