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In [[probability]] and [[statistics]],  '''point process notation''' comprises the range of [[mathematical notation]] used to symbolically represent [[random]] [[Mathematical object|objects]] known as [[point process]]es, which are used in related fields such as [[stochastic geometry]], [[spatial statistics]] and [[continuum percolation theory]] and frequently serve as [[mathematical models]] of random phenomena, representable as points, in time, space or both.

The notation varies due to the histories of certain mathematical fields and the different interpretations of point processes,<ref name="stoyan1995stochastic">D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.</ref><ref name="daleyPPI2003">{{cite doi|10.1007/b97277}}</ref> and borrows notation from mathematical areas of study such as [[measure theory]] and [[set theory]].<ref name="stoyan1995stochastic"/>

==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 assumptions can be interpreted as a random [[sequences]] of points, random [[Set (mathematics)|sets]] of points or random [[counting measure]]s.<ref name="stoyan1995stochastic"/>

===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 ''d''-dimensional [[Euclidean space]] '''R'''<sup>''d''</sup><ref name="stoyan1995stochastic"/> as well as some other more abstract [[mathematical space]]s.  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 '''R'''<sup>''d''</sup>.<ref name="daleyPPII2008">{{cite doi|10.1007/978-0-387-49835-5}}</ref>

===Random set of points===

A point process is called ''simple'' if no two (or more points) coincide in location with [[Almost surely|probability one]]. 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<ref name="stoyan1995stochastic"/><ref name="baddeley2007spatial">{{cite doi|10.1007/978-3-540-38175-4_1}}</ref> The theory of random sets was independently developed by [[David George Kendall|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 [[Limit point#Types of limit points|accumulation points]] with probability one<ref name="schneider2008stochastic">{{cite doi|10.1007/978-3-540-78859-1}}</ref>

A point process is often denoted by a single letter,<ref name="stoyan1995stochastic"/><ref name="kingman1992poisson">[[J. F. C. Kingman]]. ''Poisson processes'', volume 3. Oxford university press, 1992.</ref><ref name="moller2003statistical">{{cite doi|10.1201/9780203496930}}</ref> for example <math>  {N}</math>, and if the point process is considered as a random set, then the corresponding notation:<ref name="stoyan1995stochastic"/>

:<math> x\in {N}, </math>

is used to denote that a random point <math>x</math> is an [[Element (mathematics)|element]] of (or [[Element (mathematics)#Notation and terminology|belongs]] to) the point  process <math>  {N}</math>. The theory of random sets can be applied to point processes owing to this interpretation, which alongside the random sequence interpretation has resulted in a point process being written as:

:<math>  \{x_1, x_2,\dots \}=\{x\}_i,</math>

which highlights its interpretation as either a random sequence or random closed set of points.<ref name="stoyan1995stochastic"/>

===Random measures===

To denote the number of points of <math>  {N}</math> located in some [[Borel set]] <math>  B</math>, it is sometimes written <ref name="kingman1992poisson"/>

:<math> \Phi(B) =\#( B \cap {N}), </math>

where <math>  \Phi(B)</math> is a [[random variable]] and <math>  \#</math> is a [[counting measure]], which gives the number of points in some set. In this [[mathematical expression]] the point process is denoted by:

<math>  {N}</math>.

On the other hand, the symbol:

<math>  \Phi </math> 

represents the number of points of <math>  {N}</math> in <math>  B</math>. In the context of random measures,  one  can write:

<math>  \Phi(B)=n</math>

to denote that there is the set <math>  B</math> that contains <math>  n</math> points of <math> {N}</math>. In other words, a point process can be considered as a [[random measure]] that assigns some non-negative integer-valued [[Measure (mathematics)|measure]]  to sets.<ref name="stoyan1995stochastic"/> This interpretation has motivated a point process being considered just another name for a ''random counting measure''<ref name="molvcanov2005theory">{{cite doi|10.1007/1-84628-150-4}}</ref>{{rp|106}} and the techniques of random measure theory offering another way to study point processes,<ref name="stoyan1995stochastic"/><ref name="grandell1977point">{{cite jstor|1426111}}</ref> which also induces the use of the various notations used in [[Integral#Terminology and notation|integration]] and measure theory. {{efn|As discussed in Chapter 1 of Stoyan, Kendall and Mechke,<ref name="stoyan1995stochastic"/> 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 often used notation <ref name="stoyan1995stochastic"/><ref name="moller2003statistical"/><ref name="BB1">{{cite doi|10.1561/1300000006}}</ref> in which:

* <math> {N}</math> denotes a set of random points.

* <math> {N}(B)</math> denotes a random variable that gives the number of points  of <math>  {N}</math> in <math>  B</math> (hence it is a random counting measure).

Denoting the counting measure again with <math>  \#</math>, this dual notation implies:

:<math> {N}(B) =\#(B \cap {N}). </math>

==Sums==

If <math>f</math> is some [[measurable function]] on '''R'''<sup>''d''</sup>, then the sum of <math>  f(x)</math> over all the points <math>  x</math> in <math>  {N}</math> can<ref name="stoyan1995stochastic"/> be written as:

:<math> f(x_1) + f(x_2)+ \cdots </math>

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

:<math> \sum_{x\in {N}}f(x) </math>

or equivalently as:

:<math> \int_{\textbf{N}} f(x) {N}(dx) </math>

where <math>  \textbf{N}</math> is the [[Sample_space|space]] of all possible counting measures, hence putting an  emphasis on the interpretation of <math>  {N}</math> as a random counting measure. An alternative integration notation may be used to write this integral as:

:<math> \int_{\textbf{N}} f \, d{N} </math>

The dual interpretation of point processes is illustrated when writing the number of <math>  {N}</math> points in a set <math>  B</math> as:

:<math> {N}(B)= \sum_{x\in {N}}1_B(x) </math>

where the [[indicator function]] <math>  1_B(x) =1</math> if the point <math>  x</math> is exists in <math>  B</math> and zero otherwise, which in this setting is also known as a [[Dirac measure]].<ref name="BB1"/> 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:<ref name="stoyan1995stochastic"/>

:<math> E\left[\sum_{x\in {N}}f(x)\right] \qquad \text{or} \qquad  \int_{\textbf{N}}\sum_{x\in {N}}f(x) P(d{N}), </math>

where (in the random measure sense) <math>  P</math> is an appropriate [[probability measure]] defined on the space of [[counting measure]]s <math>  \textbf{N}</math>. The expected value of <math>  {N}(B)</math>  can be written as:<ref name="stoyan1995stochastic"/>

:<math> E[{N}(B)]=E\left( \sum_{x\in {N}}1_B(x)\right) \qquad \text{or}  \qquad  \int_{\textbf{N}}\sum_{x\in {N}}1_B(x) P(d{N}). </math>

which is also known as the first [[moment measure]] of  <math>  {N}</math>.

==Uses in other fields==

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

==See also==
* [[Mathematical Alphanumeric Symbols]]
* [[Mathematical notation]]
* [[Notation in probability]]
* [[Table of mathematical symbols]]

==References==

{{notelist}}

<references/>

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[[Category:Mathematical notation| ]]