# Point process notation

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{{Short description|Mathematical notation used in probability and statistics}}
{{ProbabilityTopics}}

In [probability](/source/probability) and [statistics](/source/statistics),  '''point process notation''' comprises the range of [mathematical notation](/source/mathematical_notation) used to symbolically represent [random](/source/random) [objects](/source/Mathematical_object) known as [point process](/source/point_process)es, which are used in related fields such as [stochastic geometry](/source/stochastic_geometry), [spatial statistics](/source/spatial_statistics) and [continuum percolation theory](/source/continuum_percolation_theory) and frequently serve as [mathematical models](/source/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'', Second Edition, Section 4.1, Wiley Chichester, 1995.</ref><ref name="daleyPPI2003">{{Cite book | doi = 10.1007/b97277 | first1 = D. J. | last1 = Daley | first2 = D. | last2 = Vere-Jones| title = An Introduction to the Theory of Point Processes | url = https://archive.org/details/introductiontoth0000dale | url-access = registration | series = Probability and its Applications | year = 2003 | isbn = 978-0-387-95541-4 }}</ref><ref name="haenggi2012stochastic">M. Haenggi. ''Stochastic geometry for wireless networks''. Chapter 2. Cambridge University Press, 2012.</ref>  and borrows notation from mathematical areas of study such as [measure theory](/source/measure_theory) and [set theory](/source/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 random [sequences](/source/sequences) of points, random [sets](/source/Set_(mathematics)) of points or [random counting measure](/source/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](/source/Euclidean_space) '''R'''<sup>''d''</sup><ref name="stoyan1995stochastic"/> as well as some other more abstract [mathematical space](/source/mathematical_space)s.  In general, whether  or not  a random sequence is equivalent to the other interpretations of a point process depends on the underlying mathematical space, but this holds true for the setting of finite-dimensional Euclidean space '''R'''<sup>''d''</sup>.<ref name="daleyPPII2008">{{Cite book | last1 = Daley | first1 = D. J. | last2 = Vere-Jones | first2 = D. | doi = 10.1007/978-0-387-49835-5 | title = An Introduction to the Theory of Point Processes | series = Probability and Its Applications | year = 2008 | isbn = 978-0-387-21337-8 }}</ref>

===Random set of points===

A point process is called ''simple'' if no two (or more points) coincide in location with [probability one](/source/Almost_surely). 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 book | doi = 10.1007/978-3-540-38175-4_1 | first1 = A. | last1 = Baddeley | first2 = I. | last2 = Barany | first3 = R. | last3 = Schneider | first4 = W. | last4 = Weil| chapter = Spatial Point Processes and their Applications | title = Stochastic Geometry | series = Lecture Notes in Mathematics | volume = 1892 | pages = 1 | year = 2007 | isbn = 978-3-540-38174-7 }}</ref> The theory of random sets was independently developed by [David Kendall](/source/David_George_Kendall) and [Georges Matheron](/source/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](/source/Limit_point) with probability one<ref name="schneider2008stochastic">{{Cite book | last1 = Schneider | first1 = R. | last2 = Weil | first2 = W. | doi = 10.1007/978-3-540-78859-1 | title = Stochastic and Integral Geometry | series = Probability and Its Applications | year = 2008 | isbn = 978-3-540-78858-4 }}</ref>

A point process is often denoted by a single letter,<ref name="stoyan1995stochastic"/><ref name="kingman1992poisson">[J. F. C. Kingman](/source/J._F._C._Kingman). ''Poisson processes'', volume 3. Oxford university press, 1992.</ref><ref name="moller2003statistical">{{Cite book | last1 = Moller | first1 = J. | last2 = Plenge Waagepetersen | first2 = R. | doi = 10.1201/9780203496930 | title = Statistical Inference and Simulation for Spatial Point Processes | series = C&H/CRC Monographs on Statistics & Applied Probability | volume = 100 | year = 2003 | isbn = 978-1-58488-265-7 | citeseerx = 10.1.1.124.1275 }}</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](/source/Element_(mathematics)) of (or [belongs](/source/Element_(mathematics)) 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"/> Furthermore, sometimes an uppercase letter denotes the point process, while a lowercase denotes a point from the process, so, for example, the point <math>\textstyle x</math> (or <math>\textstyle x_i</math>) belongs to or is a point of the point process <math>\textstyle X</math>, or with set notation, <math>\textstyle x\in X</math>.<ref name="moller2003statistical"/>

===Random measures===

To denote the number of points of <math>  {N}</math> located in some [Borel set](/source/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](/source/random_variable) and <math>  \#</math> is a [counting measure](/source/counting_measure), which gives the number of points in some set. In this [mathematical expression](/source/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](/source/random_measure) that assigns some non-negative integer-valued [measure](/source/Measure_(mathematics))  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 book | doi = 10.1007/1-84628-150-4 | title = Theory of Random Sets | url = https://archive.org/details/probabilityitsap0000unse_i5l1 | url-access = registration | first = Ilya | last = Molchanov| series = Probability and Its Applications | year = 2005 | isbn = 978-1-85233-892-3 }}</ref>{{rp|106}} and the techniques of random measure theory offering another way to study point processes,<ref name="stoyan1995stochastic"/><ref name="grandell1977point">{{cite journal | last1 = Grandell | first1 = Jan | year = 1977 | title = Point Processes and Random Measures | journal = Advances in Applied Probability  | volume = 9 | issue = 3 | pages = 502–526 | jstor = 1426111 | doi = 10.2307/1426111 | s2cid = 124650005 }}</ref> which also induces the use of the various notations used in [integration](/source/Integral) and measure theory. {{efn|As discussed in Chapter 1 of Stoyan, Kendall and Mechke,<ref name="stoyan1995stochastic"/> varying [integral](/source/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="haenggi2012stochastic"/><ref name="moller2003statistical"/><ref name="BB1">{{Cite journal | last1 = Baccelli | first1 = F. O. | title = Stochastic Geometry and Wireless Networks: Volume I Theory | doi = 10.1561/1300000006 | journal = Foundations and Trends in Networking | volume = 3 | issue = 3–4 | pages = 249–449 | year = 2009 | url = https://hal.inria.fr/inria-00403039/file/FnT1.pdf }}</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](/source/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 be written in a number of ways <ref name="stoyan1995stochastic"/><ref name="haenggi2012stochastic"/> such as:

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

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

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

or, equivalently, with integration notation as:

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

which puts an emphasis on the interpretation of <math>  {N}</math> being a random counting measure. An alternative integration notation may be used to write this integral as:

:<math> \int_{\textbf{R}^d} 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](/source/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](/source/Dirac_measure).<ref name="BB1"/> In this expression the random measure interpretation is on the [left-hand side](/source/left-hand_side) while the random set notation is used is on the right-hand side.

==Expectations==

The [average](/source/average) or [expected value](/source/expected_value) of a sum of functions over a point process is written as:<ref name="stoyan1995stochastic"/><ref name="haenggi2012stochastic"/>

:<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](/source/probability_measure) defined on the space of [counting measure](/source/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](/source/moment_measure) of  <math>  {N}</math>. The expectation of such a random sum, known as a ''shot noise process'' in the theory of point processes, can be calculated with [Campbell's theorem](/source/Campbell's_theorem_(probability)).<ref name="daleyPPI2003"/>

==Uses in other fields==

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

==See also==
* [Mathematical Alphanumeric Symbols](/source/Mathematical_Alphanumeric_Symbols)
* [Mathematical notation](/source/Mathematical_notation)
* [Notation in probability](/source/Notation_in_probability)
* [Table of mathematical symbols](/source/Table_of_mathematical_symbols)

==Notes==
{{notelist}}

==References==

<references/>

{{DEFAULTSORT:Mathematical Notation}}
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Adapted from the Wikipedia article [Point process notation](https://en.wikipedia.org/wiki/Point_process_notation) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Point_process_notation?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.