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

{{ProbabilityTopics}}

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'', volumeSecond 2Edition, Section 4.1, Wiley Chichester, 1995.</ref><ref name="daleyPPI2003">{{Cite book | doi = 10.1007/b97277 | first1 = D. J. | last1 = Daley and| first2 = D. | last2 = Vere-Jones.| title = ''An introductionIntroduction to the theoryTheory of pointPoint processes.Processes Vol.| I''url = https://archive.org/details/introductiontoth0000dale | url-access = registration | series = Probability and its Applications (New| York)year = 2003 | isbn = 978-0-387-95541-4 }}</ref><ref name="haenggi2012stochastic">M. Springer,Haenggi. New''Stochastic York,geometry secondfor editionwireless networks''. Chapter 2. Cambridge University Press, 20032012.</ref> and borrows notation from mathematical areas of study such as [[measure theory]] and [[set theory]].<ref name="stoyan1995stochastic"/>

</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'''^''d''<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 underlingunderlying mathematical space, but this holds true for the setting of finite-dimensional Euclidean space '''R'''^''d''.<ref name="daleyPPII2008">{{Cite book | last1 = Daley | first1 = D. J. Daley| andlast2 D.= Vere-Jones | first2 = D. ''| doi = 10.1007/978-0-387-49835-5 | title = An introductionIntroduction to the theoryTheory of pointPoint processes.Processes Vol.| {II''}.series = Probability and itsIts Applications (New| York).year Springer,= New2008 York,| secondisbn edition,= 2008.978-0-387-21337-8 }}</ref>.

===Random set of points===

A point process is called ''simple'' if no two (or more points) coincide in ___location with [[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 book | doi = 10.1007/978-3-540-38175-4_1 | first1 = A. | last1 = Baddeley, | first2 = I. | last2 = Barany, and| first3 = R. | last3 = Schneider | first4 = W. | last4 = Weil| chapter = Spatial pointPoint processesProcesses and their applications.Applications ''Stochastic| Geometry:title Lectures= givenStochastic atGeometry the| CIMEseries Summer= SchoolLecture heldNotes in MartinaMathematics Franca,| Italy,volume September= 13–18,1892 2004'',| pages 1–75,= 1 | year = 2007. | isbn = 978-3-540-38174-7 }}</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_pointLimit point#Types_of_limit_pointsTypes of limit points|accumulation points]] with probability one<ref name="schneider2008stochastic">{{Cite book | last1 = Schneider | first1 = R. Schneider| andlast2 = Weil | first2 = W. Weil| doi = 10.1007/978-3-540-78859-1 | title = ''Stochastic and integralIntegral geometry''.Geometry Springer,| 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]]. ''Poisson processes'', volume 3. Oxford university press, 1992.</ref><ref name="moller2003statistical">{{Cite book | last1 = Moller | first1 = J. Moller| andlast2 = Plenge Waagepetersen | first2 = R. P.| Waagepetersendoi = 10.1201/9780203496930 | title = ''Statistical inferenceInference and simulationSimulation for spatialSpatial pointPoint processes''.Processes | series = C&H/CRC Press,Monographs on Statistics & Applied Probability | volume = 100 | year = 2003 | isbn = 978-1-58488-265-7 | citeseerx = 10.1.1.124.1275 }}</ref><ref name="stoyan1995stochastic"/>, for example <math> \Phi{N}</math>, and if the point process is considered as a random set, then the corresponding notation:<ref name="stoyan1995stochastic"/>:

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

is used to denote that a random point <math>x</math> is an [[Element (mathematics)|element]] of (or [[Element_Element (mathematics)#Notation_and_terminologyNotation and terminology|belongs]] to) the point process <math> \Phi{N}</math>. The theory of random sets can be applied to point processes owing to this interpretation., which Thesealongside twothe interpretationsrandom sequence interpretation havehas 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> \Phi{N}</math> located in some [[Borel set]] <math> B</math>, it is sometimes written <ref name="kingman1992poisson"/>

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

where <math> N\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> \Phi</math> while <math> N</math> represents the number of points of <math> \Phi</math> in <math> B</math>. In the context of random measures, one can write <math> N(B)=n</math> to denote that there is the set <math> B</math> that contains <math> n</math> points of <math> \Phi</math>. In other words, <math> N</math> 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"> I. S. Mol{\vc}anov. ''Theory of random sets''. Springer, 2005.:

:<math> {N}</math>.

</ref> and the techniques of random measure theory offering another way to study point processes<ref name="stoyan1995stochastic"/><ref name="grandell1977point"> J. Grandell. Point processes and random measures. ''Advances in Applied Probability'', pages 502–526, 1977.

On the other hand, the symbol:

</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.}}

:<math> \Phi </math>

==Dual notation==

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

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"> F. Baccelli and B. B{\l}aszczyszyn. ''Stochastic Geometry and Wireless Networks, Volume I – Theory'', volume 3, No 3–4 of ''Foundations and Trends in Networking''. NoW Publishers, 2009.</ref> in which:

:<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 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 [[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="haenggi2012stochastic"/><ref name="moller2003statistical"/><ref name="BB1">{{Cite F.journal | last1 = Baccelli and| Bfirst1 = F. B{\l}aszczyszynO. ''| title = Stochastic Geometry and Wireless Networks,: Volume I – Theory'', volume| 3,doi No= 3–410.1561/1300000006 of| journal = ''Foundations and Trends in Networking''. NoW| Publishers,volume = 3 | issue = 3–4 | pages = 249–449 | year = 2009 | url = https://hal.inria.fr/inria-00403039/file/FnT1.pdf }}</ref> in which:

* <math> \Phi</math> denotes a set of random points.

* <math> \Phi{N}</math> denotes a set of random points.
* <math> {N}(B)</math> denotes a random variable that gives the number of points of <math> \Phi{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> \Phi{N}(B) =\#(B \cap \Phi{N}). </math>

==Sums==

If <math>f</math> is some [[measurable function]] on '''R'''^''d'', then the sum of <math> f(x)</math> over all the points <math> x</math> in <math> \Phi{N} </math> can be written in a number of ways <ref name="stoyan1995stochastic"/><ref bename="haenggi2012stochastic"/> writtensuch 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 \Phi{N}}f(x) </math>

or, equivalently, with integration notation as:

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

wherewhich <math> \textbf{N}</math> is the space of all counting measures, hence puttingputs an emphasis on the interpretation of <math> \Phi{N}</math> asbeing a random counting measure. An alternative integration notation may be used to write this integral as:

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

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

:<math> \Phi{N}(B)= \sum_{x\in \Phi{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"/>:<ref name="haenggi2012stochastic"/>

:<math> E\left[\sum_{x\in \Phi{N}}f(x)\right] \qquad \text{or} \qquad \int_{\textbf{N}}\sum_{x\in \Phi{N}}f(x) P(d\Phi{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> \Phi{N}(B)</math> can be written as:<ref name="stoyan1995stochastic"/>:

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

which is also known as the first [[moment measure]] of <math> \Phi{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 (probability)#Campbell's theorem: general point process|Campbell's theorem]].<ref name="daleyPPI2003"/>

==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.

==ReferencesSee also==

* [[Mathematical Alphanumeric Symbols]]

* [[Mathematical notation]]

* [[Notation in probability]]

* [[Table of mathematical symbols]]

==Notes==

{{notelist}}

==References==

<references/>

<references/>

[[Category{{DEFAULTSORT:Mathematical notation]]Notation}}

[[Category:Mathematical notation|*]]

[[Category:Point processes|N]]