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

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.

{{ProbabilityTopics}}

In [[probability]] and [[statistics]], '''point process notation''' iscomprises the varyingrange of [[mathematical notation]] used to symbolically represent stochastic[[random]] [[Mathematical object|objects]] known as [[point processesprocess]]es, which are used in related fields ofsuch 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 intertwining history 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"> D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. I''. Probability and its Applications (New York). Springer, New York, second edition, 2003.

The notation varies due to the intertwining historyhistories 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 mathematical assumptions can be interpreted as a random [[sequences]] of points, random [[Set (mathematics)|sets]] of points or [[random counting measuresmeasure]]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 <math>''d''-dimensional [[Euclidean \textbf{space]] '''R}^'''<sup>''d''</mathsup><ref name="stoyan1995stochastic"/> as well as some other more abstract [[mathematical spacesspace]]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'''<mathsup> \textbf{R}^''d''</mathsup>.<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>

</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. B{\'a}r{\'a}ny,| andlast2 = Barany | first3 = R. | last3 = Schneider | first4 = W. | last4 = Weil| chapter = Spatial pointPoint processesProcesses and their applications.Applications | title = ''Stochastic Geometry: Lectures| givenseries at= theLecture CIMENotes Summerin SchoolMathematics held| involume Martina= Franca,1892 Italy,| Septemberpages 13= 1 | year = 2007 | isbn = 978-3-18,540-38174-7 2004''}}</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, pagesa 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 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 = 978-3-75,540-78858-4 2007.}}</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. | 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"/>

</ref>. 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<ref name="schneider2008stochastic"> R. Schneider and W. Weil. ''Stochastic and integral geometry''. Springer, 2008.

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

</ref>.

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> \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> to highlight its interpretation as either a sequence or random closed set of points<ref name="stoyan1995stochastic"/>.:

A point process is often denoted by a single letter <ref name="kingman1992poisson"> J. F. C. Kingman. ''Poisson processes'', volume 3. Oxford university press, 1992.

:<math> \Phi(B) =\#(B{x_1, x_2,\capdots \Phi). }=\{x\}_i,</math>

</ref><ref name="moller2003statistical"> J. Moller and R. P. Waagepetersen. ''Statistical inference and simulation for spatial point processes''. CRC Press, 2003.

</ref>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> \Phitextstyle x</math>, and if(or <math>\textstyle \Phix_i</math>) isbelongs consideredto asor is a randompoint setof the point process <math>\textstyle X</math>, thenor thewith correspondingset notation, <math>\textstyle x\in X</math>.<ref name="stoyan1995stochasticmoller2003statistical"/>:

===Random measures===

:<math> x\in \Phi, </math>

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

is used to denote that a random point <math> x</math> belongs to the point process <math> \Phi</math>. 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 <math> \{x_1, x_2,\dots \}=\{x\}_i</math> to highlight its interpretation as either a sequence or random closed set of points<ref name="stoyan1995stochastic"/>.

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

===Random measures===

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:

To denote the number of points of <math> \Phi</math> located in some set Borel <math> B</math>, it is sometimes written <ref name="kingman1992poisson"/>

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

On the other hand, the symbol:

where <math> N(B)</math> is a random variable and <math> \#</math> is a counting measure, which gives the number of points in some set. In this expression 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 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> \Phi </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.

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

</ref>, also which leads to the various notations used in 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(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 popularoften 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> {N}</math> denotes a set of random points.

</ref>:\ --* <math> \Phi</math> is a set of random points.\ -- <math> \Phi{N}(B)</math> isdenotes a random variable that gives the number of points of <math> \Phi{N}</math> in <math> B</math>. (hence \it Denotingis thea random counting measure again with <math> \#</math>, this new notation would imply:).

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

:<math> \Phi(B) =\#(B \cap \Phi). </math>

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

==Sums==

If <math> f</math> is some ([[measurable) function]] on '''R'''<mathsup> \textbf{R}^''d''</mathsup>, 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)+ \dotscdots </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, d{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 functionsmeasure]]s <math> \textbf{N}</math>. The expected value of <math> \Phi{N}(B)</math>, which is the definition of <math> \Lambda(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 serveare as cornerstonesemployed 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]]

==Notes==

{{notelist}}

==References==

<references/>

{{DEFAULTSORT:Mathematical Notation}}

[[Category:Mathematical notation|*]]

[[Category:Point processes|N]]