Point process notation: Difference between revisions

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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">D.{{cite Jdoi|10. Daley1007/b97277}}</ref> and D.borrows Vere-Jones.notation ''Anfrom introductionmathematical to the theoryareas of pointstudy processes.such Vol.as I''.[[measure Probabilitytheory]] and its[[set Applications (New York)theory]].<ref Springer, New York, second edition, 2003.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==

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===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 underling mathematical space, but this holds true for the setting of finite-dimensional Euclidean space '''R'''^''d''.<ref name="daleyPPII2008">D.{{cite Jdoi|10. Daley and D. Vere1007/978-Jones. ''An introduction to the theory of point processes. Vol. {II''0-387-49835-5}}. Probability and its Applications (New York). Springer, New York, second edition, 2008.</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">A.{{cite Baddeley, I. Barany, and R. Schneider. Spatial point processes and their applications. ''Stochastic Geometry: Lectures given at the CIME Summer School held in Martina Franca, Italy, September 13–18, 2004'', pages 1–75, 2007doi|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">R.{{cite Schneider and W. Weil. ''Stochastic and integral geometry''. Springer, 2008doi|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">J. Moller and R. P. Waagepetersen. ''Statistical inference and simulation for spatial point processes''. CRC Press,{{cite 2003doi|10.1201/9780203496930}}</ref> for example <math> \Phi</math>, and if the point process is considered as a random set, then the corresponding notation:<ref name="stoyan1995stochastic"/>

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

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

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 [[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">{{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">J.{{cite Grandell.jstor|1426111‎}}</ref> Pointwhich processesalso induces the use of the various notations used in [[Integral#Terminology and randomnotation|integration]] measuresand measure theory. ''Advances{{efn|As discussed in AppliedChapter Probability''1 of Stoyan, pagesKendall 502–526and Mechke,<ref name="stoyan1995stochastic"/> varying [[integral]] notation in general applies to all integrals here and 1977elsewhere.}}

</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">F.{{cite Baccelli and Bdoi|10. B{\l1561/1300000006}}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</math> denotes a set of random points.