Point process notation: Difference between revisions

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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'', Second Edition, Section 4.1, Wiley Chichester, 1995.</ref><ref name="daleyPPI2003">{{Cite journalbook | 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 = 978-0-387-95541-0 | pmid = | pmc =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]] 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 random [[sequences]] of points, random [[Set (mathematics)|sets]] of points or random [[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 underlying mathematical space, but this holds true for the setting of finite-dimensional Euclidean space '''R'''^''d''.<ref name="daleyPPII2008">{{Cite journalbook | 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 | pmid = | pmc = }}</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 | 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 | pmid = | pmc = }}</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 journalbook | 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 | pmid = | pmc = }}</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 journalbook | 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 | pmidciteseerx = | pmc =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>

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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 journalbook | 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-X | pmid = | pmc =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 | publisher = Applied Probability Trust | jstor = 1426111 | urldoi = 10.2307/1426111 | formats2cid = | accessdate =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 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 | pmidurl = | pmc =https://hal.inria.fr/inria-00403039/file/FnT1.pdf }}</ref> in which:

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

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or, equivalently, with integration notation as:

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

wherewhich <math> \textbf{N}</math> is the [[Sample space|space]] of all possible counting measures, hence puttingputs an emphasis on the interpretation of <math> {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{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:

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{{DEFAULTSORT:Mathematical Notation}}

[[Category:Mathematical notation| *]]

[[Category:Point processes|N]]