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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 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
==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 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'''<sup>''d''</sup><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'''<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
===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
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
:<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 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
==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
* <math> {N}</math> denotes a set of random points.
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:<math> \int_{\textbf{R}^d} f(x) {N}(dx) </math>
:<math> \int_{\textbf{R}^d} f \, d{N} </math>
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{{DEFAULTSORT:Mathematical Notation}}
[[Category:Mathematical notation|*]]
[[Category:Point processes|N]]
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