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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'',
==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===
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
===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 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">{{
:<math> x\in {N}, </math>
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:<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===
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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">{{
==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">{{
* <math> {N}</math> denotes a set of random points.
* <math> {N}(B)</math> denotes a random variable that gives the number of points of <math> {N}</math> in <math> B</math> (hence it is a random counting measure).
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==Sums==
If <math>f</math> is some [[measurable function]] on '''R'''<sup>''d''</sup>, then the sum of <math> f(x)</math> over all the points <math> x</math> in <math> {N} </math> can be written in a number of ways <ref name="stoyan1995stochastic"/><ref
:<math> f(x_1) + f(x_2)+ \cdots </math>
which has the random sequence appearance, or
:<math> \sum_{x\in {N}}f(x) </math>
or, equivalently, with integration notation as:
:<math> \int_{\textbf{
:<math> \int_{\textbf{
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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==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 {N}}f(x)\right] \qquad \text{or} \qquad \int_{\textbf{N}}\sum_{x\in {N}}f(x) P(d{N}), </math>
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:<math> E[{N}(B)]=E\left( \sum_{x\in {N}}1_B(x)\right) \qquad \text{or} \qquad \int_{\textbf{N}}\sum_{x\in {N}}1_B(x) P(d{N}). </math>
which is also known as the first [[moment measure]] of <math> {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==
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* [[Table of mathematical symbols]]
==
{{notelist}}
==References==
<references/>
{{DEFAULTSORT:Mathematical Notation}}
[[Category:Mathematical notation|
[[Category:Point processes|N]]
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