Point process notation: Difference between revisions

Revision as of 20:26, 13 October 2013

Revision as of 07:56, 15 October 2013

Line 3:

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. 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.

</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 a random [[sequences]] of points, random [[Set (mathematics)|sets]] of points or 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 underling mathematical space, but this holds true for the setting of finite-dimensional Euclidean space '''R'''^''d''.<ref name="daleyPPII2008"> D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. {II''}. 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. 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, 2007.</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. Schneider and W. Weil. ''Stochastic and integral geometry''. Springer, 2008.</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, 2003.</ref><ref name="stoyan1995stochastic"/>, 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>

Line 27:

:<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"/>.

===Random measures===

Line 35:

:<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"> I. S. Mol{\vc}anov. ''Theory of random sets''. Springer, 2005.

</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.

</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. Baccelli and B. B{\l}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.

* <math> \Phi(B)</math> denotes a random variable that gives the number of points of <math> \Phi</math> in <math> B</math> (hence it is a random counting measure).

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

Line 75:

:<math> \Phi(B)= \sum_{x\in \Phi}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"/>:

:<math> E\left[\sum_{x\in \Phi}f(x)\right] \qquad \text{or} \qquad \int_{\textbf{N}}\sum_{x\in \Phi}f(x) P(d\Phi), </math>

where (in the random measure sense) <math> P</math> is an appropriate [[probability measure]] defined on the space of [[counting measure]]s <math> \textbf{N}</math>. The expected value of <math> \Phi(B)</math> can be written as:<ref name="stoyan1995stochastic"/>:

:<math> E[\Phi(B)]=E\left( \sum_{x\in \Phi}1_B(x)\right) \qquad \text{or} \qquad \int_{\textbf{N}}\sum_{x\in \Phi}1_B(x) P(d\Phi). </math>

Line 104:

<references/>

{{DEFAULTSORT:Mathematical Notation}}