Point process notation: Difference between revisions

Revision as of 03:46, 13 October 2013

Revision as of 03:51, 13 October 2013

Line 11:

===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>.

</ref>.

===Random set of points===

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--1813–18, 2004'', pages 1--751–75, 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 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="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"/>:

</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 accumulation points with probability one<ref name="schneider2008stochastic"> R. Schneider and W. Weil. ''Stochastic and integral geometry''. Springer, 2008.

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

</ref>.

A point process is often denoted by a single letter <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>

n

is used to denote that a random point <math>x</math> is an [[Element (mathematics)|element]] of (or [[Element_(mathematics)#Notation_and_terminology|belongs]] to) the point process <math> \Phi</math>. The theory of random sets can be applied to point processes owing to this interpretation. These two interpretations have resulted in a point process being written as:

:<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"/>.

Line 45 ⟶ 35:

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--526502–526, 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.}}

Line 51 ⟶ 41:

==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--43–4 of ''Foundations and Trends in Networking''. NoW Publishers, 2009.</ref> in which:

* <math> \Phi</math> denotes a set of random points.

Line 63 ⟶ 53:

==Sums==

If <math> f</math> is some [[measurable function]] on '''R'''^''d'', then the sum of <math> f(x)</math> over all the points <math> x</math> in <math> \Phi</math> can<ref name="stoyan1995stochastic"/> be written as:

:<math> f(x_1) + f(x_2)+ \dotscdots </math>

which has the random sequence appearance, or more compactly with set notation as:

Line 77 ⟶ 67:

where <math> \textbf{N}</math> is the space of all counting measures, hence putting an emphasis on the interpretation of <math> \Phi</math> as a random counting measure. An alternative integration notation may be used to write this integral as:

:<math> \int_{\textbf{N}} f \, d\Phi </math>

The dual interpretation of point processes is illustrated when writing the number of <math> \Phi</math> points in a set <math> B</math> as: