Content deleted Content added
No edit summary |
|||
Line 1:
In [[probability]] and [[statistics]], '''point process notation''' is the varying [[mathematical notation]] used to represent [[stochastic]] objects known as [[point processes]], which are used in related fields of [[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
</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
===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 <math> \textbf{R}^d</math><ref name="stoyan1995stochastic"/> as well as some other more abstract [[mathematical
</ref>.
Line 31:
:<math> x\in \Phi, </math>
is used to denote that a random point <math> x</math> 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> to highlight its interpretation as either a random sequence or random closed set of points<ref name="stoyan1995stochastic"/>.
===Random measures===
Line 39:
:<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 expression 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 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>, also which leads to the various notations used in [[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
</ref>:\ -- <math> \Phi</math> is a set of random points.\ -- <math> \Phi(B)</math> is a random variable that gives the number of points of <math> \Phi</math> in <math> B</math>. \ Denoting the counting measure again with <math> \#</math>, this new notation would imply:
Line 55:
==Sums==
If <math> f</math> is some ([[measurable]]) [[function]] on <math> \textbf{R}^d</math>, 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) \dots </math>
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 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 functions <math> \textbf{N}</math>. The expected value of <math> \Phi(B)</math>, which is the definition of <math> \Lambda(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 91:
==Uses in other fields==
Point processes serve as cornerstones in other mathematical and statistical disciplines, hence the notation may be used in fields such [[stochastic geometry]], [[spatial statistics]] or [[continuum percolation theory]], and areas which use the methods and theory from these fields.
==References==
| |||