|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{about|random point processes||Campbell's theorem (geometry)}}
| | Let me inroduce myself, my name is Obdulia Grier. One of his favorite hobbies is reading comics and he'll be starting another thing along in addition to it. Dispatching is when he supports his spouse and children. should i get a legal plan ([http://www.anolivre.com Our Web Site]) currently live in Iowa which will then never advance. If you want to recognize more away my website: http://www.anolivre.com |
| | |
| In [[probability theory]] and [[statistics]], '''Campell's theorem''' can refer to a particular [[equation]] or set of results relating to the [[Expected value|expectation]] of a [[Function (mathematics)|function]] summed over a [[point process]] to an [[integral]] involving the [[Moment_measure#First_moment_measure|intensity measure]] of the point process, which allows for the calculation of [[expected value]] and [[variance]] of the [[random]] [[sum]]. One version<ref name="kingman1992poisson">{{cite book|title=Poisson Processes|page=28|first=John|last=Kingman|authorlink=John Kingman|publisher=Oxford Science Publications|year=1993|isbn=0-19-853693-3}}</ref> of the theorem specifically relates to the [[Poisson point process]] and gives a method for calculating moments as well as [[Laplace functional]]s of the process.
| |
| | |
| Another result by the name of Campell's theorem,<ref name="stoyan1995stochastic">D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.</ref> but also known as '''Campbell's formula''',<ref name="baddeley2007spatial">{{cite doi|10.1007/978-3-540-38175-4_1}}</ref>{{rp|28}} entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process.<ref name="baddeley2007spatial"/> There also exist equations involving [[moment measure]]s and [[factorial moment measure]]s that are considered versions of Campbell's formula. All these results are employed in probability and statistics with a particular importance in the related fields of [[point process]]es,<ref name="daleyPPI2003">{{cite doi|10.1007/b97277}}</ref> [[stochastic geometry]]<ref name="stoyan1995stochastic"/> and [[continuum percolation theory]],<ref name="meester1996continuum">R. Meester and R. Roy. Continuum percolation, volume 119 of Cambridge tracts in mathematics, 1996.
| |
| | |
| </ref> [[spatial statistics]].<ref name="baddeley2007spatial"/><ref name="moller2003statistical">{{cite doi|10.1201/9780203496930}}</ref>
| |
| | |
| The theorem's name stems from the work<ref name="campbell1909study">{{Cite journal|last=Campbell|first= N.| authorlink=Norman Robert Campbell |year=1909|journal= Proc. Cambr. Phil. Soc. |volume=15 |title=The study of discontinuous phenomena|pages= 117–136|url=http://www.archive.org/details/proceedingsofcam15190810camb}}</ref><ref name="campbell1909discontinuities">{{cite journal|last=Campbell|first= N.| authorlink = Norman Robert Campbell |year=1910|journal= Proc. Cambr. Phil. Soc. |volume=15 |title=Discontinuities in light emission
| |
| |pages=310–328|url=http://www.archive.org/details/proceedingsofcam15190810camb}}</ref> by [[Norman Robert Campbell|Norman R. Campbell]] on [[shot noise]],<ref name="daleyPPI2003"/> which was partly inspired by the work of [[Ernest Rutherford]] and [[Hans Geiger]] on [[alpha particle]] detection, where the [[Poisson point process]] arose as a solution to a family of differential equations by [[Harry Bateman]].<ref name="stirzaker2000advice">{{cite jstor|3621649}}</ref> In Campbell's work, he presents the [[Moment (mathematics)|moments]] and [[Moment-generating function|generating functions]] of the random sum of a Poisson process on the real line, but remarks that the main mathematical argument was due to [[G. H. Hardy]], which has inspired the result to be sometimes called the '''Campbell-Hardy theorem'''.<ref name="stirzaker2000advice"/><ref>{{cite book|last=Grimmett G. and Stirzaker D.|title=Probability and random processes|year=2001|publisher=Oxford University Press|pages=290}}</ref>
| |
| | |
| ==Background==
| |
| | |
| For a point process <math>{N}</math> defined on (''d''-dimensional) [[Euclidean space]] <math>\textbf{R}^d </math> {{efn|It can be defined on a more general mathematical space than Euclidean space, but often this space is used for models.<ref name="daleyPPI2003"/>}}, Campbell's theorem offers a way to calculate expectations of a function <math>f</math> (with [[range (mathematics)|range]] in the real line '''R''') defined also on <math>\textbf{R}^d </math> and summed over <math>{N}</math>, namely:
| |
| | |
| :<math>E[ \sum_{x\in {N}}f(x)] </math>,
| |
| | |
| where <math>E </math> denotes the expectation and set notation is used such that <math>{N}</math> is considered as a random set (see [[Point process notation]]). For a point process <math>{N}</math>, Campbell's theorem relates the above expectation with the intensity measure ''Λ''. In relation to a [[Borel set]] ''B'' the intensity measure of <math>{N}</math> is defined as:
| |
| | |
| :<math>\Lambda(B)=E[ {N}(B) ] </math>,
| |
| | |
| where the [[Measure (mathematics)|measure]] notation is used such that <math>{N}</math> is considered a random [[counting measure]]. The quantity ''Λ(B)'' can be interpreted as the average number of points of <math>{N}</math> located in the set ''B''.
| |
| | |
| ==Campbell's theorem: Poisson point process==
| |
| | |
| One version of Campbell's theorem<ref name="kingman1992poisson"/> says that for a Poisson point process <math>{N}</math> and a measurable function <math> f: \textbf{R}^d\rightarrow \textbf{R}</math>, the random sum
| |
| | |
| :<math> \Sigma=\sum_{x\in {N}}f(x) </math>
| |
| | |
| is [[absolutely convergent]] with [[Almost surely|probability one]] [[if and only if]] the integral
| |
| | |
| :<math> \int_{ \textbf{R}^d} \min(|f(x)|,1)\Lambda (dx) < \infty. </math>
| |
| | |
| Provided that this integral is finite, then the theorem further asserts that for any [[Complex Number|complex]] value <math>\theta</math> the equation
| |
| | |
| :<math> E(e^{\theta\Sigma})=\textrm{exp} \left(\int_{\textbf{R}^d} [e^{f(x)}-1]\Lambda (dx)\right), </math>
| |
| | |
| holds if the integral on the right-hand side [[convergence (mathematics)|converges]], which is the case for purely [[Imaginary number|imaginary]] <math>\theta</math>. Moreover
| |
| | |
| :<math> \Sigma=\int_{\textbf{R}^d} f(x)\Lambda (dx), </math>
| |
| | |
| and if this integral converges, then
| |
| | |
| :<math> \text{Var}(\Sigma)=\int_{\textbf{R}^d} f(x)^2\Lambda (dx), </math>
| |
| | |
| where <math> \text{Var}(\Sigma)</math> denotes the variance of the random sum <math> \Sigma</math>.
| |
| | |
| From this theorem some expectation results for the [[Poisson point process]] follow directly including its Laplace functional.<ref name="kingman1992poisson"/> {{efn|Kingman<ref name="kingman1992poisson"/> calls it a "characteristic functional" but Daley and Vere-Jones<ref name="daleyPPI2003"/> and others call it a "Laplace functional",<ref name="stoyan1995stochastic"/><ref name="BB1">{{cite doi|10.1561/1300000006}}</ref> reserving the term "characteristic functional" for when <math> \theta</math> is imaginary.}}
| |
| | |
| ==Campbell's theorem: general point process==
| |
| | |
| A related result for a general (not necessarily simple) point process <math>{N}</math> with intensity measure:
| |
| | |
| :<math> \Lambda (B)= E[{N}(B)] , </math>
| |
| | |
| is known as '''Campbell's formula'''<ref name="baddeley2007spatial"/> or '''Campbell's theorem''',<ref name="stoyan1995stochastic"/><ref name="daleyPPII2008">{{cite doi|10.1007/978-0-387-49835-5}}</ref> which gives a method for calculating expectations of sums of [[measurable function]]s <math> f</math> with [[Range (mathematics)|ranges]] on the [[real line]]. More specifically, for a point process <math> {N}</math> and a measurable function <math> f: \textbf{R}^d\rightarrow \textbf{R}</math>, the sum of <math> f</math> over the point process is given by the equation:
| |
| | |
| :<math> E\left[\sum_{x\in {N}}f(x)\right]=\int_{\textbf{R}^d} f(x)\Lambda (dx), </math>
| |
| | |
| where if one side of the equation is finite, then so is the other side.<ref name="baddeley1999crash">A. Baddeley. A crash course in stochastic geometry. ''Stochastic Geometry: Likelihood and Computation Eds OE Barndorff-Nielsen, WS Kendall, HNN van Lieshout (London: Chapman and Hall) pp'', pages 1--35, 1999.
| |
| | |
| </ref> This equation is essentially an application of [[Fubini's theorem]]<ref name="stoyan1995stochastic"/> and coincides with the aforementioned Poisson case, but holds for a much wider class of point processes, simple or not.<ref name="baddeley2007spatial"/> Depending on the integral notation{{efn|As discussed in Chapter 1 of Stoyan, Kendall and Mecke,<ref name="stoyan1995stochastic"/> which applies to all other integrals presented here and elsewhere due to varying integral notation.}}, this integral may also be written as:<ref name="baddeley1999crash"/>
| |
| | |
| :<math> E\left[\sum_{x\in {N}}f(x)\right]=\int_{\textbf{R}^d} fd\Lambda , </math> | |
| | |
| If the intensity measure <math> \Lambda</math> of a point process <math> {N}</math> has a density <math> \lambda(x) </math>, then Campbell's formula becomes:
| |
| | |
| :<math> E\left[\sum_{x\in {N}}f(x)\right]= \int_{\textbf{R}^d} f(x)\lambda(x)dx </math>
| |
| | |
| ===Stationary point process===
| |
| | |
| For a stationary point process <math>{N}</math> with constant intensity <math> \lambda>0</math>, '''Campbell's theorem''' or '''formula''' reduces to a volume integral:
| |
| | |
| :<math> E\left[\sum_{x\in {N}}f(x)\right]=\lambda \int_{\textbf{R}^d} f(x)dx </math>
| |
| | |
| This equation naturally holds for the homogeneous Poisson point processes, which is an example of a [[stationary stochastic process]].<ref name="stoyan1995stochastic"/>
| |
| | |
| ==Applications==
| |
| | |
| ===Laplace functional of the Poisson point process===
| |
| | |
| For a Poisson point process <math> {N}</math> with intensity measure <math> \Lambda</math>, the [[Laplace functional]] is a consequence of Campbell's theorem<ref name="kingman1992poisson"/> and is given by:<ref name="BB1"/>
| |
| | |
| :<math> L_{{N}}=e^{-\int_{\textbf{R}^d}(1-e^{ f(x)})\Lambda(dx)}, </math>
| |
| | |
| which for the homogeneous case is:
| |
| | |
| :<math> L_{{N}}=e^{-\lambda\int_{\textbf{R}^d}(1-e^{ f(x)})dx}. </math>
| |
| | |
| ==Notes==
| |
| {{notelist}}
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| [[Category:Probability theorems]]
| |
Let me inroduce myself, my name is Obdulia Grier. One of his favorite hobbies is reading comics and he'll be starting another thing along in addition to it. Dispatching is when he supports his spouse and children. should i get a legal plan (Our Web Site) currently live in Iowa which will then never advance. If you want to recognize more away my website: http://www.anolivre.com