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| In [[probability theory]], an '''inhomogeneous Poisson process''' (or '''non-homogeneous Poisson process''') is a [[Poisson process]] with rate parameter <math>\lambda (t)</math> such that the rate parameter of the process is a function of time.<ref name="ross" /> Inhomogeneous Poisson process have been shown to describe numerous random phenomena<ref>{{cite web|url=http://www.math.wm.edu/~leemis/icrsa03.pdf|publisher=William and Mary Mathematics Department|date=May 2003|title=Estimating and Simulating Nonhomogeneous Poisson Processes|first=Larry|last=Leemis|accessdate=Sep 26, 2011}}</ref> including [[cyclone]] prediction,<ref>{{cite journal|title=Modeling and simulation of a nonhomogeneous poisson process having cyclic behavior|doi=10.1080/03610919108812984|first1=Sanghoon|last1=Lee|first2=James R.|last2=Wilson|first3=Melba M.|last3=Crawford|pages=777–809|journal=Communications in Statistics - Simulation and Computation|volume=20|issue=2-3|year=1991|url=http://www.ise.ncsu.edu/jwilson/files/lee91.pdf}}</ref> arrival times of calls to a call centre in a hospital laboratory<ref>{{cite journal|title=Modeling Time-Dependent Arrivals to Service Systems: A Case in Using a Piecewise-Polynomial Rate Function in a Nonhomogeneous Poisson Process|journal=[[Management Science (journal)|Management Science]]|jstor=2631999|publisher=INFORMS|volume=34|issue=11|date=November 1988|first1=Edward P. C.|last1=Kao|first2=Sheng-Lin|last2=Chang|pages=1367–1379|doi=10.1287/mnsc.34.11.1367}}</ref> and call centre,<ref>{{cite doi|10.1198/016214506000001455}}</ref> arrival times of aircraft to airspace around an airport<ref>{{cite journal|title=Nonstationary Queuing Probabilities for Landing Congestion of Aircraft|first1=Herbert P.|last1=Galliher|first2=R. Clyde|last2=Wheeler|journal=[[Operations Research: A Journal of the Institute for Operations Research and the Management Sciences|Operations Research]]|volume=6|issue=2|date=March–April 1958|pages=264–275|jstor=167618|doi=10.1287/opre.6.2.264}}</ref> and database [[transaction time]]s.<ref>{{cite journal|title=Statistical Analysis of Non-stationary Series of Events in a Data Base System|first1=P. A. W.|last1=Lewis|first2=G. S.|last2=Shedler|doi=10.1147/rd.205.0465|journal=IBM Journal of Research and Development|volume=20|issue=5|date=September 1976|id = {{citeseerx|10.1.1.84.9018}} }}</ref>
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| The [[Cox process]] is an extension of this model where ''λ''(''t'') itself can be a stochastic process.
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| ==Definition==
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| Write <math>N(t)</math> for the number of events by time <math>t</math>. A [[stochastic process]] is an inhomogeneous Poisson process for some small value <math>h</math> if:<ref name="ross">{{cite book|title=Simulation|first=Sheldon M.|last=Ross|publisher=Academic Press|year=2006|isbn=0-12-598063-9|page=32}}</ref><ref>{{cite book|title=
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| Stochastic point processes and their applications|last=Srinivasan|year=1974|chapter=Chapter 2|isbn=0-85264-223-7}}</ref>
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| # <math>N(0)=0</math>
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| # Non-overlapping increments are independent
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| # <math>P(N(t+h)-N(t)=1) = \lambda(t) h + o(h)</math>
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| # <math>P(N(t+h)-N(t)>1) = o(h)</math>
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| for all ''t'' and where, in [[little o notation]], <math>\scriptstyle \frac {o(h)}{h} \rightarrow 0\; \mathrm{as}\, h\, \rightarrow 0</math>.
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| In the case of point processes with refractoriness (e.g., neural spike trains) a stronger version of property 4 applies:<ref>{{cite journal|author = L. Citi, D. Ba, E.N. Brown, and R. Barbieri|title = Likelihood methods for point processes with refractoriness|journal=Neural Computation|year=2014|doi=10.1162/NECO_a_00548|url=http://dx.doi.org/10.1162/NECO_a_00548}}</ref> <math>P(N(t+h)-N(t)>1) = o(h^2)</math>.
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| ==Properties==
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| Write ''N''(''t'') for the number of events by time ''t'' and <math>\scriptstyle m(t) = \int_0^{t} \lambda (u)\text{d}u</math> for the mean. Then ''N''(''t'') has a [[Poisson distribution]] with parameter ''m''(''t''), that is for ''k'' = 0, 1, 2, 3….<ref>{{cite doi|10.1007/1-84628-295-0_5}}</ref>
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| :<math>\mathbb P(N(t)=k) = \frac{m(t)^k}{k!}e^{-m(t)}.</math>
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| ==Fitting==
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| Traffic on the [[AT&T]] long distance network was shown to be described by a inhomogeneous Poisson process with piecewise linear rate function.<ref>{{cite doi|10.1007/BF02112523}}</ref> Ordinary least squares, iterative weighted least squares and maximum likelihood methods were evaluated and maximum likelihood shown to perform best overall for the data.
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| ==Simulation==
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| To simulate a inhomogeneous Poisson process with intensity function ''λ''(''t''), choose a sufficiently large ''λ'' so that ''λ''(''t'') = ''λ p''(''t'') and simulate a Poisson process with rate parameter ''λ''. Accept an event from the Poisson simulation at time ''t'' with probability ''p''(''t'').<ref name="ross" /><ref>{{cite doi|10.1002/nav.3800260304}}</ref> For a [[log-linear]] rate function a more efficient method was published by Lewis and Shedler in 1975.<ref>{{cite doi|10.1093/biomet/63.3.501}}</ref>
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| ==Notes==
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| {{reflist}}
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| {{Stochastic processes}}
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| {{DEFAULTSORT:Inhomogeneous Poisson Process}}
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| [[Category:Poisson processes]]
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