Poisson clumping: Difference between revisions
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[[File:PoissonClumps.svg|thumb|When points are scattered uniformly but randomly over the plane, some clumping inevitably occurs.]] |
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| ⚫ | '''Poisson clumping''', or '''Poisson bursts''',<ref name="Yang">{{cite web|url= |
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| ⚫ | '''Poisson clumping''', or '''Poisson bursts''',<ref name="Yang">{{cite web|url=https://www.thestar.com/news/gta/2010/01/30/numbers_dont_always_tell_the_whole_story.html|title=Numbers don't always tell the whole story|first=Jennifer|last=Yang|work=Toronto Star|date=30 January 2010}}</ref> is a phenomenon where [[Randomness|random]] events may appear to occur in clusters, clumps, or [[Burstiness|bursts]]. |
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==Etymology== |
==Etymology== |
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Poisson clumping is named for |
Poisson clumping is named for 19th-century [[French people|French]] mathematician [[Siméon Denis Poisson]],<ref name="Yang"/> known for his work on [[Integral|definite integrals]], [[electromagnetic theory]], and [[probability theory]], and after whom the [[Poisson distribution]] is also named. |
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==History== |
==History== |
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The [[Poisson process]] provides a description of random independent events occurring with uniform probability through time and/or space. The expected number λ of events in a time interval or area of a given measure is proportional to that measure. The [[Probability distribution|distribution]] of the number of events follows a [[Poisson distribution]] entirely determined by the parameter λ. If λ is small, events are rare, but may nevertheless occur in clumps—referred to as Poisson clumps or bursts—purely by chance.<ref name="Sci Daily">{{cite web|url=https://www.sciencedaily.com/releases/2001/08/010823084028.htm|title=Shark Attacks May Be a "Poisson Burst"|publisher=Science Daily|date=23 August 2011}}</ref> In many cases there is no other cause behind such indefinite groupings besides the nature of randomness following this distribution.<ref>Laurent Hodges, 2 - Common Univariate Distributions, in: Methods in Experimental Physics, v. 28, 1994, p. 35-61</ref> However, obviously not all clumping in nature can be explained by this property — for example earthquakes, because of local seismic activity that causes groups of local aftershocks, in this case [[Weibull distribution]] is proposed.<ref>Min-Hao Wu, J.P. Wang, Kai-Wen Ku; Earthquake, Poisson and Weibull distributions, Physica A: Statistical Mechanics and its Applications, Volume 526, 2019, https://doi.org/10.1016/j.physa.2019.04.237.</ref> |
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{{See also|Poisson distribution}} |
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The Poisson process, developed when Poisson consulted for the [[Prussian Army]] with regard to Prussian Army officers and the frequency with which they were being killed after being kicked by horses,<ref name="Sci Daily">{{cite web|url=http://www.sciencedaily.com/releases/2001/08/010823084028.htm|title=Shark Attacks May Be a "Poisson Burst"|publisher=Science Daily|date=23 August 2011}}</ref> provides a description of random independent events occurring through time. While assumptions imply that the time between successive similar events follows an exponential distribution, events may, in fact, occur in clusters, also referred to as Poisson clumps or Poisson bursts.<ref name="Sci Daily"/> |
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==Applications== |
==Applications== |
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Poisson clumping is used to explain marked increases or decreases in the frequency of an event, such as shark attacks, "coincidences", birthdays, |
Poisson clumping is used to explain marked increases or decreases in the frequency of an event, such as shark attacks, "coincidences", birthdays, heads or tails from coin tosses, and e-mail correspondence.<ref>{{cite web|url=http://www.stat.ualberta.ca/people/schmu/preprints/poisson.pdf|title=Shark attacks and the Poisson approximation|first=Byron|last=Schmuland}}</ref><ref>Anteneodo, C.; Malmgren, R. D.; Chialvo, D. R. (2010.) "Poissonian bursts in e-mail correspondence", ''The European Physical Journal B'', 75(3):389–94.</ref> |
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===Poisson clumping heuristic=== |
===Poisson clumping heuristic=== |
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The poisson clumping heuristic (PCH), published by [[David Aldous]] in 1989,<ref>Aldous, D. (1989.) "Probability Approximations via the Poisson Clumping Heuristic", ''Applied Mathematical Sciences'', 7, Springer</ref> is a model for finding [[Order of approximation|first-order approximations]] over different areas in a large class of [[Stationary process|stationary probability models]]. The probability models have a specific [[Monotonic function|monotonicity property]] with large [[Inclusion–exclusion principle|exclusions]]. The probability that this will achieve a large value is [[Asymptotic analysis|asymptotically small]] and is distributed in a [[Poisson distribution|Poisson fashion]].<ref>Sethares, W. A. and Bucklew, J. A. (1991.) ''Exclusions of Adaptive Algorithms via the Poisson Clumping Heuristic'', University of Wisconsin.</ref> |
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==See also== |
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*[[Burstiness]] |
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*[[Clustering illusion]] |
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*[[Texas sharpshooter fallacy]] |
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==References== |
==References== |
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{{Reflist}} |
{{Reflist}} |
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[[Category:Poisson processes]] |
[[Category:Poisson point processes]] |
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[[Category:Markov processes]] |
[[Category:Markov processes]] |
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Latest revision as of 12:46, 24 October 2024
Poisson clumping, or Poisson bursts,[1] is a phenomenon where random events may appear to occur in clusters, clumps, or bursts.
Etymology
[edit]Poisson clumping is named for 19th-century French mathematician Siméon Denis Poisson,[1] known for his work on definite integrals, electromagnetic theory, and probability theory, and after whom the Poisson distribution is also named.
History
[edit]The Poisson process provides a description of random independent events occurring with uniform probability through time and/or space. The expected number λ of events in a time interval or area of a given measure is proportional to that measure. The distribution of the number of events follows a Poisson distribution entirely determined by the parameter λ. If λ is small, events are rare, but may nevertheless occur in clumps—referred to as Poisson clumps or bursts—purely by chance.[2] In many cases there is no other cause behind such indefinite groupings besides the nature of randomness following this distribution.[3] However, obviously not all clumping in nature can be explained by this property — for example earthquakes, because of local seismic activity that causes groups of local aftershocks, in this case Weibull distribution is proposed.[4]
Applications
[edit]Poisson clumping is used to explain marked increases or decreases in the frequency of an event, such as shark attacks, "coincidences", birthdays, heads or tails from coin tosses, and e-mail correspondence.[5][6]
Poisson clumping heuristic
[edit]The poisson clumping heuristic (PCH), published by David Aldous in 1989,[7] is a model for finding first-order approximations over different areas in a large class of stationary probability models. The probability models have a specific monotonicity property with large exclusions. The probability that this will achieve a large value is asymptotically small and is distributed in a Poisson fashion.[8]
See also
[edit]References
[edit]- ^ a b Yang, Jennifer (30 January 2010). "Numbers don't always tell the whole story". Toronto Star.
- ^ "Shark Attacks May Be a "Poisson Burst"". Science Daily. 23 August 2011.
- ^ Laurent Hodges, 2 - Common Univariate Distributions, in: Methods in Experimental Physics, v. 28, 1994, p. 35-61
- ^ Min-Hao Wu, J.P. Wang, Kai-Wen Ku; Earthquake, Poisson and Weibull distributions, Physica A: Statistical Mechanics and its Applications, Volume 526, 2019, https://doi.org/10.1016/j.physa.2019.04.237.
- ^ Schmuland, Byron. "Shark attacks and the Poisson approximation" (PDF).
- ^ Anteneodo, C.; Malmgren, R. D.; Chialvo, D. R. (2010.) "Poissonian bursts in e-mail correspondence", The European Physical Journal B, 75(3):389–94.
- ^ Aldous, D. (1989.) "Probability Approximations via the Poisson Clumping Heuristic", Applied Mathematical Sciences, 7, Springer
- ^ Sethares, W. A. and Bucklew, J. A. (1991.) Exclusions of Adaptive Algorithms via the Poisson Clumping Heuristic, University of Wisconsin.