When points are scattered uniformly but randomly over the plane, some clumping inevitably occurs.

Poisson clumping, or Poisson bursts,[1] is a phenomenon where random events may appear to occur in clusters, clumps, or bursts.

Etymology

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

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

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

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

References

  1. 1 2 Yang, Jennifer (30 January 2010). "Numbers don't always tell the whole story". Toronto Star.
  2. ↑ "Shark Attacks May Be a "Poisson Burst"". Science Daily. 23 August 2011.
  3. ↑ Laurent Hodges, 2 - Common Univariate Distributions, in: Methods in Experimental Physics, v. 28, 1994, p. 35-61
  4. ↑ 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.
  5. ↑ Schmuland, Byron. "Shark attacks and the Poisson approximation" (PDF).
  6. ↑ Anteneodo, C.; Malmgren, R. D.; Chialvo, D. R. (2010.) "Poissonian bursts in e-mail correspondence", The European Physical Journal B, 75(3):389–94.
  7. ↑ Aldous, D. (1989.) "Probability Approximations via the Poisson Clumping Heuristic", Applied Mathematical Sciences, 7, Springer
  8. ↑ Sethares, W. A. and Bucklew, J. A. (1991.) Exclusions of Adaptive Algorithms via the Poisson Clumping Heuristic, University of Wisconsin.
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