Poisson clumping

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

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

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. 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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