Poisson clumping

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

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

Etymology[edit]

Poisson clumping is named for the 19th-century French mathematician Siméon Denis Poisson,[1] who is known for his work on definite integrals, electromagnetic theory, and probability theory and is the namesake of the Poisson distribution.

History[edit]

The Poisson process provides a description of random independent events occurring with uniform probability through time or space (or both). 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 purely by chance they may, nevertheless, occasionally occur in clusters, also referred to as Poisson clumps or Poisson bursts.[2]

Applications[edit]

Poisson clumping is used to explain marked increases or decreases in the frequency of an event, such as shark attacks, "coincidences", birthdays, or heads or tails from coin tosses, and e-mail correspondence.[3][4]

Poisson clumping heuristic[edit]

Poisson clumping heuristic (PCH), published by David Aldous in 1989,[5] is a model for finding first-order approximations over different areas in a large class of stationary probability models that have a specific monotonicity property for large exclusions. The probability that such a process will achieve a large value is asymptotically small and is distributed in a Poisson fashion.[6]

See also[edit]

References[edit]

  1. ^ a b 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. ^ Schmuland, Byron. "Shark attacks and the Poisson approximation" (PDF).
  4. ^ Anteneodo, C.; Malmgren, R. D.; Chialvo, D. R. (2010.) "Poissonian bursts in e-mail correspondence", The European Physical Journal B, 75(3):389–94.
  5. ^ Aldous, D. (1989.) "Probability Approximations via the Poisson Clumping Heuristic", Applied Mathematical Sciences, 7, Springer
  6. ^ Sethares, W. A. and Bucklew, J. A. (1991.) Exclusions of Adaptive Algorithms via the Poisson Clumping Heuristic, University of Wisconsin.