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Point Process

A point process is a probabilistic model for random scatterings of points on some space X often assumed to be a subset of R^d for some d. Oftentimes, point processes describe the occurrence over time of random events in which the occurrences are revealed one-by-one as time evolves. For a simple point process with finitely many points in every bounded time interval, a nonempty realization on such an interval has occurrence times

{tau_1,tau_2,...,tau_n},tau_1<tau_2<...<tau_n.

Poisson processes are regarded as archetypal examples of point processes (Daley and Vere-Jones 2003, p. 19).

A locally finite simple temporal point process on [0,infty) with no point at 0 can equivalently be represented by its associated counting process, which counts the points in (0,t] for t>=0. Point processes are also known as random scatters.

See also

Counting Measure, Counting Process, Marked Point Process, Poisson Process, Self-Correcting Point Process, Self-Exciting Point Process, Simple Point Process, Sine Process, Spatial Point Process, Spatial-Temporal Point Process, Temporal Point Process

Portions of this entry contributed by Christopher Stover

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References

Brillinger, D. R.; Guttorp, P. M.; and Schoenberg, F. P. "Point Processes, Temporal." Encyclopedia of Environmetrics 3, 1577-1581, 2002.Daley, D. J. and Vere-Jones, D. An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods, 2nd ed. New York: Springer, 2003.Daley, D. J. and Vere-Jones, D. An Introduction to the Theory of Point Processes Volume II: General Theory and Structure, 2nd ed. New York: Springer, 2007.Jacobsen, M. Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Boston, MA: Birkhäuser, 2006.

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Point Process

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Point Process." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PointProcess.html

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