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Temporal Point Process


A temporal point process is a random process whose realizations consist of the times {tau_j}_(j in J) of isolated events.

Note that in some literature, the values tau_j are assumed to be arbitrary real numbers while the index set J is assumed to be the set Z of integers (Schoenberg 2002); on the other hand, some authors view temporal point processes as binary events so that tau_j takes values in a two-element set for each j, and further assume that the index set J is some finite set of points (Liam 2013). The prior perspective corresponds to viewing temporal point processes as how long events occur where the events themselves are spaced according to a discrete set of time parameters; the latter view corresponds to viewing temporal point processes as indications of whether or not a finite number of events has occurred.

The behavior of a simple temporal point process N is typically modeled by specifying its conditional intensity lambda=lambda(t). Indeed, a number of specific examples of temporal point processes are defined merely by specifying their conditional intensity functions, e.g., the Poisson and Hawkes processes.


See also

Hawkes Process, Marked Point Process, Point Process, Poisson Process, Self-Correcting Point Process, Self-Exciting Point Process, Simple Point Process, Spatial Point Process, Spatial-Temporal Point Process

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 Environments 3, 1577-1581, 2002.Paninski, L. "Chapter 2: Introduction to Point Processes." 2013. http://www.stat.columbia.edu/~liam/teaching/neurostat-fall13/uri-eden-point-process-notes.pdf.Schoenberg, F. P. "Introduction to Point Processes."

Cite this as:

Stover, Christopher. "Temporal Point Process." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TemporalPointProcess.html

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