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Intensity Function

There are at least two distinct notions of an intensity function related to the theory of point processes.

In some literature, the intensity lambda of a point process N is defined to be the quantity

lambda=lim_(hv0)(Pr{N(0,h]>0})/h
(1)

provided it exists. Here, Pr denotes probability. In particular, it makes sense to talk about point processes having infinite intensity, though when finite, lambda allows N to be rewritten so that

Pr{N(x,x+h]>0}=lambdah+o(h)
(2)

as hv0 where here, o(h) denotes little-O notation (Daley and Vere-Jones 2007).

Other authors define the function rho to be an intensity function of a point process N provided that rho is a density of the intensity measure mu associated to N relative to Lebesgue measure, i.e.,if for all Borel sets B in R^d,

mu(B)=int_Brho(xi)dxi
(3)

where xi denotes Lebesgue measure (Pawlas 2008).

See also

Conditional Intensity Function, Intensity Measure, Point Process, Probability

This entry contributed by Christopher Stover

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References

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.Pawlas, Z. "Spatial Modeling and Spatial Statistics." Course Notes. Autumn 2008. http://www.math.ku.dk/~pawlas/rumlig.pdf.

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Intensity Function

Cite this as:

Stover, Christopher. "Intensity Function." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/IntensityFunction.html

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