In probability and statistics, a spherical contact distribution function, first contact distribution function,[1] or empty space function[2] is a mathematical function that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both.
[1][3] More specifically, a spherical contact distribution function is defined as probability distribution of the radius of a sphere when it first encounters or makes contact with a point in a point process.
Spherical contact distribution functions are used in the study of point processes[2][3][4] as well as the related fields of stochastic geometry[1] and spatial statistics,[2][5] which are applied in various scientific and engineering disciplines such as biology, geology, physics, and telecommunications.
, then this can be written as:[1] and represents the point process being interpreted as a random set.
is often written as:[1][5][6] which reflects a random measure interpretation for point processes.
[1][5][6] The spherical contact distribution function is defined as: where b(o,r) is a ball with radius r centered at the origin o.
In other words, spherical contact distribution function is the probability there are no points from the point process located in a hyper-sphere of radius r. The spherical contact distribution function can be generalized for sets other than the (hyper-)sphere in
with positive volume (or more specifically, Lebesgue measure), the contact distribution function (with respect to
denotes the volume (or more specifically, the Lebesgue measure) of the ball of radius
[1] In fact, this characteristic is due to a unique property of Poisson processes and their Palm distributions, which forms part of the result known as the Slivnyak-Mecke[6] or Slivnyak's theorem.
For example, in spatial statistics the J-function is defined for all r ≥ 0 as:[1] For a Poisson point process, the J function is simply J(r)=1, hence why it is used as a non-parametric test for whether data behaves as though it were from a Poisson process.
It is, however, thought possible to construct non-Poisson point processes for which J(r)=1,[8] but such counterexamples are viewed as somewhat 'artificial' by some and exist for other statistical tests.