In probability and statistics, a nearest neighbor function, nearest neighbor distance distribution,[1] nearest-neighbor distribution function[2] or nearest neighbor distribution[3] is a mathematical function that is defined in relation to mathematical objects known as point processes, which are often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both.
A nearest neighbor function can be contrasted with a spherical contact distribution function, which is not defined in reference to some initial point but rather as the probability distribution of the radius of a sphere when it first encounters or makes contact with a point of a point process.
Nearest neighbor function are used in the study of point processes[1][5][6] as well as the related fields of stochastic geometry[4] and spatial statistics,[1][7] which are applied in various scientific and engineering disciplines such as biology, geology, physics, and telecommunications.
belongs to or is a member of a point process, denoted by
, then this can be written as:[4] and represents the point process being interpreted as a random set.
is often written as:[8][4][7] which reflects a random measure interpretation for point processes.
[4][7][8] The nearest neighbor function, as opposed to the spherical contact distribution function, is defined in relation to some point of a point process already existing in some region of space.
, the nearest neighbor function is the probability distribution of the distance from that point to the nearest or closest neighboring point.
To define this function for a point located in
, then the nearest neighbor function is defined as:[4] where
, then the nearest neighbor function, is defined as: Mathematical expressions of the nearest neighbor distribution only exist for a few point processes.
the nearest neighbor function is: which for the homogeneous case becomes where
denotes the volume (or more specifically, the Lebesgue measure) of the (hyper) ball of radius
with the reference point located at the origin, this becomes In general, the spherical contact distribution function and the corresponding nearest neighbor function are not equal.
However, these two functions are identical for Poisson point processes.
[4] 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[8] or Slivnyak's theorem.
[1] The fact that the spherical distribution function Hs(r) and nearest neighbor function Do(r) are identical for the Poisson point process can be used to statistically test if point process data appears to be that of a Poisson point process.
For example, in spatial statistics the J-function is defined for all r ≥ 0 as:[4] 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,[10] but such counterexamples are viewed as somewhat 'artificial' by some and exist for other statistical tests.
[11] More generally, J-function serves as one way (others include using factorial moment measures[1]) to measure the interaction between points in a point process.