Nearest neighbor function

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. More precisely, for some point in the point process ${\displaystyle \textstyle {N}}$, 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 ${\displaystyle \textstyle {\textbf {R}}^{d}}$ at, for example, the origin ${\displaystyle \textstyle o}$, the ${\displaystyle \textstyle d}$-dimensional ball ${\displaystyle \textstyle b(o,r)}$ of radius ${\displaystyle \textstyle r}$ centered at the origin o is considered. Given a point of ${\displaystyle \textstyle {N}}$ existing at ${\displaystyle \textstyle o}$, then the nearest neighbor function is defined as:^[4]

${\displaystyle D_{o}(r)=1-P({N}(b(o,r))=1\mid o).}$

where ${\displaystyle \textstyle P({N}(b(o,r))=1\mid o)}$ denotes the conditional probability that there is one point of ${\displaystyle \textstyle {N}}$ located in ${\displaystyle \textstyle b(o,r)}$ given there is a point of ${\displaystyle \textstyle {N}}$ located at ${\displaystyle \textstyle o}$.

The reference point need not be at the origin, and can be located at an arbitrary point ${\displaystyle \textstyle x\in {\textbf {R}}^{d}}$. Given a point of ${\displaystyle \textstyle {N}}$ existing at ${\displaystyle \textstyle x}$, then the nearest neighbor function, is defined as:

${\displaystyle D_{x}(r)=1-P({N}(b(x,r))=1\mid x).}$

Poisson point process

For a Poisson point process ${\displaystyle \textstyle {N}}$ on ${\displaystyle \textstyle {\textbf {R}}^{d}}$ with intensity measure ${\displaystyle \textstyle \Lambda }$ the nearest neighbor function is:

${\displaystyle D_{x}(r)=1-e^{-\Lambda (b(x,r))},}$

which for the homogeneous case becomes

${\displaystyle D_{x}(r)=1-e^{-\lambda |b(x,r)|},}$

where ${\displaystyle \textstyle |b(x,r)|}$ denotes the volume (or more specifically, the Lebesgue measure) of the (hyper) ball of radius ${\displaystyle \textstyle r}$. In the plane ${\displaystyle \textstyle {\textbf {R}}^{2}}$ with the reference point located at the origin, this becomes

${\displaystyle D_{x}(r)=1-e^{-\lambda \pi r^{2}}.}$

Spherical contact distribution function

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]

J-function

The fact that the spherical distribution function H_s(r) and nearest neighbor function D_o(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]

${\displaystyle J(r)={\frac {1-D_{o}(r)}{1-H_{s}(r)}}}$

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.^[4]