Spherical contact distribution function

The spherical contact distribution function is defined as:

${\displaystyle H_{s}(r)=1-P({N}(b(o,r))=0).}$

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 ${\displaystyle \textstyle {\textbf {R}}^{d}}$. For some Borel set ${\displaystyle \textstyle B}$ with positive volume (or more specifically, Lebesgue measure), the contact distribution function (with respect to ${\displaystyle \textstyle B}$) for ${\displaystyle \textstyle r\geq 0}$ is defined by the equation:[1]

${\displaystyle H_{B}(r)=P({N}(rB)=0).}$

For a Poisson point process ${\displaystyle \textstyle {N}}$ on ${\displaystyle \textstyle {\textbf {R}}^{d}}$ with intensity measure ${\displaystyle \textstyle \Lambda }$ this becomes

${\displaystyle H_{s}(r)=1-e^{-\Lambda (b(o,r))},}$

which for the homogeneous case becomes

${\displaystyle H_{s}(r)=1-e^{-\lambda |b(o,r)|},}$

where ${\displaystyle \textstyle |b(o,r)|}$ denotes the volume (or more specifically, the Lebesgue measure) of the ball of radius ${\displaystyle \textstyle r}$. In the plane ${\displaystyle \textstyle {\textbf {R}}^{2}}$, this expression simplifies to

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

In general, the spherical contact distribution function and the corresponding nearest neighbour function are not equal. However, these two functions are identical for Poisson point processes.[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.[2]

The fact that the spherical distribution function H_s(r) and nearest neighbour 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:[1]

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

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,[8] but such counterexamples are viewed as somewhat 'artificial' by some and exist for other statistical tests.[9]

More generally, J-function serves as one way (others include using factorial moment measures[2]) to measure the interaction between points in a point process.[1]