Spherical contact distribution function

In probability and statistics, a spherical contact distribution function, first contact distribution function, or empty space function is a mathematical function that is defined in relation to random mathematical objects known as point processes, which are used as mathematical models of physical phenomena representable as randomly positioned points in time or space. 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. These functions can be contrasted with nearest neighbour functions, which are defined in relation to some point in the point process as being the probability distribution of the distance from that point to its nearest neighbouring point in the same point process.

The spherical contact function is also referred to as the contact distribution function, but some authors define the contact distribution function in relation to some more general set, and not simply a sphere as in the case of the spherical contact distribution function.

Spherical contact distribution functions are used in the study of point processes as well as the related fields of stochastic geometry and spatial statistics, which are applied in various scientific and engineering disciplines such as biology, geology, physics, and telecommunications .

Point process

Main article: Point process

Point processes are mathematical objects that seek to represent collections of points randomly scattered on some underlying mathematical space. They have a number of interpretations, which is reflected by the various types of point process notation. For example, if a point $\textstyle x$ belongs to or is a member of a point process, denoted by $\textstyle \Phi$, then this can be written as $\textstyle x\in \Phi$, and represents the point process as a random set. Alternatively, the number of points of $\textstyle \Phi$ located in some Borel set $\textstyle B$ is often written as $\textstyle \Phi (B)$, which reflects a random measure interpretation for point processes; see Point process notation.

A point process needs to be defined on an underlying mathematical space. Often this space is d-dimensional Euclidean space denoted here by $\textstyle {\textbf {R}}^{d}$, although point processes can be defined on more abstract mathematical spaces.

Contact distribution function

For some Borel set $\textstyle B$ with positive volume (or more specifically, Lebesgue measure), the contact distribution function (with respect to $\textstyle B$) for $\textstyle r\geq 0$ is defined by the equation :

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

Spherical contact distribution function

For $\textstyle B$, the symmetrical case of unit (hyper-)sphere $\textstyle b(o,1)$ is of importance, where $\textstyle o$ denotes that the sphere is centered at the origin and $\textstyle 1$ is its radius. This leads to the spherical contact distribution or first contact distribution function:

$H_{s}(r)=P(\Phi (b(o,r))=0).$

Poisson point process case

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

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

which for the homogeneous case becomes

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

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

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

Relation to the nearest neighbour function

For Poisson point processes, the spherical contact distribution function coincides with the corresponding nearest neighbour function, defined in relation to a point being located at the origin $\textstyle o$. In general, these two functions not coincide for all point processes. In fact, this characteristic is due to a defining property of Poisson processes and their Palm distributions, which forms part of the result known as the Slivnyak-Mecke or Slivnyak's theorem.

$\textstyle J$-function

The fact that the spherical distribution function $\textstyle H_{s}(r)$ coincides and nearest neighbour function $\textstyle D_{o}(r)$ are identical can be used to statistically test if a point process data appears to be that of a Poisson process. For example, in spatial statistics the $\textstyle J$-function is defined for all $\textstyle r\geq 0$ as:

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

For a Poisson point process, the $\textstyle J$ funtion is simply $\textstyle J(r)=0$, hence why it is used to test whether data behaves as though it were from a Poisson process.

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