Nearest neighbour distribution

In probability and statistics, a nearest neighbor function, nearest neighbor distance distribution, nearest-neighbor distribution function or nearest neighbor distribution 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. More specifically, nearest neighbor functions are defined with respect to some point in the point process as being the probability distribution of the distance from this point to its nearest neighboring point in the same point process, hence they are used to describe the probability of another point existing within some distance of a point. 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. In probability and statistics, a nearest neighbor function, nearest neighbor distance distribution, nearest-neighbor distribution function or nearest neighbor distribution 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. More specifically, nearest neighbor functions are defined with respect to some point in the point process as being the probability distribution of the distance from this point to its nearest neighboring point in the same point process, hence they are used to describe the probability of another point existing within some distance of a point. 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 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 processes are mathematical objects that are defined on some underlying mathematical space. Since these processes are often used to represent collections of points randomly scattered in space, time or both, the underlying space is usually d-dimensional Euclidean space denoted here by R d {displaystyle extstyle { extbf {R}}^{d}} , but they can be defined on more abstract mathematical spaces. Point processes have a number of interpretations, which is reflected by the various types of point process notation. For example, if a point x {displaystyle extstyle x} belongs to or is a member of a point process, denoted by N {displaystyle extstyle {N}} , then this can be written as: and represents the point process being interpreted as a random set. Alternatively, the number of points of N {displaystyle extstyle {N}} located in some Borel set B {displaystyle extstyle B} is often written as: which reflects a random measure interpretation for point processes. These two notations are often used in parallel or interchangeably. 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 N {displaystyle extstyle {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 R d {displaystyle extstyle { extbf {R}}^{d}} at, for example, the origin o {displaystyle extstyle o} , the d {displaystyle extstyle d} -dimensional ball b ( o , r ) {displaystyle extstyle b(o,r)} of radius r {displaystyle extstyle r} centered at the origin o is considered. Given a point of N {displaystyle extstyle {N}} existing at o {displaystyle extstyle o} , then the nearest neighbor function is defined as: where P ( N ( b ( o , r ) ) = 1 ∣ o ) {displaystyle extstyle P({N}(b(o,r))=1mid o)} denotes the conditional probability that there is one point of N {displaystyle extstyle {N}} located in b ( o , r ) {displaystyle extstyle b(o,r)} given there is a point of N {displaystyle extstyle {N}} located at o {displaystyle extstyle o} .