Factorial moment measure

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 Rd, but they can be defined on more abstract mathematical spaces. For example, if a point \textstyle x belongs to or is a member of a point process, denoted by N, then this can be written as: ==Definitions==

n-th factorial power of a point process For some positive integer \textstyle n=1,2,\ldots, the \textstyle n-th factorial power of a point process \textstyle {N} on \textstyle \textbf{R}^d is defined as: and is interpreted as the expected number of points of \textstyle {N} found or located in the set \textstyle B Second factorial moment measure The second factorial moment measure for two Borel sets \textstyle A and \textstyle B is: : M^{(2)}(A\times B)=M^2(A\times B)-M^1(A\cap B). ==Name explanation==

For some Borel set \textstyle B, the namesake of this measure is revealed when the \textstyle n\,th factorial moment measure reduces to: : M^{(n)}(B\times\cdots\times B)=E [{N}(B)({N}(B)-1)\cdots ({N}(B)-n+1)], which is the \textstyle n\,-th factorial moment of the random variable \textstyle {N}(B). ==Factorial moment density==

If a factorial moment measure is absolutely continuous, then it has a density (or more precisely, a Radon–Nikodym derivative or density) with respect to the Lebesgue measure and this density is known as the factorial moment density or product density, joint intensity, correlation function, or multivariate frequency spectrum. Denoting the \textstyle n-th factorial moment density by \textstyle \mu^{(n)}(x_1,\ldots,x_n), it is defined in respect to the equation: : M^{(n)}(B_1\times\ldots\times B_n)=\int_{B_1}\cdots\int_{B_n}\mu^{(n)}(x_1,\ldots,x_n) dx_1\cdots dx_n. Furthermore, this means the following expression : E \left[ \sum_{(x_1\neq\cdots\neq x_n)\in {N} } f(x_1,\ldots,x_n) \right]= \int_{\textbf{R}^{n d}} f(x_1,\ldots,x_n) \mu^{(n)}(x_1,\ldots,x_n) dx_1\cdots dx_n, where \textstyle f is any non-negative bounded measurable function defined on \textstyle \textbf{R}^{ n}. ==Pair correlation function==

In spatial statistics and stochastic geometry, to measure the statistical correlation relationship between points of a point process, the pair correlation function of a point process {N} is defined as: : \rho(x_1,x_2)=\frac{\mu^{(2)}(x_1,x_2)}{\mu^{(1)}(x_1) \mu^{(1)}(x_2) }, where the points x_1,x_2\in R^d . In general, \rho(x_1,x_2)\geq 0 whereas \rho(x_1,x_2)=1 corresponds to no correlation (between points) in the typical statistical sense. ==Examples==

Poisson point process For a general Poisson point process with intensity measure \textstyle \Lambda the \textstyle n-th factorial moment measure is given by the expression: : M^{(n)}(B_1\times\cdots\times B_n)=\prod_{i=1}^n[\Lambda(B_i)], where \textstyle \Lambda is the intensity measure or first moment measure of \textstyle {N}, which for some Borel set \textstyle B is given by: : \Lambda(B)=M^1(B)=E[{N}(B)]. For a homogeneous Poisson point process the \textstyle n-th factorial moment measure is simply: : M^{(n)}(B_1\times\cdots\times B_n)=\lambda^n \prod_{i=1}^n |B_i|, where \textstyle |B_i| is the length, area, or volume (or more generally, the Lebesgue measure) of \textstyle B_i. Furthermore, the \textstyle n-th factorial moment density is: : \mu^{(n)}(x_1,\ldots,x_n)=\lambda^n. The pair-correlation function of the homogeneous Poisson point process is simply : \rho(x_1,x_2)=1, which reflects the lack of interaction between points of this point process. ==Factorial moment expansion==

The expectations of general functionals of simple point processes, provided some certain mathematical conditions, have (possibly infinite) expansions or series consisting of the corresponding factorial moment measures. In comparison to the Taylor series, which consists of a series of derivatives of some function, the nth factorial moment measure plays the roll as that of the n th derivative the Taylor series. In other words, given a general functional f of some simple point process, then this Taylor-like theorem for non-Poisson point processes means an expansion exists for the expectation of the function E, provided some mathematical condition is satisfied, which ensures convergence of the expansion. ==See also==