Concept
In probability and statistics, a factorial moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both. Moment measures generalize the idea of factorial moments, which are useful for studying non-negative integer-valued random variables. The first factorial moment measure of a point process coincides with its first moment measure or intensity measure, which gives the expected or average number of points of the point process located in some region of space. In general, if the number of points in some region is considered as a random variable, then the moment factorial measure of this region is the factorial moment of this random variable. Factorial moment measures completely characterize a wide class of point processes, which means they can be used to uniquely identify a point process. If a factorial moment measure is absolutely continuous, then with respect to the Lebesgue measure it is said to have a density (which is a generalized form of a derivative), and this density is known by a number of names such as factorial moment density and product density, as well as coincidence density, joint intensity correlation function or multivariate frequency spectrum The first and second factorial moment densities of a point process are used in the definition of the pair correlation function, which gives a way to statistically quantify the strength of interaction or correlation between points of a point process. Factorial moment measures serve as useful tools 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 notation Point processes are mathematical objects that are defined on some underlying mathematical space.