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.
[1] The first factorial moment measure of a point process coincides with its first moment measure or intensity measure,[2] 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.
[3] 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,[1] joint intensity[4] , correlation function or multivariate frequency spectrum[5] 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.
[6] Factorial moment measures serve as useful tools in the study of point processes[1][6][7] as well as the related fields of stochastic geometry[3] and spatial statistics,[6][8] which are applied in various scientific and engineering disciplines such as biology, geology, physics, and telecommunications.
belongs to or is a member of a point process, denoted by N, then this can be written as:[3] and represents the point process being interpreted as a random set.
Alternatively, the number of points of N located in some Borel set B is often written as:[2][3][8] which reflects a random measure interpretation for point processes.
-th factorial power of a point process
is a collection of not necessarily disjoint Borel sets in
-fold Cartesian product of sets denoted by: The symbol
-tuples of distinct points, including permutations, which can be contrasted with the definition of the n-th power of a point process.
denotes multiplication while the existence of various point process notation means that the n-th factorial power of a point process is sometimes defined using other notation.
[2] The n th factorial moment measure or n th order factorial moment measure is defined as: where the E denotes the expectation (operator) of the point process N. In other words, the n-th factorial moment measure is the expectation of the n th factorial power of some point process.
The n th factorial moment measure of a point process N is equivalently defined[3] by: where
tuples of distinct points, including permutations.
Consequently, the factorial moment measure is defined such that there are no points repeating in the product set, as opposed to the moment measure.
The second factorial moment measure for two Borel sets
, it is defined in respect to the equation:[3] Furthermore, this means the following expression where
is any non-negative bounded measurable function defined on
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
corresponds to no correlation (between points) in the typical statistical sense.
[6] For a general Poisson point process with intensity measure
is given by: For a homogeneous Poisson point process the
is the length, area, or volume (or more generally, the Lebesgue measure) of
-th factorial moment density is:[3] The pair-correlation function of the homogeneous Poisson point process is simply which reflects the lack of interaction between points of this point process.
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.
[11][12] 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.