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.
Contents
- Point process notation
- n th factorial power of a point process
- n th factorial moment measure
- First factorial moment measure
- Second factorial moment measure
- Name explanation
- Factorial moment density
- Pair correlation function
- Poisson point process
- Factorial moment expansion
- References
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 including factorial moment density, product density, 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. 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.
Point processes have a number of interpretations, which is reflected by the various types of point process notation. For example, if a point
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:
which reflects a random measure interpretation for point processes. These two notations are often used in parallel or interchangeably.
n th factorial power of a point process
For some positive integer
where
The symbol
n th factorial moment measure
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 by:
where f is any non-negative measurable function on Rnd, and the above summation is performed over all n 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.
First factorial moment measure
The first factorial moment measure
where
Second factorial moment measure
The second factorial moment measure for two Borel sets
Name explanation
For some Borel set
which is the
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
Furthermore, this means the following expression
where
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
where the points
Poisson point process
For a general Poisson point process with intensity measure
where
For a homogeneous Poisson point process the
where
The pair-correlation function of the homogeneous Poisson point process is simply
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.