In probability and statistics, a 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 (raw) moments of random variables, hence arise often in the study of point processes and related fields.
[1] An example of a moment measure is the first moment measure of a point process, often called mean measure or intensity measure, which gives the expected or average number of points of the point process being located in some region of space.
[2] In other words, if the number of points of a point process located in some region of space is a random variable, then the first moment measure corresponds to the first moment of this random variable.
[3] Moment measures feature prominently in the study of point processes[1][4][5] as well as the related fields of stochastic geometry[3] and spatial statistics[5][6] whose applications are found in numerous scientific and engineering disciplines such as biology, geology, physics, and telecommunications.
[3][4][7] 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 physical space, time or both, the underlying space is usually d-dimensional Euclidean space denoted here by
, but they can be defined on more abstract mathematical spaces.
[1] Point processes have a number of interpretations, which is reflected by the various types of point process notation.
belongs to or is a member of a point process, denoted by
, then this can be written as:[3] and represents the point process being interpreted as a random set.
is often written as:[2][3][6] which reflects a random measure interpretation for point processes.
is a collection of not necessarily disjoint Borel sets (in
-fold Cartesian product of sets denoted by
reflects the interpretation of the point process
can be equivalently defined as:[3] where summation is performed over all
This definition can be contrasted with the definition of the n-factorial power of a point process for which each n-tuples consists of n distinct points.
-th moment measure is defined as: where the E denotes the expectation (operator) of the point process
In other words, the n-th moment measure is the expectation of the n-th power of some point process.
th moment measure of a point process
For some Borel set B, the first moment of a point process N is: where
is known, among other terms, as the intensity measure[3] or mean measure,[8] and is interpreted as the expected or average number of points of
The second moment measure for two Borel sets
denotes the variance of the random variable
The previous variance term alludes to how moments measures, like moments of random variables, can be used to calculate quantities like the variance of point processes.
A further example is the covariance of a point process
, which is given by:[2] For a general Poisson point process with intensity measure
the first moment measure is:[2] which for a homogeneous Poisson point process with constant intensity
is the length, area or volume (or more generally, the Lebesgue measure) of
the second moment measure defined on the product set