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.
[3]
[7] For example, if a point 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. Alternatively, the number of points of located in some
Borel set is often written as:
[2]
[3]
[6]
which reflects a
random measure interpretation for point processes. These two notations are often used in parallel or interchangeably.
[2]
[3]
[6]
n-th power of a point process
For some
integer , the -th power of a point process is defined as:
[2]
where is a collection of not necessarily disjoint Borel sets (in ), which form a -fold
Cartesian product of sets denoted by . The symbol denotes standard
multiplication.
The notation reflects the interpretation of the point process as a random measure.
[3]
The -th power of a point process can be equivalently defined as:
[3]
where
summation is performed over all -
tuples of (possibly repeating) points, and denotes an
indicator function such that is a
Dirac measure. 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.
The -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.
The th moment measure of a point process is equivalently defined
[3] as:
where is any
non-negative
measurable function on and the sum is over -
tuples of points for which repetition is allowed.
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 found or located in the set .
The second moment measure for two Borel sets and is:
which for a single Borel set becomes
where 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 for two Borel sets and , which is given by:
[2]
Example: Poisson point process
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 means:
where is the length, area or volume (or more generally, the
Lebesgue measure) of .
For the Poisson case with measure the second moment measure defined on the product set is:
[5]
which in the homogeneous case reduces to
- ^
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b
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- ^
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f F. Baccelli and B. Błaszczyszyn. Stochastic Geometry and Wireless Networks, Volume I – Theory, volume 3, No 3-4 of Foundations and Trends in Networking. NoW Publishers, 2009.
- ^
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- ^
a
b D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Vol. I. Probability and its Applications (New York). Springer, New York, second edition, 2003.
- ^
a
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c J. Moller and R. P. Waagepetersen. Statistical inference and simulation for spatial point processes. CRC Press, 2003.
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Foundations and Trends in Networking. NoW Publishers, 2009.
-
^ J. F. C. Kingman. Poisson processes, volume 3. Oxford university press, 1992.