Moment measure explained

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 process notation

See main article: 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 physical space, time or both, the underlying space is usually d-dimensional Euclidean space denoted here by

style bf{R} , 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

style x belongs to or is a member of a point process, denoted by style {N} , then this can be written as:^[3] style x\in {N},

and represents the point process being interpreted as a random set. Alternatively, the number of points of

style {N} located in some Borel set style B is often written as:^{[2] [3] [6]} style {N}(B),

which reflects a random measure interpretation for point processes. These two notations are often used in parallel or interchangeably.^{[2] [3] [6]}

Definitions

n-th power of a point process

style n =1 , 2 , ... , the style n -th power of a point process style {N} is defined as:^[2] {N} ^{n(B} _{1 x … x} B _{n)=} \prod ^{n{N}(B} _{i)}

where

style B _{1,...,B} _{n} is a collection of not necessarily disjoint Borel sets (in style bf{R} ), which form a style n -fold Cartesian product of sets denoted by B _{1 x … x} B _{n} . The symbol style \Pi denotes standard multiplication.

The notation

style {N}(B _{i)} reflects the interpretation of the point process style {N} as a random measure.^[3]

The

style n -th power of a point process style {N} can be equivalently defined as:^[3] {N} ^{n} (B _{1 x … x} B _{n)=} \sum _{(x _{1,...,} x _{n)\in} {N} } \prod ^{n} _{i =1} 1 _{B _{i}} (x _{i)}

where summation is performed over all

style n -tuples of (possibly repeating) points, and style 1 denotes an indicator function such that style 1 _{B _{1}} 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.

n-th moment measure

The

style n -th moment measure is defined as:M ^{n(B} _{1 x \ldots x} B _{n)=E} [{N} ^{n(B} _{1 x \ldots x} B _{n)],}

where the E denotes the expectation (operator) of the point process

style {N} . In other words, the n-th moment measure is the expectation of the n-th power of some point process.

The

style n th moment measure of a point process style {N} is equivalently defined^[3] as:\int ^{n d} _{bf{R} }f(x_1,\dots,x_n) M^n(dx_1,\dots,dx_n)=E \left[\sum_{(x_1,\dots,x_n)\in {N} } f(x_1,\dots,x_n) \right],

where

style f is any non-negative measurable function on style bf{R} ^{n} and the sum is over style n -tuples of points for which repetition is allowed.

First moment measure

For some Borel set B, the first moment of a point process N is:

M ^{1(B)=E} [{N}(B)],

where

style M ^{1} 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 style {N} found or located in the set style B .

Second moment measure

The second moment measure for two Borel sets

style A and style B is:M ^{2(A x} B) = E [{N}(A){N}(B)],

which for a single Borel set

style B becomesM ^{2(B x} B) = (E [{N}(B)]) ^{2+Var[{N}(B)],}

where

style Var [{N}(B)] denotes the variance of the random variable style {N}(B) .

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

style {N} for two Borel sets style A and style B , which is given by:^[2] Cov [{N}(A),{N}(B)] = M ^{2(A x} B) - M ^{1(A)M} ^{1(B)}

Example: Poisson point process

For a general Poisson point process with intensity measure

style Λ the first moment measure is:^[2] M ^{1(B)=Λ(B),}

which for a homogeneous Poisson point process with constant intensity

style λ means:M ^{1(B)=λ|B|,}

where

style |B| is the length, area or volume (or more generally, the Lebesgue measure) of style B .

For the Poisson case with measure

style Λ the second moment measure defined on the product set (B x B) is:^[5] M ^{2(B} x B) =Λ (B) +Λ (B) ^{2.}

which in the homogeneous case reduces to

M ^{2(B} x B) =λ |B| + ( λ |B|) ^{2.}

See also

• Factorial moment
• Factorial moment measure
• Moment

References

1. D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Vol. . Probability and its Applications (New York). Springer, New York, second edition, 2008.
2. 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.
3. D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. Stochastic geometry and its applications, volume 2. Wiley Chichester, 1995.
4. 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.
5. A. Baddeley, I. Bárány, and R. Schneider. Spatial point processes and their applications. Stochastic Geometry: Lectures given at the CIME Summer School held in Martina Franca, Italy, September 13–18, 2004, pages 1-75, 2007.
6. J. Moller and R. P. Waagepetersen. Statistical inference and simulation for spatial point processes. CRC Press, 2003.
7. F. Baccelli and B. Błaszczyszyn. Stochastic Geometry and Wireless Networks, Volume II – Applications, volume 4, No 1-2 of Foundations and Trends in Networking. NoW Publishers, 2009.
8. J. F. C. Kingman. Poisson processes, volume 3. Oxford university press, 1992.