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In [[probability]] and [[statistics]], a '''moment measure''' is a [[mathematical]] quantity, [[Function (mathematics)|function]] or, more precisely, [[Measure (mathematics)|measure]]  that is defined in relation to [[mathematical objects]] known as [[point process]]es, which are types of [[stochastic processes]] often used as [[mathematical model]]s of physical phenomena representable as [[random]]ly positioned [[Point (geometry)|points]] in [[time]],[[space]] or both. Moment measures generalize the idea of (raw) [[Moment (mathematics)|moments]] of [[random variable]]s, hence arise often in the study of point processes and related fields.<ref name="daleyPPII2008">D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. {II''}. Probability and its Applications (New York). Springer, New York, second edition, 2008.

</ref>

An example of a moment measure is the ''first moment measure'' or ''intensity measure'' of a point process, which gives the [[Expected value|expected]] or average number of points of the point process being located in some region of space<ref name="BB1">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.</ref>.  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.<ref name="stoyan1995stochastic">D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.</ref>

Moment measures feature prominently in the study of point processes<ref name="daleyPPII2008"/><ref name="daleyPPI2003">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.

</ref><ref name="baddeley2007spatial">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.

</ref> as well as the related fields of [[stochastic geometry]]<ref name="stoyan1995stochastic"/> and [[spatial statistics]]<ref name="baddeley2007spatial"/><ref name="moller2003statistical">J. Moller and R. P. Waagepetersen. ''Statistical inference and simulation for spatial point processes''. CRC Press, 2003.

</ref> whose applications are found in numerous [[scientific]] and [[engineering]] disciplines such as [[biology]], [[geology]], [[physics]], and [[telecommunications]].<ref name="stoyan1995stochastic"/><ref name="daleyPPI2003"/><ref name="BB2">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.</ref>

==Point process notation==
{{Main|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 <math>\textstyle \textbf{R}^{ d}</math>, but they can be defined on more [[Abstraction (mathematics)|abstract]] mathematical spaces.<ref name="daleyPPII2008"/>

Point processes have a number of interpretations, which is reflected by the various types of [[point process notation]].<ref name="stoyan1995stochastic"/><ref name="BB2">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.

</ref>  For example, if a point <math>\textstyle x</math> belongs to or is a member of a point process, denoted by <math>\textstyle {N}</math>, then this can be written as:<ref name="stoyan1995stochastic"/>

: <math>\textstyle x\in {N},</math>

and represents the point process being interpreted as a random [[Set (mathematics)|set]]. Alternatively, the number of points of <math>\textstyle {N}</math> located in some [[Borel set]] <math>\textstyle B</math> is often written as:<ref name="BB1"/><ref name="stoyan1995stochastic"/><ref name="moller2003statistical">{{cite doi|10.1201/9780203496930}}</ref>

: <math>\textstyle {N}(B), </math>

which reflects a [[random measure]] interpretation for point processes. These two notations are often used in parallel or interchangeably.<ref name="stoyan1995stochastic"/><ref name="moller2003statistical"/><ref name="BB1">{{cite doi|10.1561/1300000006}}</ref>

==Definitions==

===''n''-th power of a point process===

For some [[integer]] <math>\textstyle n=1,2,\dots</math>, the <math>\textstyle n</math>-th power of a point process <math>\textstyle {N}</math> is defined as:<ref name="BB1">F. Baccelli and B. Blaszczyszyn. ''Stochastic Geometry and Wireless Networks, Volume I - Theory'', volume 3, No 3-4 of ''Foundations and Trends in Networking''. NoW Publishers, 2009.

</ref>

:<math> {N}^n(B_1\times,\dots,\times B_n)= \prod_{i=1}^n{N}(B_i) </math>

where <math>\textstyle B_1,...,B_n</math> is a collection of not necessarily disjoint Borel sets (in  <math>\textstyle \textbf{R}^{ d}</math>), which form a <math>\textstyle n</math>-fold [[Cartesian product]] of sets denoted by <math> B_1\times,\dots,\times B_n</math>. The symbol  <math>\textstyle \Pi</math> denotes standard [[multiplication#Capital Pi notation|multiplication]].

The notation <math>\textstyle {N}(B_i)</math> reflects the interpretation of the point process <math>\textstyle {N}</math> as a random measure.<ref name="stoyan1995stochastic"/>

The   <math>\textstyle n</math>-th power of a point process <math>\textstyle {N}</math> can be equivalently defined as:<ref name="stoyan1995stochastic"/>

:<math> {N}^{n}(B_1\times,\dots,\times B_n)= \sum_{(x_1,\dots, x_n)\in {N} } \prod_{i=1}^n \mathbf{1}_{B_i}(x_i) </math>

where  [[summation]] is performed over all <math>\textstyle n</math>-[[tuples]] of (possibly repeating) points, and  <math>\textstyle \mathbf{1}</math> denotes an [[indicator function]] such that <math>\textstyle \mathbf{1}_{B_1} </math>is a [[Dirac measure]]. This definition can be contrasted with the definition of the [[Factorial moment measure#n-th factorial power of a point process|''n''-factorial power of a point process]] for which each ''n''-[[tuples]] consists of ''n'' points.

===''n''-th moment measure===

The <math>\textstyle n</math>-th moment measure is defined as:

:<math> M^n(B_1\times,\dots,\times B_n)=E [{N}^n(B_1\times,\dots,\times B_n)], </math>

where the ''E'' denotes the [[Expected value|expectation]] ([[Operator (mathematics)#Probability theory|operator]]) of the point process <math>\textstyle {N}</math>. In other words, the ''n''-th moment measure is the expectation of the ''n''-th power of some point process.

The <math>\textstyle n\,</math>th moment measure of a point process <math>\textstyle {N}</math> is equivalently defined<ref name="stoyan1995stochastic"/> as:

:<math> \int_{\textbf{R}^{n d}}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], </math>

where <math>\textstyle f</math> is any [[non-negative]] [[measurable function]] on <math>\textstyle \textbf{R}^{n d}</math> and the sum is over <math>\textstyle n</math>-[[tuple]]s of points for which repetition is allowed.

===First moment measure===

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

:<math> M^1(B)=E [{N}(B)], </math>

where <math>\textstyle M^1</math> is known, among other terms, as the ''intensity measure''<ref name="stoyan1995stochastic"/> or ''mean measure'',<ref name="kingman1992poisson">J. F. C. Kingman. ''Poisson processes'', volume 3. Oxford university press, 1992.

</ref> and is interpreted as the expected or average number of points of <math>\textstyle {N}</math> found or located in the set <math>\textstyle B</math>.

===Second moment measure===

The second moment measure for two Borel sets <math>\textstyle A</math> and <math>\textstyle B</math> is:

:<math> M^2(A\times B)=E [{N}(A){N}(B)], </math>

which for a single Borel set <math>\textstyle B</math> becomes

:<math> M^2(B)=(E [{N}(B)])^2+\text{Var}[{N}(B)], </math>

where <math>\textstyle \text{Var}[{N}(B)]</math> denotes the [[variance]] of the random variable <math>\textstyle {N}(B)</math>.

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 <math>\textstyle {N}</math> for two Borel sets <math>\textstyle A</math> and <math>\textstyle B</math>, which is given by:<ref name="BB1"/>

:<math> \text{Cov}[{N}(A),{N}(B)]=M^2(A\times B)-M^1(A)M^1(B) </math>

==Example: Poisson point process==

For a [[general Poisson point process]] with intensity measure <math>\textstyle \Lambda</math> the first moment measure is:<ref name="BB1"/>

:<math> M^1(B)=\Lambda(B), </math>

which for a [[homogeneous Poisson point process]] with [[Mathematical constant|constant]] intensity <math>\textstyle \lambda</math> means:

:<math> M^1(B)=\lambda|B|, </math>

where <math>\textstyle |B|</math> is the length, area or volume (or more generally, the [[Lebesgue measure]]) of <math>\textstyle B</math>.

For the Poisson case with measure <math>\textstyle \Lambda</math> the second moment measure is:<ref name="baddeley2007spatial"/>

:<math> M^2(B)=\Lambda(B)+\Lambda(B)^2. </math>

which in the homogeneous case reduces to

:<math> M^2(B)=\lambda|B|+(\lambda|B|)^2. </math>

==See also==

* [[Factorial moment]]
* [[Factorial moment measure]]
* [[Moment (mathematics)|Moment]]

==References==
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[[Category:Probability theory]]
[[Category:Spatial data analysis]]