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| In probability and statistics, a '''spherical contact distribution function''', '''first contact distribution function''',<ref name="stoyan1995stochastic">D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.</ref> or '''empty space function'''<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.
| | 45 yr old Naturopath Rave from Sheet Harbour, enjoys to spend some time illusion, [http://hardmagic.com/groups/need-to-reside-on-mars-private-martian-colony-undertaking-seeks-astronauts/ property sale singapore] developers in singapore and handball. Recently has visited Mapungubwe Cultural Landscape. |
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| </ref> is a [[mathematical function]] 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.<ref name="stoyan1995stochastic"/><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.
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| </ref> More specifically, a spherical contact distribution function is defined as probability distribution of the radius of a sphere when it first encounters or makes contact with a point in a point process. This function can be contrasted with the [[nearest neighbour function]], which is defined in relation to some point in the point process as being the probability distribution of the distance from that point to its nearest neighbouring point in the same point process.
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| The spherical contact function is also referred to as the '''contact distribution function''',<ref name="baddeley2007spatial"/> but some authors<ref name="stoyan1995stochastic"/> define the contact distribution function in relation to a more general set, and not simply a sphere as in the case of the spherical contact distribution function.
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| Spherical contact distribution functions are used in the study of point processes<ref name="baddeley2007spatial"/><ref name="daleyPPI2003"/><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.
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| </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.
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| </ref> which are applied in various [[scientific]] and [[engineering]] disciplines such as [[biology]], [[geology]], [[physics]], and [[telecommunications]].<ref name="stoyan1995stochastic"/><ref name="daleyPPI2003"/><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><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>
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| ==Point process notation==
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| {{Main|Point process notation}}
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| 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"/>
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| 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.
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| </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"/>
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| : <math>\textstyle x\in {N},</math>
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| 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="stoyan1995stochastic"/><ref name="moller2003statistical">{{cite doi|10.1201/9780203496930}}</ref><ref name="BB1"/>
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| : <math>\textstyle {N}(B), </math>
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| 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>
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| ==Definitions==
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| ===Spherical contact distribution function===
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| The '''spherical contact distribution function''' is defined as:
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| :<math> H_s(r)=P({N}(b(o,r))=0). </math>
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| where ''b(o,r)'' is a [[Ball (mathematics)|ball]] with radius ''r'' centered at the origin ''o''. In other words, spherical contact distribution function is the probability there are no points from the point process located in a hyper-sphere of radius ''r''.
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| ===Contact distribution function===
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| The spherical contact distribution function can be generalized for sets other than the (hyper-)sphere in <math>\textstyle \textbf{R}^{ d}</math>. For some Borel set <math>\textstyle B</math> with positive volume (or more specifically, Lebesgue measure), the ''contact distribution function'' (''with respect to'' <math>\textstyle B</math>) for <math>\textstyle r\geq0</math> is defined by the equation:<ref name="stoyan1995stochastic"/>
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| :<math> H_B(r)=P({N}(rB)=0). </math>
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| ==Examples==
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| ===Poisson point process===
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| For a Poisson point process <math>\textstyle {N}</math> on <math>\textstyle \textbf{R}^d</math> with intensity measure <math>\textstyle \Lambda</math> this becomes
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| :<math> H_s(r)=1-e^{-\Lambda(b(o,r))}, </math>
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| which for the homogeneous case becomes
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| :<math> H_s(r)=1-e^{-\lambda |b(o,r)|}, </math>
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| where <math>\textstyle |b(o,r)|</math> denotes the volume (or more specifically, the Lebesgue measure) of the ball of radius <math>\textstyle r</math>. In the plane <math>\textstyle \textbf{R}^2</math>, this expression simplifies to
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| :<math> H_s(r)=1-e^{-\lambda \pi r^2}. </math>
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| ==Relationship to other functions==
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| ===Nearest neighbour function===
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| In general, the spherical contact distribution function and the corresponding [[nearest neighbour function]] are not equal. However, these two functions are identical for Poisson point processes.<ref name="stoyan1995stochastic"/> In fact, this characteristic is due to a unique property of Poisson processes and their [[Palm distribution]]s, which forms part of the result known as the ''Slivnyak-Mecke''<ref name="BB1"/> or ''Slivnyak's theorem''.<ref name="baddeley2007spatial"/>
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| ==={{mvar|J}}-function===
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| The fact that the spherical distribution function {{mvar| H<sub>s</sub>(r)}} and nearest neighbour function {{mvar| D<sub>o</sub>(r)}} are identical for the Poisson point process can be used to statistically test if point process data appears to be that of a Poisson point process. For example, in spatial statistics the {{mvar|J}}-function is defined for all {{mvar|r}} ≥ 0 as:<ref name="stoyan1995stochastic"/>
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| :<math> J(r)=\frac{1-D_o(r)}{1-H_s(r)} </math>
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| For a Poisson point process, the {{mvar|J}} function is simply {{math|''J''(''r'')}}=1, hence why it is used as a [[Non-parametric statistics|non-parametric]] test for whether data behaves as though it were from a Poisson process. It is, however, thought possible to construct non-Poisson point processes for which {{math|''J''(''r'')}}=1,<ref name="bedford1997remark">{{cite journal| author=Bedford, T, Van den Berg, J| title=A remark on the Van Lieshout and Baddeley J-function for point processes| journal=Advances in Applied Probability| year=1997| pages=19–25| publisher=JSTOR| accessdate=10 January 2014}}</ref> but such counterexamples are viewed as somewhat 'artificial' by some and exist for other statistical tests.<ref name="foxall2002nonparametric">{{cite journal| author=Foxall, Rob, Baddeley, Adrian| title=Nonparametric measures of association between a spatial point process and a random set, with geological applications| journal=Journal of the Royal Statistical Society: Series C (Applied Statistics)| year=2002| volume=51| number=2| pages=165–182| publisher=Wiley Online Library| accessdate=10 January 2014}}</ref>
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| More generally, {{mvar|J}}-function serves as one way (others include using [[factorial moment measure]]s<ref name="baddeley2007spatial"/>) to measure the interaction between points in a point process.<ref name="stoyan1995stochastic"/>
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| ==See also==
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| * [[Nearest neighbour function]]
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| * [[Factorial moment measure]]
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| * [[Moment measure]]
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| ==References==
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| {{notelist}}
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| <references/>
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| [[Category:Probability theory]]
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| [[Category:Spatial data analysis]]
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45 yr old Naturopath Rave from Sheet Harbour, enjoys to spend some time illusion, property sale singapore developers in singapore and handball. Recently has visited Mapungubwe Cultural Landscape.