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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.
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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.
 
</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.
 
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.
 
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.
 
</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> 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>
 
==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="stoyan1995stochastic"/><ref name="moller2003statistical">{{cite doi|10.1201/9780203496930}}</ref><ref name="BB1"/>
 
: <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==
 
===Spherical contact distribution function===
 
The '''spherical contact distribution function''' is defined as:
 
:<math> H_s(r)=P({N}(b(o,r))=0). </math>
 
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''.
 
===Contact distribution function===
 
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"/>
 
:<math> H_B(r)=P({N}(rB)=0). </math>
 
==Examples==
 
===Poisson point process===
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
 
:<math> H_s(r)=1-e^{-\Lambda(b(o,r))}, </math>
 
which for the homogeneous case becomes
 
:<math> H_s(r)=1-e^{-\lambda |b(o,r)|}, </math>
 
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
 
:<math> H_s(r)=1-e^{-\lambda \pi r^2}. </math>
 
==Relationship to other functions==
 
===Nearest neighbour function===
 
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"/>
 
==={{mvar|J}}-function===
 
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}}&nbsp;≥&nbsp;0 as:<ref name="stoyan1995stochastic"/>
 
:<math> J(r)=\frac{1-D_o(r)}{1-H_s(r)} </math>
 
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>
 
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"/>
 
==See also==
* [[Nearest neighbour function]]
* [[Factorial moment measure]]
* [[Moment measure]]
 
==References==
{{notelist}}
<references/>
 
[[Category:Probability theory]]
[[Category:Spatial data analysis]]

Latest revision as of 19:28, 9 January 2015

Singapore has elevated a tax on international property buyers as a part of new non permanent measures to chill its residential housing market which has seen continued sturdy demand regardless of previous efforts to curb prices.

JLL ranked high real estate investment advisor in Asia Pacific Newest data from Real Capital Analytics (RCA) reveals real property advisory agency advised on probably the most offers by value in the area in 2013 JLL named Best Performing Property Model for second 12 months operating JLL has been named Greatest Performing property sale singapore Model at the twelfth Annual MPF Awards for Management Excellence. By its wholly-ownded subsidiary, Frasers Centrepoint Limited ("FCL"), Fraser and Neave, Limited ("F&N") has a global portfolio of high quality residential properties, serviced residences, actual estate and fund management, and business properties which include retail malls, places of work and business parks. FCL is certainly one of Singapore's top three property companies with whole belongings of over $8 billion.

Rare Industrial Improvement inside Pandan Meals Zone area up for sale A house loan or mortgage loan is a loan to purchase property and secured on the property that you purchase. A house mortgage is normally repayable in monthly instalments. Before taking over a house loan, just remember to can afford the repayments. Do ask for a repayment schedule that can assist you estimate prices. Starting from March 2012, your bank also needs to give you a residential property loan reality sheet to help you understand the terms of the mortgage. The Topiary EC @ Fernvale any unit in an permitted condominium improvement under the Planning Act; and Approval must be obtained from the Minister for Legislation to purchase a restricted residential property. GETTING YOUR PROPERTY SOLD / LEASED? SLA web site

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