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{{Other uses|Nearest neighbor (disambiguation){{!}}Nearest neighbor}}
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In probability and statistics, a '''nearest neighbour function''', '''nearest neighbour distance distribution''',<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> '''nearest-neighbour distribution function'''<ref name="torquato1990nearest">{{cite journal| author=Torquato, S, Lu, B, Rubinstein, J| title=Nearest-neighbour distribution function for systems on interacting particles| journal=Journal of Physics A: Mathematical and General| year=1990| volume=23| number=3| pages=L103| publisher=IOP Publishing| accessdate=10 January 2014}}</ref> or '''nearest neighbour distribution'''<ref name="doguwa1992estimation">{{cite journal| author=Doguwa, Sani I| title=On the estimation of the point-object nearest neighbour distribution F (y) for point processes| journal=Journal of statistical computation and simulation| year=1992| volume=41| number=1-2| pages=95–107| publisher=Taylor \& Francis| accessdate=10 January 2014}}</ref> is a [[mathematical function]] that is defined in relation to [[random]] [[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">D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.</ref><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, nearest neighbour functions are defined with respect to some point in the point process as being the probability distribution of the distance from this point to its nearest neighbouring point in the same point process, hence they are used to describe the probability of another point existing within some distance of a point. A nearest neighbour function can be contrasted with a [[spherical contact distribution function]], which is not defined in reference to some initial point but rather as the probability distribution of the radius of a sphere when it first encounters or makes contact with a point in a point process.
 
Nearest neighbour function 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="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==
 
===Nearest neighbour function===
 
The nearest neighbour function, as opposed to the [[spherical contact distribution function]], is defined in relation to some point of a point process already existing in some region of space. More precisely, for some point in the point process <math>\textstyle {N}</math>, the nearest neighbour function is the probability distribution of the distance from that point to the nearest or closest neighbouring point.
 
To define this function for a point located in <math>\textstyle \textbf{R}^d</math> at, for example, the [[Origin (mathematics)|origin]] <math>\textstyle o</math>, the <math>\textstyle d</math>-dimensional [[Ball (mathematics)|ball]] <math>\textstyle b(o,r)</math> of radius <math>\textstyle r</math> centered at the [[Origin (mathematics)|origin]] ''o'' is considered. Given a point of <math>\textstyle {N}</math> existing at <math>\textstyle o</math>, then the '''nearest neighbour function''' is defined as:<ref name="stoyan1995stochastic"/>
 
:<math> D_o(r)=1-P({N}(b(o,r))=1\mid o). </math>
 
where <math>\textstyle P({N}(b(o,r))=1\mid o)</math> denotes the conditional probability that there is one point of <math>\textstyle {N}</math> located in <math>\textstyle b(o,r)</math> given there is a point of <math>\textstyle {N}</math> located at <math>\textstyle o</math>.
 
The reference point need not be at the origin, and can be located at an arbitrary point <math>\textstyle x\in\textbf{R}^d</math>. Given a point of <math>\textstyle {N}</math> existing at <math>\textstyle o</math>, then the '''nearest neighbour function''', is defined as:
 
:<math> D_x(r)=1-P({N}(b(x,r))=1\mid x). </math>
 
==Examples==
 
===Poisson point process===
 
For a [[Poisson point process]] <math>\textstyle {N}</math> on <math>\textstyle \textbf{R}^d</math> with [[Moment_measure#First_moment_measure|intensity measure]] <math>\textstyle \Lambda</math> the nearest neighbour function is:
 
:<math> D_x(r)=1-e^{-\Lambda(b(x,r))}, </math>
 
which for the homogeneous case becomes
 
:<math> D_x(r)=1-e^{-\lambda |b(x,r)|}, </math>
 
where <math>\textstyle |b(x,r)|</math> denotes the volume (or more specifically, the [[Lebesgue measure]]) of the (hyper) ball of radius <math>\textstyle r</math>. In the plane <math>\textstyle \textbf{R}^2</math>, this becomes
 
:<math> D_x(r)=1-e^{-\lambda \pi r^2}. </math>
 
==Relationship to other functions==
 
===Spherical contact distribution 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 distributions, 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 ''H''<sub>''s''</sub>(''r'') and nearest neighbour function ''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'')}}&nbsp;=&nbsp;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'')}}&nbsp;=&nbsp;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==
* [[Factorial moment]]
* [[Local feature size]]
* [[Moment measure]]
* [[Spherical contact distribution function]]
 
==References==
{{notelist}}
<references/>
 
[[Category:Probability theory]]
[[Category:Spatial data analysis]]

Latest revision as of 16:39, 5 May 2014

To secure the long run, there are numerous funding options, one out of which is investing in new launch apartment property. In case you are planning to make investment in Singapore condominiums to safeguard your future, then we've got give you sure execs & cons of creating investment within the rental property for your assist. Learn these pros and cons earlier than you decide to make the funding at Duo Residences , Thomson View , or different condos.

When prospects ask in regards to the subsequent batch of units for launch, gross sales representatives might hint that subsequent phases will provide items with poorer view or decrease quality finishings, despite the fact that they are offered at larger prices. Tempted buyers are sometimes unaware that the value of these goodies can simply be offset by a slight drop in the property's market worth. It's due to this fact more sensible to get an instantaneous discount off the list worth. Ask the sales representative the estimated value of that branded appliance or furnishing package deal, then request for a direct deduction from the unit's asking worth in lieu of the developer's goodie. December 4, 2013 by iskandarinsider November four, 2013 by iskandarinsider October 23, 2013 by iskandarinsider

SpaceX can be gearing as much as launch from Vandenberg Air Drive Base in California. This web site offers entry to polar orbits, which many Air Drive payloads and environmental monitoring satellites require. A satellite tv for pc in polar orbit can expose its sensors to changing slices of the Earth rotating beneath it. On this approach, your complete globe might be seen in a couple of days or much less. Better spaceflight economics — particularly lowered launch costs — have long been thought of essential for strong improvement of the photo voltaic system's potentially habitable frontiers. Musk has repeatedly and publically said his want, and his firm's aim, to explore and settle Mars District 19, Freehold strata cluster landed Shutter bus service to MRT station Freehold Combined improvement

New launch condo La Fiesta @ Sengkang Central New launch condo Jade Residences stroll to Nex Hallmark new launch apartment at prime Bukit Timah New launch condominium The Hallmark Residences, fittingly nestled at Ewe Boon Street within the splendid enclave of Bukit Timah.Experience the excitement of dwelling precise bliss with all of the comforts for a superb city way of life are just a blink away. Take a look at the likeable Central Enterprise District of singapore new condominium which mainly requires a short drive away from home or go to the Orchard Street Belt for a load of procuring choices. New launch condominium near Mrt Jewel @ Buangkok by CDL New launch condominium Tembusu by Wing Tai New launch condo near Thomson Plaza selling fast New Launch Condominiums District 19, Freehold Condominium New Condo SKIES MILTONIA

Lush Acres Govt Condominium is going to launch soon at Sengkang West Method By Metropolis Developments Restricted. Lush Acres displays the large expense of house, environmentally-inexperienced constructing and plush panorama of this low density development with solely 22% of web site protection. There were students, retirees and property brokers napping, snacking and enjoying cards in the queue to kill time. Such a scene made passers-by wonder whether the crowd actually consisted of real patrons, or simply half-timers being paid to fake how hot the brand new undertaking was. The brand was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was additionally recorded in France on the INPI (Nationwide Institute of Intellectual Property) Coastal business spaceports

New Rental Daisy Suites (Freehold) Preview New launch rental Jewel @ Buangkok is positioned at 2 minutes walk to Buangkok MRT station. Other than the luxurious living, The Tembusu Kovan can be encircled by retail and dining amenities, entertainment, recreation and transport convenience. It's endowed with good connectivity to major roads like Upper Serangoon Highway and expressways similar to Central Expressway (CTE) and Kallang-Paya Lebar Expressway (KPE). With Kovan MRT Station 6 minutes stroll away, getting to any a part of Singapore is just a breeze. New launch condos are fashionable investments As soon as the gross sales representatives know that you are shopping for the property for investment, they may declare that their units may be rented out at a sure market rate.

Duo Residences is a blended development of a deliberate luxury residences, lodge, office and retail complex. It will be situated on Ophir-Rochor Highway in the Downtown Core of Singapore. It can characteristic 660 units of prime residences, 21 storeys of grade A offices and 5 Star Hotel. Duo Residences is designed by world-famend architect Ole Scheeren. This 2.7 ha prime website is strategically situated within the heart of the Metropolis Centre along the Seaside Road / Ophir-Rochor Hall and is a pure extension from the established conference, office, resort hub at Marina Centre.

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