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{{Expert-subject|statistics|date=June 2009}}
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'''Complete spatial randomness (CSR)''' describes a [[point process]] whereby point events occur within a given study area in a completely random fashion. It is synonymous with a homogeneous [[Spatial Poisson Process]].<ref name="Omaimon2010DataM">O. Maimon, L. Rokach, ''Data Mining and Knowledge Discovery Handbook'' , Second Edition, Springer  2010, pages 851-852
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</ref> Such a process is modeled using only one parameter <math>\rho</math>, i.e. the density of points within the defined area. The term complete spatial randomness is commonly used in Applied Statistics in the context of examining certain point patterns, whereas in most other statistical contexts it is referred to the concept of a spatial Poisson process.<ref name="Omaimon2010DataM">O. Maimon, L. Rokach, ''Data Mining and Knowledge Discovery Handbook'' , Second Edition, Springer  2010, pages 851-852
</ref>
 
== Model ==
Data in the form of a set of points, irregularly distributed within a region of space, arise in many different contexts; examples include locations of trees in a forest, of nests of birds, of nuclei in tissue, of ill people in a population at risk. We call any such data-set a spatial point pattern and refer to the locations as events, to distinguish these from arbitrary points of the region in question. The hypothesis of complete spatial randomness for a spatial point pattern asserts that the number of events in any region follows a [[Poisson distribution]] with given mean count per uniform subdivision. The events of a pattern are independently and uniformly distributed over space; in other words, the events are equally likely to occur anywhere and do not interact with each other.
 
"Uniform" is used in the sense of following a uniform probability distribution across the study region, not in the sense of “evenly” dispersed across the study region.<ref name="waller2004ApSpSt"> L. A. Waller, C. A. Gotway, ''Applied Spatial Statistics for Public Health Data'', volume 1 Wiley Chichester, 2004, pages 119–121,
123–127, 137, 139–141, 146–148,
150–151, 157, 203. </ref> There are no interactions amongst the events, as the intensity of events does not vary over the plane. For example, the independence assumption would be violated if the existence of one event either encouraged or inhibited the occurrence of other events in the neighborhood.
 
== Distribution ==
The probability of finding exactly <math>k</math> points within the area <math>V</math> with event density <math>\rho</math> is therefore:
 
:<math>P(k,\rho,V) = \frac{(V\rho)^k e^{-(V\rho) }}{k!} . \,\!</math>
 
The first moment of which, the average number of points in the area, is simply <math>\rho V</math>. This value is intuitive as it is the Poisson rate parameter.
 
The probability of locating the <math>N^{\mathrm{th}}</math> neighbor of any given point, at some radial distance <math>r</math> is:
 
:<math>P_N(r) = \frac{D}{(N-1)!}  {\lambda}^N r^{DN-1} e^{- \lambda r^D} ,</math>
 
where <math>D</math> is the number of dimensions, <math>\lambda</math> is a density dependent parameter given by <math>\lambda = \frac{\rho \pi ^{\frac{D}{2}}}{\Gamma (\frac{D}{2} +1)} </math> and <math>\Gamma</math> is the [[gamma function]], which when its argument is integral, is simply the [[factorial]] function.
 
The expected value of <math> P_N(r)</math> can be derived via the use of the gamma function using statistical moments. The first moment is the mean distance between randomly distributed particles in <math>D</math> dimensions.
 
== Applications ==
The study of CSR is essential for the comparison of measured point data from experimental sources. As a statistical testing method, the test for CSR has many applications in the [[social sciences]] and in astronomical examinations.<ref>{{cite web|url=http://www.galaxy.gmu.edu/interface/I02/I2002Proceedings/HauckSteven/HauckSteven.presentation.ppt|title=Statistics on Venus: Craters and Catastrophes}}</ref> CSR is often the standard against which data sets are tested. Roughly described one approach to test the CSR hypothesis is the following:<ref name="Okabe2012SpAn"> A. Okabe, K. Sugihara, "Spatial Analysis along Networks- Statistical and Computational Methods", volume 1 Wiley Chichester, 2012, pages 135-136 </ref>
# Use [[statistics]] that are a function of the distance from every event to the next nearest event.
# Firstly focus on a specific event and formulate a method for testing whether the event and the next nearest event are significantly close (or distant).
# Next consider all events and formulate a method for testing whether the average distance from every event to the next nearest event is significantly short (or long).
 
In cases where computing test statistics analytically is difficult, numerical methods, such as the [[Monte Carlo method]] simulation are employed, by simulating a stochastic process a large number of times.<ref name="Okabe2012SpAn"> A. Okabe, K. Sugihara, ''Spatial Analysis along Networks- Statistical and Computational Methods'', volume 1 Wiley Chichester, 2012, pages 135-136 </ref>
 
== References ==
{{Reflist}}
 
== Further reading ==
*{{cite book |last=Diggle |first=P. J. |year=2003 |title=Statistical Analysis of Spatial Point Patterns |edition=2nd |publisher=Academic Press |location=New York |isbn=0340740701 }}
 
== External links ==
*[http://handle.dtic.mil/100.2/ADA291151 Improvement of Inter-event Distance Tests of Randomness in Spatial Point Processes]
 
[[Category:Spatial data analysis]]
[[Category:Point processes]]
[[Category:Statistical randomness]]
[[Category:Spatial processes]]

Latest revision as of 08:36, 14 August 2014

If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Should you go with simple HTML or use a platform like Wordpress. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. You can customize the appearance with PSD to Word - Press conversion .

You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. While direct advertising is limited to few spots in your site and tied to fixed monthly payment by the advertisers, affiliate marketing can give you unlimited income as long as you can convert your traffic to sales. Several claim that Wordpress just isn't an preferred tool to utilise when developing a professional site. Now, I want to anxiety that not every single query will be answered. That's a total of 180$ for each Wordpress theme if you sell 3 links and keep the designer link for your own website, or 240$ if you sell all links.

The entrepreneurs can easily captivate their readers by using these versatile themes. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. This platform can be customizedaccording to the requirements of the business. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. If you've hosted your Word - Press website on a shared hosting server then it'll be easier for you to confirm the restricted access to your site files.

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