World Library  
Flag as Inappropriate
Email this Article

Complete spatial randomness

Article Id: WHEBN0021447866
Reproduction Date:

Title: Complete spatial randomness  
Author: World Heritage Encyclopedia
Language: English
Subject: Point pattern analysis, Statistical randomness, Spatial data analysis, Stochastic geometry, Ecological correlation
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Complete spatial randomness

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.[1] Such a process is modeled using only one parameter \rho, 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.[1]

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.[2] 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 k points within the area V with event density \rho is therefore:

P(k,\rho,V) = \frac{(V\rho)^k e^{-(V\rho) }}{k!} . \,\!

The first moment of which, the average number of points in the area, is simply \rho V. This value is intuitive as it is the Poisson rate parameter.

The probability of locating the N^{\mathrm{th}} neighbor of any given point, at some radial distance r is:

P_N(r) = \frac{D}{(N-1)!} {\lambda}^N r^{DN-1} e^{- \lambda r^D} ,

where D is the number of dimensions, \lambda is a density dependent parameter given by \lambda = \frac{\rho \pi ^{\frac{D}{2}}}{\Gamma (\frac{D}{2} +1)} and \Gamma is the gamma function, which when its argument is integral, is simply the factorial function.

The expected value of P_N(r) 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 D 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.[3] CSR is often the standard against which data sets are tested. Roughly described one approach to test the CSR hypothesis is the following:[4]

  1. Use statistics that are a function of the distance from every event to the next nearest event.
  2. 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).
  3. 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.[4]

References

  1. ^ a b O. Maimon, L. Rokach, Data Mining and Knowledge Discovery Handbook , Second Edition, Springer 2010, pages 851-852
  2. ^ 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.
  3. ^ "Statistics on Venus: Craters and Catastrophes". 
  4. ^ a b A. Okabe, K. Sugihara, "Spatial Analysis along Networks- Statistical and Computational Methods", volume 1 Wiley Chichester, 2012, pages 135-136

Further reading

  • Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns (2nd ed.). New York: Academic Press.  

External links

  • Improvement of Inter-event Distance Tests of Randomness in Spatial Point Processes
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from Hawaii eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.