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<title>Complete spatial randomness</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Complete spatial randomness</span></span>

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<p><b>Complete spatial randomness</b> (<b>CSR</b>) describes a <a href="Point_process" title="Point process">point process</a> whereby point events occur within a given study area in a completely random fashion. It is synonymous with a <i>homogeneous <a href="Spatial_Poisson_process" class="mw-redirect" title="Spatial Poisson process">spatial Poisson process</a></i>.<sup id="cite_ref-Omaimon2010DataM_1-0" class="reference"><a href="#cite_note-Omaimon2010DataM-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Such a process is modeled using only one parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math></span><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span>, 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.<sup id="cite_ref-Omaimon2010DataM_1-1" class="reference"><a href="#cite_note-Omaimon2010DataM-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Model">Model</h2></div>

<p>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 <a href="Poisson_distribution" title="Poisson distribution">Poisson distribution</a> 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.

</p><p>"Uniform" is used in the sense of following a <a href="Uniform_probability_distribution" class="mw-redirect" title="Uniform probability distribution">uniform probability distribution</a> across the study region, not in the sense of “evenly” dispersed across the study region.<sup id="cite_ref-waller2004ApSpSt_2-0" class="reference"><a href="#cite_note-waller2004ApSpSt-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> There are no interactions amongst the events, and 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.

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<div class="mw-heading mw-heading2"><h2 id="Distribution">Distribution</h2></div>

<p>The probability of finding exactly <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> points within the area <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> with event density <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math></span><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> is therefore:

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<mfrac>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle P(k,\rho ,V)={\frac {(V\rho )^{k}e^{-(V\rho )}}{k!}}.\,\!}</annotation>

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</math></span><img src="./ee98a690db440e58c959a7e26843ebe45118f5df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:26.666ex; height:6.176ex;" alt="{\displaystyle P(k,\rho ,V)={\frac {(V\rho )^{k}e^{-(V\rho )}}{k!}}.\,\!}" loading="lazy"></span></dd></dl>

<p>The first moment of which, the <a href="Average" title="Average">average</a> number of points in the area, is simply <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho V}</annotation>

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</math></span><img src="./ce7502f1e16a46a4f2d070c6ddc2247e2ed4f44a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.989ex; height:2.676ex;" alt="{\displaystyle \rho V}" loading="lazy"></span>. This value is intuitive as it is the Poisson rate parameter.

</p><p>The probability of locating the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{\mathrm {th} }}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N^{\mathrm {th} }}</annotation>

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</math></span><img src="./2b27675c5574522fc60c30994b13d1035f055099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.909ex; height:2.676ex;" alt="{\displaystyle N^{\mathrm {th} }}" loading="lazy"></span> neighbor of any given point, at some radial distance <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

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</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> is:

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<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle P_{N}(r)={\frac {D}{(N-1)!}}{\lambda }^{N}r^{DN-1}e^{-\lambda r^{D}},}</annotation>

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</math></span><img src="./7e1be2e7a7be7b5efba8f94d72aa58128c69481c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.957ex; height:6.009ex;" alt="{\displaystyle P_{N}(r)={\frac {D}{(N-1)!}}{\lambda }^{N}r^{DN-1}e^{-\lambda r^{D}},}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> is the number of dimensions, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> is a density-dependent parameter given by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {\rho \pi ^{\frac {D}{2}}}{\Gamma ({\frac {D}{2}}+1)}}}">

<mfrac>

<mrow>

<msup>

<mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {\rho \pi ^{\frac {D}{2}}}{\Gamma ({\frac {D}{2}}+1)}}}</annotation>

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</math></span><img src="./e6477296e3b380415463407c73aa05dcc4b23297.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:14.752ex; height:8.509ex;" alt="{\displaystyle \lambda ={\frac {\rho \pi ^{\frac {D}{2}}}{\Gamma ({\frac {D}{2}}+1)}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> is the <a href="Gamma_function" title="Gamma function">gamma function</a>, which when its argument is integer, is simply the <a href="Factorial" title="Factorial">factorial</a> function - i.e. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (n+1)=n!}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (n+1)=n!}</annotation>

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</math></span><img src="./2fcf8541920a9f7b0ad3ae3ffaf8870022cddb29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.8ex; height:2.843ex;" alt="{\displaystyle \Gamma (n+1)=n!}" loading="lazy"></span> for integral <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>.

</p><p>The expected value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{N}(r)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{N}(r)}</annotation>

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</math></span><img src="./347b20d8e91eb9ddad88e70643bf1a443a5ccdc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.042ex; height:2.843ex;" alt="{\displaystyle P_{N}(r)}" loading="lazy"></span> 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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> dimensions.

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>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 <a href="Social_sciences" class="mw-redirect" title="Social sciences">social sciences</a> and in astronomical examinations.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> CSR is often the standard against which data sets are tested. Roughly described one approach to test the CSR hypothesis is the following:<sup id="cite_ref-Okabe2012SpAn_4-0" class="reference"><a href="#cite_note-Okabe2012SpAn-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<ol><li>Use <a href="Statistics" title="Statistics">statistics</a> that are a function of the distance from every event to the next nearest event.</li>

<li>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).</li>

<li>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).</li></ol>

<p>In cases where computing test statistics analytically is difficult, numerical methods, such as the <a href="Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a> simulation are employed, by simulating a stochastic process a large number of times.<sup id="cite_ref-Okabe2012SpAn_4-1" class="reference"><a href="#cite_note-Okabe2012SpAn-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-Omaimon2010DataM-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Omaimon2010DataM_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Omaimon2010DataM_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">O. Maimon, L. Rokach, <i>Data Mining and Knowledge Discovery Handbook</i>, Second Edition, Springer 2010, pages 851-852</span>

</li>

<li id="cite_note-waller2004ApSpSt-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-waller2004ApSpSt_2-0">^</a></b></span> <span class="reference-text">L. A. Waller, <a href="Carol_A._Gotway_Crawford" title="Carol A. Gotway Crawford">C. A. Gotway</a>, <i>Applied Spatial Statistics for Public Health Data</i>, volume 1 Wiley Chichester, 2004, pages 119–121,

123–127, 137, 139–141, 146–148,

150–151, 157, 203.</span>

</li>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.galaxy.gmu.edu/interface/I02/I2002Proceedings/HauckSteven/HauckSteven.presentation.ppt">"Statistics on Venus: Craters and Catastrophes"</a>.</cite></span>

</li>

<li id="cite_note-Okabe2012SpAn-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Okabe2012SpAn_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Okabe2012SpAn_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">A. Okabe, K. Sugihara, "Spatial Analysis along Networks- Statistical and Computational Methods", volume 1 Wiley Chichester, 2012, pages 135-136</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFDiggle2003" class="citation book cs1">Diggle, P. J. (2003). <i>Statistical Analysis of Spatial Point Patterns</i> (2nd ed.). New York: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0340740701</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140116072512/http://handle.dtic.mil/100.2/ADA291151">Improvement of Inter-event Distance Tests of Randomness in Spatial Point Processes</a></li></ul></div><div><div class="zim-footer">

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