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			'''Spatial [[descriptive statistics]]''' are used for a variety of purposes in geography, particularly in quantitative data analyses involving [[GIS\|Geographic Information Systems (GIS)]].

			==Types of spatial data==

			The simplest forms of spatial data are ''gridded data'', in which a scalar quantity is measured for each point in a regular grid of points, and ''point sets'', in which a set of coordinates (e.g. of points in the plane) is observed. An example of gridded data would be a satellite image of forest density that has been digitized on a grid. An example of a point set would be the latitude/longitude coordinates of all elm trees in a particular plot of land. More complicated forms of data include marked point sets and spatial time series.

			==Measures of spatial central tendency==

			The coordinate-wise mean of a point set is the [[centroid]], which solves the same [[central tendency#solutions to variational problems\|variational problem]] in the plane (or higher dimensional Euclidean space) that the familiar average solves on the real line — that is, the centroid has the smallest possible average squared distance to all points in the set.

			==Measures of spatial dispersion==

			[[statistical dispersion\|Dispersion]] captures the degree to which points in a point set are separated from each other. For most applications, spatial dispersion should be quantified in a way that is invariant to rotations and reflections. Several simple measures of spatial dispersion for a point set can be defined using the [[covariance matrix]] of the coordinates of the points. The [[trace (linear algebra)\|trace]], the [[determinant]], and the largest [[eigenvalue]] of the covariance matrix can be used as measures of spatial dispersion.

			A measure of spatial dispersion that is not based on the covariance matrix is the average distance between nearest neighbors.<ref>{{cite journal \| last = Clark\| first = Philip \| coauthors = Evans, Francis \| year = 1954 \| title = Distance to nearest neighbor as a measure of spatial relationships in populations \| journal = Ecology \| volume = 35 \| pages = 445–453 \| doi = 10.2307/1931034 \| issue = 4 \| publisher = Ecological Society of America \| jstor= 1931034}}</ref>

			==Measures of spatial autocorrelation==
			See the [[Spatial analysis#Spatial autocorrelation\|spatial autocorrelation]] section from Wikipedia's spatial analysis page.

			==Measures of spatial homogeneity==

			A homogeneous set of points in the plane is a set that is distributed such that approximately the same number of points occurs in any circular region of a given area. A set of points that lacks homogeneity is ''spatially clustered''. A simple probability model for spatially homogeneous points is the [[Poisson process]] in the plane with constant intensity function.

			===Ripley's K and L functions===

			Ripley's ''K'' and ''L'' functions <ref>{{cite journal \| last = Ripley\| first = B.D. \| year = 1976 \| title = The second-order analysis of stationary point processes \| journal = Journal of Applied Probability \| volume = 13 \| pages = 255–266 \| }}</ref> are closely related descriptive statistics for detecting deviations from spatial homogeneity. The ''K'' function (technically its sample-based estimate) is defined as

			:<math>
			\hat{K}(t) = \lambda^{-1}\sum_{i\ne j} I(d_{ij}<t)/n,
			</math>

			where ''d''<sub>ij</sub> is the Euclidean distance between the ''i''<sup>th</sup> and ''j''<sup>th</sup> points in a data set of ''n'' points, t is the search radius and λ is the average density of points, generally estimated as ''n''/''A'', where ''A'' is the area of the region containing all points. If the points are approximately homogeneous, <math>\hat{K}(s)</math> should be approximately equal to π''t''<sup>2</sup>.

			For data analysis, the variance stabilized Ripley ''K'' function called the ''L'' function is generally used. The sample version of the ''L'' function is defined as

			:<math>
			\hat{L}(t) = \Big( \hat{K}(t)/\pi\Big)^{1/2}.
			</math>

			For approximately homogeneous data, the ''L'' function has expected value ''t'' and its variance is approximately constant in ''t''. A common plot is a graph of <math>t - \hat{L}(t)</math> against ''t'', which will approximately follow the horizontal zero-axis with constant dispersion if the data follow a homogeneous Poisson process.

			==See also==

			* [[Geostatistics]]
			* [[Variogram]]
			* [[Correlogram]]
			* [[Kriging]]
			* [[Cuzick–Edwards test]] for clustering of sub-populations within clustered populations

			==References==

			<references/>

			[[Category:Spatial data analysis]]

en>David Eppstein: /* References */ start class, not stub

2012-08-29T04:57:34Z

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