94:
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
274:
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122:
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 may be
534:
356:
169:
83:) 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.
716:
Wilschut, L.I.; Laudisoit, A.; Hughes, N.K.; Addink, E.A.; de Jong, S.M.; Heesterbeek, J.A.P.; Reijniers, J.; Eagle, S.; Dubyanskiy, V.M.; Begon, M. (2015).
383:
693:
540:, which will approximately follow the horizontal zero-axis with constant dispersion if the data follow a homogeneous Poisson process.
59:
of forest density that has been digitized on a grid. An example of a point set would be the latitude/longitude coordinates of all
775:
114:
A measure of spatial dispersion that is not based on the covariance matrix is the average distance between nearest neighbors.
547:
function it can be determined whether points have a random, dispersed or clustered distribution pattern at a certain scale.
601:
Clark, Philip; Evans, Francis (1954). "Distance to nearest neighbor as a measure of spatial relationships in populations".
718:"Spatial distribution patterns of plague hosts: point pattern analysis of the burrows of great gerbils in Kazakhstan"
32:
55:, in which a set of coordinates (e.g. of points in the plane) is observed. An example of gridded data would be a
770:
63:
in a particular plot of land. More complicated forms of data include marked point sets and spatial time series.
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320:(i.e. 1 if its operand is true, 0 otherwise). In 2 dimensions, if the points are approximately homogeneous,
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24:
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are closely related descriptive statistics for detecting deviations from spatial homogeneity. The
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at a certain spatial scale. A simple probability model for spatially homogeneous points is the
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points, t is the search radius, λ is the average density of points (generally estimated as
269:{\displaystyle {\widehat {K}}(t)=\lambda ^{-1}\sum _{i\neq j}{\frac {I(d_{ij}<t)}{n}},}
156:
128:
80:
56:
51:, in which a scalar quantity is measured for each point in a regular grid of points, and
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470:{\displaystyle {\widehat {L}}(t)=\left({\frac {{\widehat {K}}(t)}{\pi }}\right)^{1/2}.}
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of the covariance matrix can be used as measures of spatial dispersion.
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function (technically its sample-based estimate) is defined as
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for clustering of sub-populations within clustered populations
36:
684:. In El-Shaarawi, Abdel H.; Piegorsch, Walter W. (eds.).
638:"The second-order analysis of stationary point processes"
373:
function is generally used. The sample version of the
27:; these methods are used for a variety of purposes in
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is the area of the region containing all points) and
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469:
365:For data analysis, the variance stabilized Ripley
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131:in the plane with constant intensity function.
71:The coordinate-wise mean of a point set is the
488:and its variance is approximately constant in
688:. John Wiley & Sons. pp. 1796–1803.
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47:The simplest forms of spatial data are
288:is the Euclidean distance between the
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79:in the plane (or higher-dimensional
67:Measures of spatial central tendency
37:Geographic Information Systems (GIS)
529:{\displaystyle t-{\widehat {L}}(t)}
358:should be approximately equal to π
14:
351:{\displaystyle {\widehat {K}}(t)}
134:
492:. A common plot is a graph of
118:Measures of spatial homogeneity
686:Encyclopedia of Environmetrics
642:Journal of Applied Probability
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87:Measures of spatial dispersion
17:Spatial descriptive statistics
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484:function has expected value
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33:quantitative data analyses
677:Dixon, Philip M. (2002).
296:points in a data set of
155:functions introduced by
75:, which solves the same
722:Journal of Biogeography
583:Spatial autocorrelation
377:function is defined as
19:is the intersection of
776:Descriptive statistics
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471:
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25:descriptive statistics
679:"Ripley's K function"
636:Ripley, B.D. (1976).
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472:
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43:Types of spatial data
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369:function called the
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577:Cuzick–Edwards test
125:spatially clustered
77:variational problem
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318:indicator function
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107:, and the largest
31:, particularly in
21:spatial statistics
734:10.1111/jbi.12534
695:978-0-471-89997-6
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97:covariance matrix
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771:Spatial analysis
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728:(7): 1281–1292.
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543:Using Ripley's
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157:Brian D. Ripley
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129:Poisson process
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81:Euclidean space
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57:satellite image
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699:. Retrieved
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49:gridded data
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567:Correlogram
105:determinant
765:Categories
589:References
109:eigenvalue
92:Dispersion
53:point sets
35:involving
701:April 25,
562:Variogram
512:^
503:−
442:π
426:^
394:^
334:^
219:≠
212:∑
203:−
199:λ
180:^
147:Ripley's
143:functions
135:Ripley's
61:elm trees
29:geography
752:26877580
551:See also
536:against
308:, where
73:centroid
743:4737218
664:3212829
623:1931034
603:Ecology
572:Kriging
316:is the
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662:
621:
279:where
103:, the
682:(PDF)
660:JSTOR
619:JSTOR
101:trace
748:PMID
703:2014
690:ISBN
292:and
249:<
151:and
139:and
23:and
738:PMC
730:doi
650:doi
611:doi
767::
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736:.
726:42
724:.
720:.
658:.
646:13
644:.
640:.
617:.
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605:.
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285:ij
39:.
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732::
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666:.
652::
625:.
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545:K
538:t
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509:L
500:t
490:t
486:t
482:L
465:.
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456:/
452:1
447:)
438:)
435:t
432:(
423:K
414:(
409:=
406:)
403:t
400:(
391:L
375:L
371:L
367:K
360:t
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343:t
340:(
331:K
314:I
310:A
306:A
304:/
302:n
298:n
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281:d
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252:t
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241:i
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233:(
230:I
222:j
216:i
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195:=
192:)
189:t
186:(
177:K
161:K
153:L
149:K
141:L
137:K
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.