Analysing spatial point patterns in R - CSIRO

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124 Distance methods for point patterns > par(pty = "s") > plot(Gest(cells)) Gest(cells) G(r) 0.0 0.2 0.4 0.6 0.8 km rs han theo 0.00 0.05 0.10 0.15 r The estimate of G(r) suggests strongly that the pattern is regular. Indeed, Ĝ(r) is zero for r ≤ 0.07 which indicates that there are no nearest-neighbour distances shorter than 0.07. Common ways of plotting Ĝ include: Ĝ(r) and G pois (r) plotted against r plot(Gest(X)) Ĝ(r) − G pois (r) plotted against r plot(Gest(X), . - theo ~ r) Ĝ(r) plotted against G pois (r) in P–P style plot(Gest(X), . ~ theo) and Fisher’s variance-stabilising transformation φ(G(t)) = sin −1 ( √ G(t)) applied to the P–P plot: > fisher plot(Gest(cells), fisher(.) ~ fisher(theo)) Gest(cells) fisher(G(r)) 0.0 0.5 1.0 1.5 km rs han theo 0.0 0.5 1.0 1.5 fisher(G pois(r)) CopyrightCSIRO 2010
19.4 Pairwise distances and the K function 125 19.4 Pairwise distances and the K function The observed pairwise distances s ij = ||x i −x j || in the data pattern x constitute a biased sample of pairwise distances in the point process, with a bias in favour of smaller distances. For example, we can never observe a pairwise distance greater than the diameter of the window. Ripley [54] defined the K-function for a stationary point process so that λK(r) is the expected number of other points of the process within a distance r of a typical point of the process. Formally K(r) = 1 E [n(X ∩ b(u,r) \ {u}) | u ∈ X]. (21) λ For a homogeneous Poisson process, intuitively, the knowledge that u is a point of X does not affect the other points of the process, so that X\{u} is conditionally a Poisson process. The expected number of points falling in b(u,r) is λπr 2 . Thus for a homogeneous Poisson process K pois (r) = πr 2 (22) regardless of the intensity. Numerous estimators of K have been proposed. Most of them are weighted and renormalised empirical distribution functions of the pairwise distances, of the general form ̂K(r) = 1 ̂λ 2 area(W) ∑∑ 1 {||x i − x j || ≤ r}e(x i ,x j ;r) (23) i j≠i where e(u,v,r) is an edge correction weight. The choice of estimator does not seem to be very important, as long as some edge correction is applied. Again we usually compare the estimate ̂K(r) with the Poisson K function. Values ̂K(r) > πr 2 suggest clustering, while ̂K(r) < πr 2 suggests a regular pattern. In spatstat the function Kest computes several estimates of the K-function. > Gc Gc Function value object (class fv) for the function r -> K(r) Entries: id label description -- ----- ----------- r r distance argument r theo K[pois](r) theoretical Poisson K(r) border K[bord](r) border-corrected estimate of K(r) trans K[trans](r) translation-corrected estimate of K(r) iso K[iso](r) Ripley isotropic correction estimate of K(r) -------------------------------------- Default plot formula: . ~ r Recommended range of argument r: [0, 0.25] Available range of argument r: [0, 0.25] CopyrightCSIRO 2010
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Analysing spatial point patterns in
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CONTENTS 3 Contents PART I. OVERVIE
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CONTENTS 5 PART I. OVERVIEW The fir
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130 1.1 Types of data 7 The mark co
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1.2 Typical scientific questions 9
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1.2 Typical scientific questions 11
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13 We’ll cover both classical and
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2.4 Marks and covariates 15 Data ar
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2.4 Marks and covariates 17 example
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3.3 Contributed libraries for R 19
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4.4 Licence 21 The response will be
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0.005 4.6 Exploratory data analysis
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4.7 Models 25 E K(r) 0 500 1000 150
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4.8 Multitype point patterns 27 4.8
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4.8 Multitype point patterns 29 e.g
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4.9 Installed datasets 31 PART II.
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5.2 Classes in spatstat 33 Point pa
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5.3 Return values 35 density(swedis
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5.3 Return values 37 Many of the fu
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6.2 Creating a ppp object 39 To rea
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6.4 Categorical marks 41 longleaf T
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6.6 Checking data 43 > par(mfrow =
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45 7 Converting from GIS formats Th
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8.1 Making windows by hand 47 8.1.3
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8.2 Converting from GIS formats 49
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8.5 Creating a point pattern in any
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53 9 Manipulating point patterns Be
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9.2 Operations on ppp objects 55 [1
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9.2 Operations on ppp objects 57 ma
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9.3 Example 59 You may also notice
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9.5 List of operations on point pat
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63 10 Pixel images in spatstat An o
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10.2 Inspecting an image 65 > f w
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10.2 Inspecting an image 67 > plot(
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10.3 Manipulating images 69 0 200 6
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71 > W V 3) > U 3, 42, Z)) Other
Page 73 and 74: 11.3 Operations involving a tessell
Page 79 and 80: 12.2 Inhomogeneous intensity 79 12.
Page 81 and 82: 12.2 Inhomogeneous intensity 81 den
Page 83 and 84: 13.3 Relative distribution estimate
Page 85 and 86: 1 1 2 4 4 1 2 13.4 Distance map 85
Page 87 and 88: 13.4 Distance map 87 PART IV. POISS
Page 89 and 90: 14.2 Quadrat counting tests for CSR
Page 91 and 92: 14.4 Kolmogorov-Smirnov test of CSR
Page 93 and 94: 14.5 Using covariate data 93 Becaus
Page 95 and 96: 95 15 Maximum likelihood for Poisso
Page 97 and 98: 15.3 Fitting Poisson processes in s
Page 99 and 100: 15.4 Fitted models 99 > ppm(bei, ~s
Page 101 and 102: 15.4 Fitted models 101 > predict(fi
Page 103 and 104: 15.4 Fitted models 103 effectfun(fi
Page 105 and 106: 15.5 Simulating the fitted model 10
Page 107 and 108: 16.2 Validation using residuals 107
Page 113 and 114: 113 PART V. INTERACTION Part V of t
Page 115 and 116: 115 independent + + + + + + + + + +
Page 117 and 118: 19.2 Empty space distances 117 Anot
Page 119 and 120: 19.2 Empty space distances 119 cell
Page 121 and 122: 19.2 Empty space distances 121 Fest
Page 123: 19.3 Nearest neighbour distances 12
Page 127 and 128: 19.4 Pairwise distances and the K f
Page 129 and 130: 19.6 Manipulating and plotting summ
Page 131 and 132: 19.7 Caveats 131 Kest(X) X K(r) 0.0
Page 133 and 134: 20.1 Envelopes and Monte Carlo test
Page 139 and 140: 139 21 Spatial bootstrap methods Sp
Page 141 and 142: 22.2 Cox processes 141 22.2 Cox pro
Page 143 and 144: 22.4 Sequential models 143 rSSI(0.0
Page 145 and 146: 23.1 Fitting a cluster process 145
Page 147 and 148: 23.4 Generic algorithm for minimum
Page 149 and 150: 149 In data sharpening [27] the poi
Page 151 and 152: 25.2 Inhomogeneous cluster models 1
Page 153 and 154: 25.4 Local scaling 153 > data(bei)
Page 155 and 156: 25.4 Local scaling 155 PART VI. GIB
Page 157 and 158: 26.3 Pairwise interaction models 15
Page 159 and 160: 26.4 Higher-order interactions 159
Page 161 and 162: 26.6 Simulating Gibbs models 161 St
Page 163 and 164: 27.2 Fitting Gibbs models in spatst
Page 165 and 166: 27.3 Interpoint interactions 165 27
Page 167 and 168: 27.4 Fitted point process models 16
Page 169 and 170: 27.6 Dealing with nuisance paramete
Page 171 and 172: 171 Stationary Strauss process Firs
Page 173 and 174: 28.2 Residuals for Gibbs processes
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28.2 Residuals for Gibbs processes
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28.3 New methods 177 PART VII. MARK
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29.3 Methodological issues 179 A ma
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181 Chorley−Ribble Data For furth
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30.2 Inspecting a marked point patt
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30.3 Manipulating data 185 > data(l
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187 31 Exploratory tools for multit
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31.2 Simple summaries of neighbouri
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31.3 Distance methods and summary f
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31.4 Randomisation tests 197 31.4 R
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31.4 Randomisation tests 199 > plot
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32.1 Numeric marks: distribution an
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32.3 Reverse conditional moments 20
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33.2 Simulation 205 33.2 Simulation
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33.3 Fitting Poisson models 207 33.
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33.3 Fitting Poisson models 209 whe
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34.1 Gibbs models 211 34.1.3 Pairwi
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34.3 Fitting Gibbs models to multit
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34.3 Fitting Gibbs models to multit
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217 There are also the usual method
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219 lty col key label meaning iso 1
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221 38 Replicated data and hyperfra
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223 H H(r) 0.0 0.2 0.4 0.6 0.8 km h
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REFERENCES 225 References [1] F.P.
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REFERENCES 227 [32] P.J. Diggle. Bi
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Index analysis of deviance, 103 are
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INDEX 231 models, 25, 224 Monte Car
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Analysing spatial point patterns in R - CSIRO

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