Analysing spatial point patterns in R - CSIRO

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144 Model-fitting using summary statistics 23 Model-fitting using summary statistics Summary statistics can also be used to fit models to data. In the ‘method of moments’ we estimate a parameter θ by solving E θ [S(X)] = S(x) where S(x) is the observed value of a statistic S for our data x, and the left side is the theoretical mean of S for the model governed by parameter θ. The analogue for point process models is to fit the model by matching a summary statistic such as the K function to its theoretical value under the model. 23.1 Fitting a cluster process In a precious few cases, the K function of a point process is known exactly, as an analytic expression in terms of the model parameters. These happy cases include many Neyman-Scott cluster processes. For example, the K-function of the Thomas process (Section 22.1, page 140) with parameters θ = (κ,µ,σ) is K θ (r) = πr 2 + 1 κ r2 (1 − exp(− 4σ2)). (26) We can use this to fit a Thomas model to data. We determine the values of the parameters θ = (κ,µ,σ) to achieve the best match between K θ (r) and the estimated K-function of the data, ̂K(r). The best match is determined by minimising the discrepancy between the two functions over some range [a,b]: ∫ b D(θ) = ∣ ̂K(r) ∣ q − K θ (r) q ∣∣ p dr (27) a where 0 ≤ a < b, and where p,q > 0 are indices. This method was originally advocated by Peter Diggle and collaborators, and is now known as the method of minimum contrast. See [35]. The command kppm fits cluster point process models by the method of minimum contrast. To fit the Thomas model to the redwood data: > data(redwood) > fit fit Stationary cluster point process model Fitted to point pattern dataset redwood Fitted using the K-function Cluster model: Thomas process Fitted parameters: kappa sigma mu 23.55389789 0.04704965 2.63226071 CopyrightCSIRO 2010
23.1 Fitting a cluster process 145 > plot(fit) fit K−function K(r) 0.00 0.05 0.10 0.15 0.20 fit iso trans border theo 0.00 0.05 0.10 0.15 0.20 0.25 r The plot shows the theoretical K function of the fitted Thomas process (fit), three nonparametric estimates of the K function (iso, trans, border) and the Poisson K function (theo). At present, the only cluster process models that can be fitted using kppm are the Thomas process and the Matérn cluster process. To fit the Matérn cluster process to the redwood data, > fitM plot(simulate(fit, nsim = 4)) The command simulate is generic; here we have used the method simulate.kppm. Simulation envelopes are also available, using the method envelope.kppm. > plot(envelope(fit, Lest, nsim = 39)) 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
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11.3 Operations involving a tessell
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12.2 Inhomogeneous intensity 79 12.
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12.2 Inhomogeneous intensity 81 den
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13.3 Relative distribution estimate
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1 1 2 4 4 1 2 13.4 Distance map 85
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13.4 Distance map 87 PART IV. POISS
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14.2 Quadrat counting tests for CSR
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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 and 124: 19.3 Nearest neighbour distances 12
Page 125 and 126: 19.4 Pairwise distances and the K f
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: 22.4 Sequential models 143 rSSI(0.0
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
Page 175 and 176: 28.2 Residuals for Gibbs processes
Page 177 and 178: 28.3 New methods 177 PART VII. MARK
Page 179 and 180: 29.3 Methodological issues 179 A ma
Page 181 and 182: 181 Chorley−Ribble Data For furth
Page 183 and 184: 30.2 Inspecting a marked point patt
Page 185 and 186: 30.3 Manipulating data 185 > data(l
Page 187 and 188: 187 31 Exploratory tools for multit
Page 189 and 190: 31.2 Simple summaries of neighbouri
Page 191 and 192: 31.3 Distance methods and summary f
Page 193 and 194: 31.3 Distance methods and summary f
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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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