Analysing spatial point patterns in R - CSIRO

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168 Fitting Gibbs models 27.5 Simulation from fitted models A fitted Gibbs model can also be simulated automatically using rmh. > fit Xsim plot(Xsim, main = "Simulation from fitted Strauss model") Simulation from fitted Strauss model The envelope command will also generate simulation envelopes for a fitted model. > plot(envelope(fit, nsim = 39)) envelope(fit, nsim = 39) K(r) 0 500 1000 1500 obs mmean hi lo 0 5 10 15 20 r (one unit = 0.1 metres) 27.6 Dealing with nuisance parameters Irregular parameters, such as the interaction radius r in the Strauss process, cannot be estimated directly using ppm. Indeed the statistical theory for estimating such parameters is unclear. For some special cases, a maximum likelihood estimator of the nuisance parameter is available. For example, for the ‘hard core process’ (Strauss process with interaction parameter γ = 0) with interaction radius r, the maximum likelihood estimator is the minimum nearest-neighbour distance. Thus the following is a reasonable approach to the cells dataset: > rhat rhat ppm(cells, ~1, Strauss(r = rhat)) CopyrightCSIRO 2010
27.6 Dealing with nuisance parameters 169 Stationary Strauss process First order term: beta 301.0949 Interaction: Strauss process interaction distance: 0.0836293018068393 Fitted interaction parameter gamma: 0 Relevant coefficients: Interaction -20.77031 The analogue of profile likelihood, profile pseudolikelihood, provides a general solution which may or may not perform well. If θ = (φ,η) where φ denotes the nuisance parameters and η the regular parameters, define the profile log pseudolikelihood by PPL(φ,x) = max log PL ((φ,η);x) . η The right hand side can be computed, for each fixed value of φ, by the algorithm ppm. Then we just have to maximise PPL(φ) over φ. This is done by the command profilepl: > data(simdat) > df pfit pfit Profile log pseudolikelihood values for model: ppm(simdat, ~1, interaction = Strauss) fitted with rbord= 2 Interaction: Strauss with irregular parameter r in [0.05, 2] Optimum value of irregular parameter: r = 0.275 The result is an object of class profilepl containing the profile log pseudolikelihood function, the optimised value of the irregular parameter r, and the final fitted model. To plot the profile log pseudolikelihood, > plot(pfit) 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
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14.5 Using covariate data 93 Becaus
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95 15 Maximum likelihood for Poisso
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15.3 Fitting Poisson processes in s
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15.4 Fitted models 99 > ppm(bei, ~s
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15.4 Fitted models 101 > predict(fi
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15.4 Fitted models 103 effectfun(fi
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15.5 Simulating the fitted model 10
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16.2 Validation using residuals 107
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113 PART V. INTERACTION Part V of t
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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 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: 27.4 Fitted point process models 16
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 197 and 198: 31.4 Randomisation tests 197 31.4 R
Page 199 and 200: 31.4 Randomisation tests 199 > plot
Page 201 and 202: 32.1 Numeric marks: distribution an
Page 203 and 204: 32.3 Reverse conditional moments 20
Page 205 and 206: 33.2 Simulation 205 33.2 Simulation
Page 207 and 208: 33.3 Fitting Poisson models 207 33.
Page 209 and 210: 33.3 Fitting Poisson models 209 whe
Page 211 and 212: 34.1 Gibbs models 211 34.1.3 Pairwi
Page 213 and 214: 34.3 Fitting Gibbs models to multit
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Page 217 and 218: 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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