Analysing spatial point patterns in R - CSIRO

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208 Multitype Poisson models You’ll notice that the parameter estimates ̂β m coincide with those obtained fromsummary.ppp above. That is a consequence of the fact that the maximum likelihood estimates (obtained by ppm) are also the method-of-moments estimates (obtained by summary.ppp). A more complicated example is > ppm(lansing, ~marks + x) Nonstationary multitype Poisson process Possible marks: blackoak hickory maple misc redoak whiteoak Trend formula: ~marks + x Fitted coefficients for trend formula: (Intercept) markshickory marksmaple marksmisc marksredoak 4.94294727 1.65008211 1.33694849 -0.25131442 0.94116400 markswhiteoak x 1.19951845 -0.07581624 This is the marked Poisson process whose intensity function λ((x,y,m)) at location (x,y) and mark m satisfies log λ((x,y,m)) = α m + βx where α 1 ,...,α 6 and β are parameters. The intensity is loglinear in x, with a different intercept for each mark, but the same slope (“parallel loglinear regression”). In the printout above, the fitted slope parameter β is ˆβ =-0.07581624. As discussed in Section 15.3 on page 98, the fitted coefficients α m for the categorical mark are interpreted in the light of the ‘contrasts’ in force. The default is the treatment contrasts, and the first level of the mark is blackoak, so in this case the fitted coefficient for m=blackoak is 4.942947, while the fitted coefficient for m=hickory is 4.942947 + 1.650082 = 6.593029 and so on. > ppm(lansing, ~marks * x) Nonstationary multitype Poisson process Possible marks: blackoak hickory maple misc redoak whiteoak Trend formula: ~marks * x Fitted coefficients for trend formula: (Intercept) markshickory marksmaple marksmisc marksredoak 5.2378062 1.4424915 0.6795604 -0.8482907 0.6916392 markswhiteoak x markshickory:x marksmaple:x marksmisc:x 1.0901772 -0.7063987 0.4511157 1.3243326 1.2138278 marksredoak:x markswhiteoak:x 0.5380413 0.2421379 The symbol * here is an ‘interaction’ in the usual sense for linear models. The fitted model is the marked Poisson process with log λ((x,y,m)) = α m + β m x CopyrightCSIRO 2010
33.3 Fitting Poisson models 209 where α 1 ,... ,α 6 and β 1 ,...,β 6 are parameters. The intensity is loglinear in x with a different slope and intercept for each mark. The result of ppm is again an object of class "ppm" representing a fitted point process model. To plot the fitted intensity and conditional intensity of the fitted model, use plot.ppm. For a multitype point process you will get a separate plot for each possible mark value. More complicated examples are: > ppm(lansing, ~marks * polynom(x, y, 2)) > ppm(lansing, ~marks * harmonic(x, y, 2)) 33.3.4 Facilities available A fitted multitype Poisson process model can be manipulated using any of the methods available for the class ppm: print print basic information summary print detailed summary information plot plot the fitted (conditional) intensity predict fitted (conditional) intensity fitted fitted (conditional) intensity at data points update re-fit the model coef extract the fitted coefficient vector ̂θ vcov variance-covariance matrix of ̂θ anova analysis of deviance logLik evaluate log-pseudolikelihood model.matrix extract design matrix formula extract trend formula of model terms extract terms in model formula The following functions are also available: step stepwise model selection drop1 one step backward in model selection model.images compute images of canonical covariates in model effectfun fitted intensity as function of one covariate A fitted multitype Poisson process model can be simulated automatically using rmh.ppm or simulate.ppm. 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 + + + + + + + + + +
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19.2 Empty space distances 117 Anot
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19.2 Empty space distances 119 cell
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19.2 Empty space distances 121 Fest
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19.3 Nearest neighbour distances 12
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19.4 Pairwise distances and the K f
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19.4 Pairwise distances and the K f
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19.6 Manipulating and plotting summ
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19.7 Caveats 131 Kest(X) X K(r) 0.0
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20.1 Envelopes and Monte Carlo test
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139 21 Spatial bootstrap methods Sp
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22.2 Cox processes 141 22.2 Cox pro
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22.4 Sequential models 143 rSSI(0.0
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23.1 Fitting a cluster process 145
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23.4 Generic algorithm for minimum
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149 In data sharpening [27] the poi
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25.2 Inhomogeneous cluster models 1
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25.4 Local scaling 153 > data(bei)
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25.4 Local scaling 155 PART VI. GIB
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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 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: 33.3 Fitting Poisson models 207 33.
Page 211 and 212: 34.1 Gibbs models 211 34.1.3 Pairwi
Page 213 and 214: 34.3 Fitting Gibbs models to multit
Page 215 and 216: 34.3 Fitting Gibbs models to multit
Page 217 and 218: 217 There are also the usual method
Page 219 and 220: 219 lty col key label meaning iso 1
Page 221 and 222: 221 38 Replicated data and hyperfra
Page 223 and 224: 223 H H(r) 0.0 0.2 0.4 0.6 0.8 km h
Page 225 and 226: REFERENCES 225 References [1] F.P.
Page 227 and 228: REFERENCES 227 [32] P.J. Diggle. Bi
Page 229 and 230: Index analysis of deviance, 103 are
Page 231 and 232: INDEX 231 models, 25, 224 Monte Car
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