Analysing spatial point patterns in R - CSIRO

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108 Checking a fitted Poisson model These residuals are closely related to the residuals for quadrat counts that were used above. Taking the set B to be one of our quadrats, the ‘observed’ quadrat count is n(x ∩ B). The ‘expected’ quadrat count is ̂λarea(B) if the model is CSR, or more generally ∫ B ̂λ(u)du if the model is an inhomogeneous Poisson process. Hence the ‘raw residual’ is observed -- expected = n(x ∩ B) − ∫ B ̂λ(u)du. 16.2.2 Residual measure Equation (11) defines the total residual for any region B, large or small. Intuitively the residuals can be visualised as an electric charge, with unit positive charge at each data point, and a diffuse negative charge at all other locations u, with density ˆλ(u). If the model is true, then these charges should approximately cancel. If we’d like to visualise this electric charge, one way is to plot the observed points and the fitted intensity function together: > data(bei) > fit plot(predict(fit)) > plot(bei, add = TRUE, pch = "+") predict(fit) ++ ++ + + ++ + ++ + +++ ++ + +++ ++ ++ + ++ ++ ++ ++ + + ++ + ++++ ++ ++ ++ + + ++ ++ + +++ + + + +++ + ++ ++ + + +++ ++ + + + ++ ++ ++ + ++ + ++ ++ ++ ++ +++++ + ++ ++ + ++ +++ ++ + ++ ++ + ++ ++ ++ ++ ++ + ++ ++ + + + + ++ + ++ ++++ + ++ ++ + ++ + ++ ++ + ++ ++ ++ ++++ ++ +++ ++ ++ ++ ++ + +++ ++ ++ + ++ + ++ +++++ + ++ ++ ++ ++ + +++ ++ + ++ + ++ + ++ +++ ++ + +++ ++ ++ + + + + + + + ++ + ++ ++ +++ +++ +++ ++ + +++ ++++ + ++ + + + + ++ ++ + ++ + + + ++ ++ + + ++++ + + + + + +++ ++ ++ + + + + ++++ + + + + + + ++ + + + + + + + + + + + + ++ + +++ + + ++ + + + ++ + + + + + + + + + + + ++ + + ++ ++ + + + + + + + ++ + ++ + ++ + +++ ++ + ++ + +++ ++ + + ++ ++++ ++ + + +++ + + ++ ++ ++ + + + + +++ ++ + + + + ++ ++ ++++ + ++ ++ ++ ++ ++ + ++ + +++ + ++ ++ ++ + ++ + ++ + ++ ++ + + ++ + ++ +++ ++ ++ ++ + + + + + +++ + +++ ++ ++ ++ + + ++ + ++ + ++ + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + ++ ++++ + +++ ++ ++ +++ +++ ++ ++ + + + + + +++ + + ++ ++ + ++ + + + ++ + ++ ++ + ++ ++ ++ + + + +++ ++ + + ++++ + + + + + + ++ + ++++++ +++ + ++ + ++ + +++ +++ + ++ + +++ +++ + + + ++ ++ ++ ++ + + + + + + ++ + + + + ++ + + + + + + + ++ + + + + + ++ + + + + + + + + + + + ++++ + + + + + + ++ + + + ++ + + + + ++ + + + ++ + + + + + + + + + + ++ + + ++ + + ++ +++ ++++ ++ + ++ ++ ++ +++ + ++ + +++ +++ ++ + + + + + + + + ++ ++ + + ++ + + + + + + +++ ++ ++ ++ ++ ++++ +++ + + ++ + +++ ++ + + ++ ++ ++ ++ +++ ++ + ++ + ++ ++ ++ + + + + ++ +++ ++ ++ +++ ++ ++++ ++ ++ ++ + ++ + ++ + ++ + ++ + +++ + + +++ ++ ++ + + + + + + + ++ + + + + + ++ + + + + + + + ++ + + + + +++ + ++ + ++ + + + + + ++ +++ ++++ ++ + + + ++ + + + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + ++ + + + ++ +++ + + + + + + ++ + + + + + + + + + 0.004 0.006 0.008 0.01 0.012 Each data point should be visualised as a charge of +1, while the colour image indicates a negative charge density. If the model is true then these positive and negative charges should even out to zero. 16.2.3 Smoothed residuals A more useful way to visualise the residuals is to smooth them. > data(bei) > fitx diagnose.ppm(fitx, which = "smooth") Smoothed raw residuals 0.006 0.005 0.003 0.004 0.003 0.002 0.001 0 −0.001 −0.002 −0.003 −0.004 −0.003 −0.002 −0.001 0 0.001 0.003 0.002 0.002 0.001 0 −0.001 0.004 0.005 0.006 CopyrightCSIRO 2010
16.2 Validation using residuals 109 This is an image plot of the ‘smoothed residual field’ where ̂λ(u) is the nonparametric, kernel estimate of the intensity, s(u) = ̂λ(u) − λ † (u) (12) n(x) ∑ ̂λ(u) = e(u) κ(u − x i ) i=1 while λ † (u) is a correspondingly-smoothed version of the parametric estimate of the intensity according to the fitted model, ∫ λ † (u) = e(u) κ(u − v)λˆθ(v)dv. W Here κ is the smoothing kernel and e(u) is the edge correction (2) on page 79. The difference (12) should be approximately zero if the model is true. In this example the smoothed residual image contains a visible trend, suggesting that the model is inappropriate. 16.2.4 Lurking variable plot If there is a spatial covariate Z(u) that plays an important role in the analysis, it may be useful to display a lurking variable plot of the residuals against Z. This is a plot of C(z) = R(B(z)) against z, where B(z) = {u ∈ W : Z(u) ≤ z} is the region of space where the covariate value is less than or equal to z. > grad lurking(fitx, grad, type = "raw") cumulative raw residuals −600 −400 −200 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 covariate Note that the lurking variable plot typically starts and ends at the horizontal axis, since (for any model with an intercept term) the total residual for the entire window W must equal zero. This is analogous to the fact that the residuals in linear regression sum to zero. The plot also shows approximate 5% significance bands for the cumulative residual C(x) or C(y), obtained from the asymptotic variance under the model. 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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Page 59 and 60: 9.3 Example 59 You may also notice
Page 61 and 62: 9.5 List of operations on point pat
Page 63 and 64: 63 10 Pixel images in spatstat An o
Page 65 and 66: 10.2 Inspecting an image 65 > f w
Page 67 and 68: 10.2 Inspecting an image 67 > plot(
Page 69 and 70: 10.3 Manipulating images 69 0 200 6
Page 71 and 72: 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: 16.2 Validation using residuals 107
Page 111 and 112: 16.2 Validation using residuals 111
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 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
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26.4 Higher-order interactions 159
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26.6 Simulating Gibbs models 161 St
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27.2 Fitting Gibbs models in spatst
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27.3 Interpoint interactions 165 27
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27.4 Fitted point process models 16
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27.6 Dealing with nuisance paramete
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171 Stationary Strauss process Firs
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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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