Analysing spatial point patterns in R - CSIRO

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82 Dependence of intensity on a covariate 13 Dependence of intensity on a covariate 13.1 Spatial covariates Often we want to know whether the intensity of points depends on the values of a covariate. For example, the tropical rainforest point pattern dataset bei comes with an extra set of covariate data bei.extra, which contains a pixel image of terrain elevation bei.extra$elev and a pixel image of terrain slope bei.extra$grad. It is of interest to determine whether the trees prefer steep or flat terrain, and whether they prefer a particular altitude. > data(bei) > slope par(mfrow = c(1, 2)) > plot(bei) > plot(slope) > par(mfrow = c(1, 1)) bei slope 0 0.05 0.1 0.15 0.2 0.25 0.3 13.2 Quadrats determined by a covariate In quadrat counting methods, any choice of quadrats is permissible. From a theoretical viewpoint, the quadrats do not have to be rectangles of equal area, and could be regions of any shape. Quadrat counting is more useful if we choose the quadrats in a meaningful way. One way to do this is to define the quadrats using covariate information. For the tropical rainforest data bei, it might be useful to split the study region into several sub-regions according to the terrain slope. > data(bei) > Z b Zcut V plot(V) > plot(bei, add = TRUE, pch = "+") CopyrightCSIRO 2010
13.3 Relative distribution estimate 83 V ++ ++ + + ++ + + ++ +++ ++ + + + + + ++ ++ + ++ +++ + + ++ + ++++ ++ ++ ++ ++ +++ ++ + + ++ + + + ++ ++ + + ++ +++ ++ + + +++ ++ + ++ ++ + + + + + + ++ + ++ ++ ++ + + + ++ ++ ++ + + ++ ++ + ++ ++ + ++ ++ +++ ++ + + ++ ++ ++++ ++ ++ + +++ + + + + ++ + ++ + ++++ ++ + + + + + ++ ++ +++ ++ + + + + ++ + ++ + + + + ++ ++ + ++ +++ + ++ ++ + + + +++ + ++ + + + + + + + + ++ ++ ++ ++ ++++ + ++ ++ + + +++ + + + + + + + + + ++ +++ ++ + + + + + + + + + ++ + + ++ + ++ + + + + + + + ++++++ + + ++ ++ +++ + ++ + + ++++ + + + + + ++ + +++ ++ + ++ ++ ++ ++ + ++ +++ ++ + + + ++ ++ ++++ ++ + ++ +++ + ++ ++ + + + + ++ +++ ++ + + ++ + + +++ ++ + ++ ++ + + ++ ++ + +++ + ++ ++ +++ ++ + +++ + +++ ++ + + ++ + ++ ++ ++ ++ ++ ++ + ++ + ++++ ++ ++ ++ + ++ +++ + ++ +++++ + ++ + ++ +++ + + ++ +++ +++ + + ++ +++ +++ +++ ++ + +++ + ++ +++ + + + + ++ ++ + ++ ++ + + + + + + ++ + ++ ++ + + ++++ + + + ++ + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++++ + + + + ++ ++ + + + + ++ + + ++ + + +++ + + + ++ + ++ + + + + + + ++ + ++ + ++ + + +++ +++ ++ ++ + + ++ + ++ ++ ++ + ++ + ++ +++ ++ ++ ++ ++ ++ ++ +++ ++ +++ +++ ++ ++ ++ + + + ++ + ++ + +++ + + + + + ++ + + + + + + + + + + + ++++ + + ++ ++ + + + + + + + + + + + + + + + + + + + + + + + ++ + + + +++ +++ ++ + + + + + + + + ++ + + + + + + + + + + + + + ++ + + + + + + + + + + + + ++ + + + ++ + ++ ++ ++ + + + + + + +++ + + + + + ++ ++ + ++ + + +++ + + ++ ++ + + + + + + + ++ + + + + + + + + + + ++ + ++ + + + + + ++ + + + + ++ + + + + + + + + ++ +++ + ++ ++ + + + + ++ + ++ + +++ + ++ +++ ++++ +++ ++ + + + + + + ++ ++++ + + + + + ++ + + + + ++ ++ + + ++ ++ +++ ++ ++ ++ + + + ++ ++ ++ ++ ++ + + + ++ + + +++ ++ + ++++ ++ ++ + ++ ++ +++ ++ ++ + +++ + ++ + + + + + +++ +++ + ++ + ++ ++ + + + + ++ + ++ + ++ + + + + + + + + + ++ + + + + + + +++ + + + + + + +++ ++++ + ++ ++ ++ +++ + + ++ + + + + + ++ + + + + + + + + + + + +++ + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + ++ ++ + + + + 1 2 3 4 The call to quantile gave us the quartiles of the slope values, so the four tiles in the tessellation V have equal area (ignoring discretisation effects). In other words, we have divided the study region into four zones of equal area according to the terrain slope. We can now use this tessellation to study the point pattern bei. We could invoke the commands split, cut or by to divide the points according to this tessellation and manipulate the sub-patterns. The command quadratcount also works with tessellations: > qb qb tile 1 2 3 4 271 984 1028 1321 > plot(qb) qb 1028 984 271 1321 1 2 3 4 The text annotations show the number of trees in each region. Since the four regions have equal area, the counts should be approximately equal if there is a uniform density of trees. Obviously they are not equal; there appears to be a strong preference for steeper slopes. 13.3 Relative distribution estimate Let us assume that the intensity of the point process is a function of the covariate Z. At any spatial location u, let λ(u) be the intensity of the point process, and Z(u) the value of the covariate. Then we are assuming λ(u) = ρ(Z(u)) where ρ is a function that we want to investigate, telling us how the intensity of points depends on the value of the covariate. Kernel smoothing can be used to estimate the function ρ, using methods of relative distribution or relative risk, as explained in [11, 5]. 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
Page 31 and 32: 4.9 Installed datasets 31 PART II.
Page 33 and 34: 5.2 Classes in spatstat 33 Point pa
Page 35 and 36: 5.3 Return values 35 density(swedis
Page 37 and 38: 5.3 Return values 37 Many of the fu
Page 39 and 40: 6.2 Creating a ppp object 39 To rea
Page 41 and 42: 6.4 Categorical marks 41 longleaf T
Page 43 and 44: 6.6 Checking data 43 > par(mfrow =
Page 45 and 46: 45 7 Converting from GIS formats Th
Page 47 and 48: 8.1 Making windows by hand 47 8.1.3
Page 49 and 50: 8.2 Converting from GIS formats 49
Page 51 and 52: 8.5 Creating a point pattern in any
Page 53 and 54: 53 9 Manipulating point patterns Be
Page 55 and 56: 9.2 Operations on ppp objects 55 [1
Page 57 and 58: 9.2 Operations on ppp objects 57 ma
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: 12.2 Inhomogeneous intensity 81 den
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 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
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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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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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