Analysing spatial point patterns in R - CSIRO

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180 Marked point patterns independence of components: the sub-processes X m of points of each mark m, are independent point processes; complete spatial randomness and independence (CSRI): the locations X are a uniform Poisson point process, and the marks are independent and identically distributed. (This implies both random labelling and independence of components). These null hypotheses are not equivalent. The properties of random labelling and independence of components are not equivalent. For example, take a point process X where nearest neighbour distances are always larger than a threshold r, and attach random marks to the points. The resulting marked point process cannot be generated using the independence construction, because if points with different marks are independent, they can come arbitrarily close to one another. Example 18 (Ant nests data) Two species of ants build nests in a desert. We want to investigate ecological interaction between the species, and between different nests of the same species. The locations of all nests are mapped, and marked by the species. These data can be analysed as a marked point process consisting of two different types of points. The ‘mark’ attached to each point is its species (a categorical variable). The most natural kind of modelling and analysis is either joint [X,M] or split by species [M] [X|M]. We could also treat one of the species as a covariate and analyse the other species conditional on it. Example 19 Trees in an orchard are examined and their disease status (infected/not infected) is recorded. We are interested in the spatial characteristics of the disease, such as contagion between neighbouring trees. These data probably should not be treated as a point process. The response is ‘disease status’. We can think of disease status as a label applied to the trees after their locations have been determined. Since we are interested in the spatial correlation of disease status, the tree locations are effectively fixed covariate values. It would probably be best to treat these data as a discrete random field (of disease status values) observed at a finite known set of sites (the trees). 29.3.3 Grey areas There are some ‘grey areas’ which permit several alternative choices of analysis. It could be appropriate either to analyse the locations and marks jointly (denoted [X,M]), or to analyse the marks conditional on the locations ([M|X]) or to analyse the locations given the marks ([X|M]). One grey area occurs when the locations are random, but may be ancillary for the parameters of interest. Example 20 Case-control study of cancer [34, 38]. The domicile locations of all new cases of a rare cancer are mapped. To allow for spatial variation in the density of the susceptible population, domicile locations are recorded for a random sample of (matched) controls. This can be analysed either as a marked point pattern (where the mark is the case/control label) or, by conditioning on locations, as a random field of case/control values attached to the known domicile locations. CopyrightCSIRO 2010
181 Chorley−Ribble Data For further discussion of these issues, see [3]. 30 Handling marked point pattern data This section explains how to create a marked point pattern dataset in spatstat, and how to manipulate it. 30.1 Creating marked point pattern datasets The marks attached to a point pattern may be stored in a vector (with one entry for each point) or in a matrix or data frame (with one row for each point and one column for each mark variable). The mark values can be of any atomic type: numeric, integer, character, factor, logical or complex. A marked point pattern dataset can be created using any of the following tools: ppp create point pattern dataset as.ppp convert other data to point pattern superimpose combine several point patterns into a marked point pattern marks extract marks from a point pattern marks ppp(x, y, ..., marks = m) wherexand y are vectors of equal length containing the (x,y) coordinates, m is either a vector of the same length as x containing the mark values for each point, or a matrix or data frame with nrow(m) = length(x) containing the multivariate mark for each point, and ... are arguments that determine the window for the point pattern. 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
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 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: 29.3 Methodological issues 179 A ma
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
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
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INDEX 231 models, 25, 224 Monte Car
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