Analysing spatial point patterns in R - CSIRO

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24 Introduction to spatstat X 7 3 6 5 5 9 7 7 4 3 6 9 Another common example is Ripley’s K function. I’ll explain more about the K function later. For now, we’ll just demonstrate how easy it is to compute and plot it. To compute the K function for a point pattern X, type Kest(X). This returns an object which can be plotted. > K plot(K) K K(r) 0 500 1000 1500 iso trans border theo 0 5 10 15 20 r (one unit = 0.1 metres) In this plot, the empirical K function (solid lines) deviates from the theoretical expected value assuming the points are completely random (dashed lines). To test whether this deviation is statistically significant, the standard approach is to use a Monte Carlo test based on envelopes of the K function obtained from simulated point patterns. In spatstat this is done with the envelope function: > E plot(E) CopyrightCSIRO 2010
4.7 Models 25 E K(r) 0 500 1000 1500 2000 obs theo hi lo 0 5 10 15 20 r (one unit = 0.1 metres) 4.7 Models The main strength of spatstat is that it supports statistical models of point patterns. Models can be fitted to point pattern data; the fitted models can be used to summarise the data or make predictions; the fitted models can be simulated (i.e. a random pattern can be generated according to the model); and there are facilities for model selection, for testing whether a term in the model is required (like analysis of variance), and for model criticism (like residuals, regression diagnostics, and goodness-of-fit tests). Participants in this workshop often say “I’m not interested in modelling my data; I only want to analyse it.” However, any kind of data analysis or data manipulation is equivalent to imposing assumptions. We can’t say something is ‘statistically significant’ unless we assume a model, because the p-value is the probability according to a model. The purpose of statistical modelling is to make these assumptions or hypotheses explicit. By doing so, we are able to determine the best and most powerful way to analyse data, we can subject the assumptions to criticism, and we are more aware of the potential pitfalls of analysis. In statistical usage, a model is always tentative; it is assumed for the sake of argument; we might even want it to be wrong. In the famous words of George Box: “All models are wrong, but some are useful.” If you only want to do data analysis without statistical models, your results will be less informative and more vulnerable to critique. A statistical model for a point pattern is technically termed a point process model. Think of a point process as a black box that generates a random spatial point pattern according to some rules. To fit a point process model to a point pattern dataset in spatstat, use the function ppm (point process model). This is analogous to the standard functions in R for fitting linear models (lm), generalized linear models (glm) and so on. > data(swedishpines) > X fit fit Stationary Strauss process First order term: beta 0.04378316 CopyrightCSIRO 2010
Page 1 and 2: Analysing spatial point patterns in
Page 3 and 4: CONTENTS 3 Contents PART I. OVERVIE
Page 5 and 6: CONTENTS 5 PART I. OVERVIEW The fir
Page 7 and 8: 130 1.1 Types of data 7 The mark co
Page 9 and 10: 1.2 Typical scientific questions 9
Page 11 and 12: 1.2 Typical scientific questions 11
Page 13 and 14: 13 We’ll cover both classical and
Page 15 and 16: 2.4 Marks and covariates 15 Data ar
Page 17 and 18: 2.4 Marks and covariates 17 example
Page 19 and 20: 3.3 Contributed libraries for R 19
Page 21 and 22: 4.4 Licence 21 The response will be
Page 23: 0.005 4.6 Exploratory data analysis
Page 27 and 28: 4.8 Multitype point patterns 27 4.8
Page 29 and 30: 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
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11.3 Operations involving a tessell
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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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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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