Analysing spatial point patterns in R - CSIRO

More documents

Recommendations

Info

94 Tests of Complete Spatial Randomness 14.6 Berman’s tests Berman [20] proposed two tests for the dependence of a point process on a spatial covariate. These tests are optimal against a certain class of alternatives. They are performed by the command bermantest which is analogous to kstest. > B plot(B) > B Berman Z1 test of CSR data: covariate Z evaluated at points of bei Z1 = 10.844, p-value < 2.2e-16 alternative hypothesis: two-sided Berman Z1 test of CSR based on distribution of covariate ...Z... Z1 statistic = 10.84 p−value= 2.13e−27 probability 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Z Two vertical lines show the mean values of these distributions. If the model is correct, the two curves should be close; the test is based on comparing the two vertical lines. When the covariate Z is the distance to a spatial pattern, another useful diagnostic is Foxall’s J-function [40], available using Jfox. CopyrightCSIRO 2010
95 15 Maximum likelihood for Poisson processes If we are willing to assume (tentatively) that the points are independent, then we can apply some decent statistical methods to the investigation of the intensity. 15.1 Inhomogeneous Poisson process The inhomogeneous Poisson process with intensity function λ(u), u ∈ R 2 , is a modification of the homogeneous Poisson process, in which properties (PP2) and (PP4) above are replaced by (PP2’): the number N(X ∩ B) of points falling in a region B has expectation ∫ E[N(X ∩ B)] = λ(u)du. (PP4’): given that N(X ∩ B) = n, the n points are independent and identically distributed, with common probability density f(u) = λ(u)/I, where I = ∫ B λ(u)du. This process can also be simulated using rpoispp using the same properties. The intensity argumentlambda can be a constant, a function(x,y) giving the values of the intensity function at coordinates x,y, or a pixel image containing the intensity values at a grid of locations. > lambda plot(rpoispp(lambda)) B rpoispp(lambda) If we discretise an inhomogeneous Poisson process, the indicators I are independent, but have unequal success probabilities, P {I(u) = 1} = λ(u)dA. The inhomogeneous Poisson process is a plausible model for point patterns under several scenarios. One is random thinning: suppose that a homogeneous Poisson process of intensity β is generated, and that each point is either deleted or retained, independently of other points. Suppose the probability of retaining a point at the location u is p(u). Then the resulting process of retained points is inhomogeneous Poisson, with intensity λ(u) = βp(u). Consider, for example, a model of plant propagation which assumes that seeds are randomly dispersed according to a Poisson process, and seeds randomly germinate or do not germinate, independently of each other, with a germination probability that depends on the local soil conditions. The resulting pattern of plants is an inhomogeneous Poisson process. 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 and 24:
0.005 4.6 Exploratory data analysis
Page 25 and 26:
4.7 Models 25 E K(r) 0 500 1000 150
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
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: 14.5 Using covariate data 93 Becaus
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
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 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 193 and 194:
Page 195 and 196:
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
Page 231 and 232:
INDEX 231 models, 25, 224 Monte Car
show all

Analysing spatial point patterns in R - CSIRO

Create successful ePaper yourself

Delete template?

Save as template?