Cox process

Abstract: Preface List of Examples 1 Introduction 11 Point process statistics 12 Examples of point process data 121 A pattern of amacrine cells 122 Gold particles 123 A pattern of Western Australian plants 124 Waterstriders 125 A sample of concrete 13 Historical notes 131 Determination of number of trees in a forest 132 Number of blood particles in a sample 133 Patterns of points in plant communities 134 Formulating the power law for the pair correlation function for galaxies 14 Sampling and data collection 141 General remarks 142 Choosing an appropriate study area 143 Data collection 15 Fundamentals of the theory of point processes 16 Stationarity and isotropy 161 Model approach and design approach 162 Finite and infinite point processes 163 Stationarity and isotropy 164 Ergodicity 17 Summary characteristics for point processes 171 Numerical summary characteristics 172 Functional summary characteristics 18 Secondary structures of point processes 181 Introduction 182 Random sets 183 Random fields 184 Tessellations 185 Neighbour networks or graphs 19 Simulation of point processes 2 The Homogeneous Poisson point process 21 Introduction 22 The binomial point process 221 Introduction 222 Basic properties 223 The periodic binomial process 224 Simulation of the binomial process 23 The homogeneous Poisson point process 231 Introduction 232 Basic properties 233 Characterisations of the homogeneous Poisson process 24 Simulation of a homogeneous Poisson process 25 Model characteristics 251 Moments and moment measures 252 The Palm distribution of a homogeneous Poisson process 253 Summary characteristics of the homogeneous Poisson process 26 Estimating the intensity 27 Testing complete spatial randomness 271 Introduction 272 Quadrat counts 273 Distance methods 274 The J-test 275 Two index-based tests 276 Discrepancy tests 277 The L-test 278 Other tests and recommendations 3 Finite point processes 31 Introduction 32 Distributions of numbers of points 321 The binomial distribution 322 The Poisson distribution 323 Compound distributions 324 Generalised distributions 33 Intensity functions and their estimation 331 Parametric statistics for the intensity function 332 Non-parametric estimation of the intensity function 333 Estimating the point density distribution function 34 Inhomogeneous Poisson process and finite Cox process 341 The inhomogeneous Poisson process 342 The finite Cox process 35 Summary characteristics for finite point processes 351 Nearest-neighbour distances 352 Dilation function 353 Graph-theoretic statistics 354 Second-order characteristics 36 Finite Gibbs processes 361 Introduction 362 Gibbs processes with fixed number of points 363 Gibbs processes with a random number of points 364 Second-order summary characteristics of finite Gibbs processes 365 Further discussion 366 Statistical inference for finite Gibbs processes 4 Stationary point processes 41 Basic definitions and notation 42 Summary characteristics for stationary point processes 421 Introduction 422 Edge-correction methods 423 The intensity lambda 424 Indices as summary characteristics 425 Empty-space statistics and other morphological summaries 426 The nearest-neighbour distance distribution function 427 The J-function 43 Second-order characteristics 431 The three functions: K, L and g 432 Theoretical foundations of second-order characteristics 433 Estimators of the second-order characteristics 434 Interpretation of pair correlation functions 44 Higher-order and topological characteristics 441 Introduction 442 Third-order characteristics 443 Delaunay tessellation characteristics 444 The connectivity function 45 Orientation analysis for stationary point processes 451 Introduction 452 Nearest-neighbour orientation distribution 453 Second-order orientation analysis 46 Outliers, gaps and residuals 461 Introduction 462 Simple outlier detection 463 Simple gap detection 464 Model-based outliers 465 Residuals 47 Replicated patterns 471 Introduction 472 Aggregation recipes 48 Choosing appropriate observation windows 481 General ideas 482 Representative windows 49 Multivariate analysis of series of point patterns 410 Summary characteristics for the non-stationary case 4101 Formal application of stationary characteristics and estimators 4102 Intensity reweighting 4103 Local rescaling 5 Stationary marked point processes 51 Basic definitions and notation 511 Introduction 512 Marks and their properties 513 Marking models 514 Stationarity 515 First-order characteristics 516 Mark-sum measure 517 Palm distribution 52 Summary characteristics 521 Introduction 522 Intensity and mark-sum intensity 523 Mean mark, mark df and mark probabilities 524 Indices for stationary marked point processes 525 Nearest-neighbour distributions 53 Second-order characteristics for marked point processes 531 Introduction 532 Definitions for qualitative marks 533 Definitions for quantitative marks 534 Estimation of second-order characteristics 54 Orientation analysis for marked point processes 541 Introduction 542 Orientation analysis for non-isotropic processes with angular marks 543 Orientation analysis for isotropic processes with angular marks 544 Orientation analysis with constructed marks 6 Modelling and simulation of stationary point processes 61 Introduction 62 Operations with point processes 621 Thinning 622 Clustering 623 Superposition 63 Cluster processes 631 General cluster processes 632 Neyman-Scott processes 64 Stationary Cox processes 641 Introduction 642 Properties of stationary Cox processes 65 Hard-core point processes 651 Introduction 652 Matern hard-core processes 653 The dead leaves model 654 The RSA model 655 Random dense packings of hard spheres 66 Stationary Gibbs processes 661 Basic ideas and equations 662 Simulation of stationary Gibbs processes 663 Statistics for stationary Gibbs processes 67 Reconstruction of point patterns 671 Reconstructing point patterns without a specified model 672 An example: reconstruction of Neyman-Scott processes 673 Practical application of the reconstruction algorithm 68 Formulas for marked point process models 681 Introduction 682 Independent marks 683 Random field model 684 Intensity-weighted marks 69 Moment formulas for stationary shot-noise fields 610 Space-time point processes 6101 Introduction 6102 Space-time Poisson processes 6103 Second-order statistics for completely stationary event processes 6104 Two examples of space-time processes 611 Correlations between point processes and other random structures 6111 Introduction 6112 Correlations between point processes and random fields 6113 Correlations between point processes and fibre processes 7 Fitting and testing point process models 71 Choice of model 72 Parameter estimation 721 Maximum likelihood method 722 Method of moments 723 Trial-and-error estimation 73 Variance estimation by bootstrap 74 Goodness-of-fit tests 741 Envelope test 742 Deviation test 75 Testing mark hypotheses 751 Introduction 752 Testing independent marking, test of association 753 Testing geostatistical marking 76 Bayesian methods for point pattern analysis Appendix A Fundamentals of statistics Appendix B Geometrical characteristics of sets Appendix C Fundamentals of geostatistics References Notation index Author index Subject index