Janine B. Illian

Bio: Janine B. Illian is an academic researcher from University of St Andrews. The author has contributed to research in topics: Cox process & Point process. The author has an hindex of 25, co-authored 77 publications receiving 4828 citations. Previous affiliations of Janine B. Illian include Abertay University & University of Glasgow.

Statistical Analysis and Modelling of Spatial Point Patterns: Illian/Statistical Analysis and Modelling of Spatial Point Patterns

Abstract: Preface. List of Examples. 1. Introduction. 1.1 Point process statistics. 1.2 Examples of point process data. 1.2.1 A pattern of amacrine cells. 1.2.2 Gold particles. 1.2.3 A pattern of Western Australian plants. 1.2.4 Waterstriders. 1.2.5 A sample of concrete. 1.3 Historical notes. 1.3.1 Determination of number of trees in a forest. 1.3.2 Number of blood particles in a sample. 1.3.3 Patterns of points in plant communities. 1.3.4 Formulating the power law for the pair correlation function for galaxies. 1.4 Sampling and data collection. 1.4.1 General remarks. 1.4.2 Choosing an appropriate study area. 1.4.3 Data collection. 1.5 Fundamentals of the theory of point processes. 1.6 Stationarity and isotropy. 1.6.1 Model approach and design approach. 1.6.2 Finite and infinite point processes. 1.6.3 Stationarity and isotropy. 1.6.4 Ergodicity. 1.7 Summary characteristics for point processes. 1.7.1 Numerical summary characteristics. 1.7.2 Functional summary characteristics. 1.8 Secondary structures of point processes. 1.8.1 Introduction. 1.8.2 Random sets. 1.8.3 Random fields. 1.8.4 Tessellations. 1.8.5 Neighbour networks or graphs. 1.9 Simulation of point processes. 2. The Homogeneous Poisson point process. 2.1 Introduction. 2.2 The binomial point process. 2.2.1 Introduction. 2.2.2 Basic properties. 2.2.3 The periodic binomial process. 2.2.4 Simulation of the binomial process. 2.3 The homogeneous Poisson point process. 2.3.1 Introduction. 2.3.2 Basic properties. 2.3.3 Characterisations of the homogeneous Poisson process. 2.4 Simulation of a homogeneous Poisson process. 2.5 Model characteristics. 2.5.1 Moments and moment measures. 2.5.2 The Palm distribution of a homogeneous Poisson process. 2.5.3 Summary characteristics of the homogeneous Poisson process. 2.6 Estimating the intensity. 2.7 Testing complete spatial randomness. 2.7.1 Introduction. 2.7.2 Quadrat counts. 2.7.3 Distance methods. 2.7.4 The J-test. 2.7.5 Two index-based tests. 2.7.6 Discrepancy tests. 2.7.7 The L-test. 2.7.8 Other tests and recommendations. 3. Finite point processes. 3.1 Introduction. 3.2 Distributions of numbers of points. 3.2.1 The binomial distribution. 3.2.2 The Poisson distribution. 3.2.3 Compound distributions. 3.2.4 Generalised distributions. 3.3 Intensity functions and their estimation. 3.3.1 Parametric statistics for the intensity function. 3.3.2 Non-parametric estimation of the intensity function. 3.3.3 Estimating the point density distribution function. 3.4 Inhomogeneous Poisson process and finite Cox process. 3.4.1 The inhomogeneous Poisson process. 3.4.2 The finite Cox process. 3.5 Summary characteristics for finite point processes. 3.5.1 Nearest-neighbour distances. 3.5.2 Dilation function. 3.5.3 Graph-theoretic statistics. 3.5.4 Second-order characteristics. 3.6 Finite Gibbs processes. 3.6.1 Introduction. 3.6.2 Gibbs processes with fixed number of points. 3.6.3 Gibbs processes with a random number of points. 3.6.4 Second-order summary characteristics of finite Gibbs processes. 3.6.5 Further discussion. 3.6.6 Statistical inference for finite Gibbs processes. 4. Stationary point processes. 4.1 Basic definitions and notation. 4.2 Summary characteristics for stationary point processes. 4.2.1 Introduction. 4.2.2 Edge-correction methods. 4.2.3 The intensity lambda. 4.2.4 Indices as summary characteristics. 4.2.5 Empty-space statistics and other morphological summaries. 4.2.6 The nearest-neighbour distance distribution function. 4.2.7 The J-function. 4.3 Second-order characteristics. 4.3.1 The three functions: K, L and g. 4.3.2 Theoretical foundations of second-order characteristics. 4.3.3 Estimators of the second-order characteristics. 4.3.4 Interpretation of pair correlation functions. 4.4 Higher-order and topological characteristics. 4.4.1 Introduction. 4.4.2 Third-order characteristics. 4.4.3 Delaunay tessellation characteristics. 4.4.4 The connectivity function. 4.5 Orientation analysis for stationary point processes. 4.5.1 Introduction. 4.5.2 Nearest-neighbour orientation distribution. 4.5.3 Second-order orientation analysis. 4.6 Outliers, gaps and residuals. 4.6.1 Introduction. 4.6.2 Simple outlier detection. 4.6.3 Simple gap detection. 4.6.4 Model-based outliers. 4.6.5 Residuals. 4.7 Replicated patterns. 4.7.1 Introduction. 4.7.2 Aggregation recipes. 4.8 Choosing appropriate observation windows. 4.8.1 General ideas. 4.8.2 Representative windows. 4.9 Multivariate analysis of series of point patterns. 4.10 Summary characteristics for the non-stationary case. 4.10.1 Formal application of stationary characteristics and estimators. 4.10.2 Intensity reweighting. 4.10.3 Local rescaling. 5. Stationary marked point processes. 5.1 Basic definitions and notation. 5.1.1 Introduction. 5.1.2 Marks and their properties. 5.1.3 Marking models. 5.1.4 Stationarity. 5.1.5 First-order characteristics. 5.1.6 Mark-sum measure. 5.1.7 Palm distribution. 5.2 Summary characteristics. 5.2.1 Introduction. 5.2.2 Intensity and mark-sum intensity. 5.2.3 Mean mark, mark d.f. and mark probabilities. 5.2.4 Indices for stationary marked point processes. 5.2.5 Nearest-neighbour distributions. 5.3 Second-order characteristics for marked point processes. 5.3.1 Introduction. 5.3.2 Definitions for qualitative marks. 5.3.3 Definitions for quantitative marks. 5.3.4 Estimation of second-order characteristics. 5.4 Orientation analysis for marked point processes. 5.4.1 Introduction. 5.4.2 Orientation analysis for non-isotropic processes with angular marks. 5.4.3 Orientation analysis for isotropic processes with angular marks. 5.4.4 Orientation analysis with constructed marks. 6. Modelling and simulation of stationary point processes. 6.1 Introduction. 6.2 Operations with point processes. 6.2.1 Thinning. 6.2.2 Clustering. 6.2.3 Superposition. 6.3 Cluster processes. 6.3.1 General cluster processes. 6.3.2 Neyman-Scott processes. 6.4 Stationary Cox processes. 6.4.1 Introduction. 6.4.2 Properties of stationary Cox processes. 6.5 Hard-core point processes. 6.5.1 Introduction. 6.5.2 Matern hard-core processes. 6.5.3 The dead leaves model. 6.5.4 The RSA model. 6.5.5 Random dense packings of hard spheres. 6.6 Stationary Gibbs processes. 6.6.1 Basic ideas and equations. 6.6.2 Simulation of stationary Gibbs processes. 6.6.3 Statistics for stationary Gibbs processes. 6.7 Reconstruction of point patterns. 6.7.1 Reconstructing point patterns without a specified model. 6.7.2 An example: reconstruction of Neyman-Scott processes. 6.7.3 Practical application of the reconstruction algorithm. 6.8 Formulas for marked point process models. 6.8.1 Introduction. 6.8.2 Independent marks. 6.8.3 Random field model. 6.8.4 Intensity-weighted marks. 6.9 Moment formulas for stationary shot-noise fields. 6.10 Space-time point processes. 6.10.1 Introduction. 6.10.2 Space-time Poisson processes. 6.10.3 Second-order statistics for completely stationary event processes. 6.10.4 Two examples of space-time processes. 6.11 Correlations between point processes and other random structures. 6.11.1 Introduction. 6.11.2 Correlations between point processes and random fields. 6.11.3 Correlations between point processes and fibre processes. 7. Fitting and testing point process models. 7.1 Choice of model. 7.2 Parameter estimation. 7.2.1 Maximum likelihood method. 7.2.2 Method of moments. 7.2.3 Trial-and-error estimation. 7.3 Variance estimation by bootstrap. 7.4 Goodness-of-fit tests. 7.4.1 Envelope test. 7.4.2 Deviation test. 7.5 Testing mark hypotheses. 7.5.1 Introduction. 7.5.2 Testing independent marking, test of association. 7.5.3 Testing geostatistical marking. 7.6 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.

Statistical Analysis and Modelling of Spatial Point Patterns

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

Bayesian Computing with INLA: A Review

Abstract: The key operation in Bayesian inference is to compute high-dimensional integrals. An old approximate technique is the Laplace method or approximation, which dates back to Pierre-Simon Laplace (1774). This simple idea approximates the integrand with a second-order Taylor expansion around the mode and computes the integral analytically. By developing a nested version of this classical idea, combined with modern numerical techniques for sparse matrices, we obtain the approach of integrated nested Laplace approximations (INLA) to do approximate Bayesian inference for latent Gaussian models (LGMs). LGMs represent an important model abstraction for Bayesian inference and include a large proportion of the statistical models used today. In this review, we discuss the reasons for the success of the INLA approach, the R-INLA package, why it is so accurate, why the approximations are very quick to compute, and why LGMs make such a useful concept for Bayesian computing.

Spatial modeling with R-INLA: A review

Abstract: Coming up with Bayesian models for spatial data is easy, but performing inference with them can be challenging. Writing fast inference code for a complex spatial model with realistically-sized datasets from scratch is time-consuming, and if changes are made to the model, there is little guarantee that the code performs well. The key advantages of R-INLA are the ease with which complex models can be created and modified, without the need to write complex code, and the speed at which inference can be done even for spatial problems with hundreds of thousands of observations. R-INLA handles latent Gaussian models, where fixed effects, structured and unstructured Gaussian random effects are combined linearly in a linear predictor, and the elements of the linear predictor are observed through one or more likelihoods. The structured random effects can be both standard areal model such as the Besag and the BYM models, and geostatistical models from a subset of the Matern Gaussian random fields. In this review, we discuss the large success of spatial modeling with R-INLA and the types of spatial models that can be fitted, we give an overview of recent developments for areal models, and we give an overview of the stochastic partial differential equation (SPDE) approach and some of the ways it can be extended beyond the assumptions of isotropy and separability. In particular, we describe how slight changes to the SPDE approach leads to straight-forward approaches for nonstationary spatial models and nonseparable space–time models. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical Models > Bayesian Models Data: Types and Structure > Massive Data.

Going off grid: computationally efficient inference for log-Gaussian Cox processes

Abstract: This paper introduces a new method for performing computational inference on log-Gaussian Cox processes. The likelihood is approximated directly by making novel use of a continuously specified Gaussian random field. We show that for sufficiently smooth Gaussian random field prior distributions, the approximation can converge with arbitrarily high order, while an approximation based on a counting process on a partition of the domain only achieves first-order convergence. The given results improve on the general theory of convergence of the stochastic partial differential equation models, introduced by Lindgren et al. (2011). The new method is demonstrated on a standard point pattern data set and two interesting extensions to the classical log-Gaussian Cox process framework are discussed. The first extension considers variable sampling effort throughout the observation window and implements the method of Chakraborty et al. (2011). The second extension constructs a log-Gaussian Cox process on the world's oceans. The analysis is performed using integrated nested Laplace approximation for fast approximate inference.

An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach

Abstract: Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R-2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matern class, provide an explicit link, for any triangulation of R-d, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere. (Less)

Riemann manifold Langevin and Hamiltonian Monte Carlo methods

Abstract: The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for Metropolis-Hastings or indeed Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain. The methodology proposed exploits the Riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density. The performance of these Riemann manifold Monte Carlo methods is rigorously assessed by performing inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models and Bayesian estimation of dynamic systems described by non-linear differential equations. Substantial improvements in the time-normalized effective sample size are reported when compared with alternative sampling approaches. MATLAB code that is available from http://www.ucl.ac.uk/statistics/research/rmhmc allows replication of all the results reported.

Stochastic Geometry and Wireless Networks: Volume I Theory

Abstract: Volume I first provides a compact survey on classical stochastic geometry models, with a main focus on spatial shot-noise processes, coverage processes and random tessellations. It then focuses on signal to interference noise ratio (SINR) stochastic geometry, which is the basis for the modeling of wireless network protocols and architectures considered in Volume II. It also contains an appendix on mathematical tools used throughout Stochastic Geometry and Wireless Networks, Volumes I and II.

Statistics for Spatio-Temporal Data

Abstract: Noel Cressie and Christopher WikleHardcover: 624 pagesYear: 2011Publisher: John WileyISBN-13: 978-0471692744Here is the new reference book about spatial and spatio-temporal statistical modeling! No...