Dietrich Stoyan

Comparison methods for queues and other stochastic models

Studies stochastic models of queueing, reliability, inventory, and sequencing in which random influences are considered. One stochastic mode--rl is approximated by another that is simpler in structure or about which simpler assumptions can be made. After general results on comparison properties of random variables and stochastic processes are given, the properties are illustrated by application to various queueing models and questions in experimental design, renewal and reliability theory, PERT networks and branching processes.

Fractals, random shapes and point fields : methods of geometrical statistics

FRACTALS AND METHODS FOR THE DETERMINATION OF FRACTAL DIMENSIONS. Hausdorff Measure and Dimension. Deterministic Fractals. Random Fractals. Methods for the Empirical Determination of Fractal Dimension. THE STATISTICS OF SHAPES AND FORMS. Fundamental Concepts. Representation of Contours. Set Theoretic Analysis. Point Description of Figures. Examples. POINT FIELD STATISTICS. Fundamentals. Finite Point Fields. Poisson Point Fields. Fundamentals of the Theory of Point Fields. Statistics for Homogeneous Point Fields. Point Field Models. Appendices. References. Index.

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

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 of Simulated Random Packings of Spheres

This paper describes two algorithms for the generation of random packings of spheres with arbitrary diameter distribution. The first algorithm is the force-biased algorithm of Mościnski and Bargiel. It produces isotropic packings of very high density. The second algorithm is the Jodrey-Tory sedimentation algorithm, which simulates successive packing of a container with spheres following gravitation. It yields packings of a lower density and of weak anisotropy. The results obtained with these algorithms for the cases of log-normal and two-point sphere diameter distributions are analysed statistically, i. e. standard characteristics of spatial statistics such as porosity (or volume fraction), pair correlation function of the system of sphere centres and spherical contact distribution function of the set-theoretical union of all spheres are determined. Furthermore, the mean coordination numbers are analysed. These results are compared for both algorithms and with data from the literature based on other numerical simulations or from experiments with real spheres.

On Parameter Estimation for Pairwise Interaction Point Processes

Summary Pairwise interaction point processes form a useful class of models for spatial point patterns, especially patterns for which the spatial distribution of points is more regular than for a homogeneous planar Poisson process. Several authors have proposed methods for estimating the parameters of a pairwise interaction point process. However, there appears to be no general theory which provides grounds for preferring a particular method, nor have any extensive empirical comparisons been published. In this paper, we review three general methods of estimation which have been proposed in the literature and present the results of a comparative simulation study of the three methods.

Statistical physics and spatial statistics

Statistical physics and spatial statistics , Statistical physics and spatial statistics , کتابخانه دیجیتال جندی شاپور اهواز

Improving Ratio Estimators of Second Order Point Process Characteristics

Ripleys K function, the L function and the pair correlation function are important second order characteristics of spatial point processes. These functions are usually estimated by ratio estimators, where the numerators are Horvitz–Thompson edge corrected estimators and the denominators estimate the intensity or its square. It is possible to improve these estimators with respect to bias and estimation variance by means of adapted distance dependent intensity estimators. Further improvement is possible by means of refined estimators of the square of intensity. All this is shown by statistical analysis of simulated Poisson, cluster and hard core processes.

Estimating the fruit dispersion of anemochorous forest trees

Abstract This paper studies the problem of modelling fruit dispersal of anemochorous forest trees by the statistical analysis of fruit collections in traps. The statistical model of fruit dispersion and the choice of trap positions turn out to be very important. It is shown that the lognormal distribution of ‘fruit–tree’ distance seems to be of great practical interest. The trap positions have considerable influence on the precision of the estimates, particularly on the total number of fruits per tree.

Case studies in spatial point process modeling

Point process statistics is successfully used in fields such as material science, human epidemiology, social sciences, animal epidemiology, biology, and seismology. Its further application depends greatly on good software and instructive case studies that show the way to successful work. This book satisfies this need by a presentation of the spatstat package and many statistical examples. Researchers, spatial statisticians and scientists from biology, geosciences, materials sciences and other fields will use this book as a helpful guide to the application of point process statistics. No other book presents so many well-founded point process case studies.

Non‐Homogeneous Gibbs Process Models for Forestry — A Case Study

Forestry statistics needs realistic models for non-homogeneously distributed trees in forests. For elder trees, non-homogeneous Poisson or Cox processes are not realistic models. Instead, non-homogeneous Gibbs processes are suggested, which are described by a fixed pair potential describing the short-range interaction of trees and a location dependent intensity function describing long-range variation of tree density. The application of this model is demonstrated for a hickory forest in North Carolina and an artificial forest stand.