Helga Stoyan

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.

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.

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.

Comparing Estimators of the Galaxy Correlation Function

We present a systematic comparison of some of the usual estimators of the two-point correlation function, some of them currently used in cosmology, others extensively employed in the field of the statistical analysis of point processes. At small scales it is known that the correlation function follows reasonably well a power-law expression ξ(r) ∝ r-γ. The accurate determination of the exponent γ (the order of the pole) depends on the estimator used for ξ(r); on the other hand, its behavior at large scales gives information on a possible trend to homogeneity. We study the concept, the possible bias, the dependence on random samples, and the errors of each estimator. Errors are computed by means of artificial catalogs of Cox processes for which the analytical expression of the correlation function is known. We also introduce a new method for extracting simulated galaxy samples from cosmological simulations.

On some partial orderings of random closed sets

Three variants of stochastic ordering of random closed sets are suggested and used in order to prove monotonieity properties. Furthermore, a class of partial orderings of distribution functions is discussed which enable the comparison of distribution functions with equal n-th moment.

Umweltdaten — Visualisierung — Monitoring

Umweltstatistische Untersuchungen sind oft grosere Aufgabenstellungen. Somit liegen die Messung, Sammlung, Speicherung, Aufbereitung und Archivierung sowie die Auswertung der Daten nicht in einer Hand. Nur in Ausnahmefallen gibt es eine Person, die alle Details der Datengewinnung und -aufbereitung kennt, die uber die umweltwissenschaftlichen Hintergrunde Bescheid weis und zugleich auch die Algorithmen und mathematischen Verfahren zur Auswertung und Darstellung der Daten und Ergebnisse beherrscht. Das fuhrt bei der Datenanalyse zu Problemen, auf die in diesem Kapitel aufmerksam gemacht werden soll.

Weitere statistische Methoden

Kein Umweltstatistiker, der mit Stoffen zu tun hat, kommt am Problem der Probennahme vorbei. Dabei besteht vielfach eine sehr unbefriedigende Situation: Es werden teure, hochgenaue Analysenverfahren angewendet, die kleinste Probenmengen mit ausgeklugelten chemischen oder physikalischen Verfahren bewerten konnen. Die analysierten Proben erhalt man aus viel groseren Proben, die vorbereitet, homogenisiert und geteilt werden. Uber die ursprungliche Probennahme und die Probenvorbehandlung und -Vorbereitung wurde und wird aber nur selten grundlich nachgedacht, und es wird oft sehr naiv gehandelt. Dabei ist mit grosen Fehlern zu rechnen, die sehr eindrucksvoll in Markert (1993) beschrieben worden sind, vgl. Tabelle 6.1 auf der nachsten Seite.

Folgen von Ereignissen, Punktprozesse und Punktfelder

Der Umweltstatistiker beobachtet nicht selten Folgen von Ereignissen, z. B. von Unwettern, Sturmen, Lawinensturzen, Hochwassern, Erdbeben, Temperaturmaxima, Erkrankungen oder Uberschreitungen von Emissionsgrenzwerten. Auch solche Folgen konnen mit statistischen Methoden analysiert werden, wobei aber die Verfahren der Zeitreihenanalyse allein nicht ausreichend sind. Dort wird ja angenommen, dass die Zeitpunkte fest vorgegeben sind, und statistisch analysiert werden die in diesen Zeitpunkten beobachteten Werte.

Kombinatorik bei Agricola — über Gemische und Verbindungen in «De natura fossilium libri X»

In Buch X versucht Agricola, Minerale nach der Art ihrer Zusammensetzung aus gewissen Komponenten zu klassifizieren. Er betrachtet die verschiedenen Moglichkeiten der Zusammensetzung in Abhangigkeit von der Anzahl der Komponenten und lost hierbei kombinatorische Probleme. Der Text von Buch X gilt als kompliziert und wenig verstandlich. Das hegt unserer Meinung nach vor allem an der Darstellung ohne Formeln sowie an der unubersichtlichen und ungegliederten Textgestaltung. Auserdem tragen die vielen Beispiele und Abschweifungen wenig zum Verstandnis bei. Sicherlich hat diese Arbeit von Agricola auf die Wissenschaftsentwicklung kaum Einflus genommen. Dennoch mag es interessant sein, den Inhalt von Liber X in moderner Sprache knapp darzustellen. Das wird sicherlich dazu beitragen, das Denken Agricolas verstandlicher zu machen. Wie es sich zeigen wird, hat er im wesentlichen mathematisch richtig gearbeitet; die vorhandenen Rechenfehler sind wohl nur eine Folge von Fluchtigkeit.