Stochastic geometry, spatial statistics and random fields: models and algorithms

Table of Contents:

1.Stein's Method for Approximating Complex Distributions, with a View towards Point Processes / Dominic Schuhmacher

1.1.Introduction
1.2.Normal Approximation
1.2.1.Two Important Lemmas
1.2.2.Independent Random Variables
1.2.3.Dependent Random Variables
1.2.4.Kolmogorov Distance
1.3.Approximation by General Distributions
1.3.1.The Key Steps of Stein's Method
1.3.2.The Generator Approach
1.4.Point Processes
1.4.1.Distances between Point Patterns
1.4.2.Statistical Applications of Distances between Point Patterns
1.4.3.Distances between Point Process Distributions
1.5.Poisson Process Approximation of Point Process Distributions
1.5.1.The Coupling Strategy
1.5.2.Two Upper Bounds for Poisson Process Approximation
1.6.Gibbs Process Approximation of Point Process Distributions

2.Clustering Comparison of Point Processes, with Applications to Random Geometric Models / Dhandapani Yogeshwaran

2.2.Examples of Point Processes
2.2.1.Elementary Models
2.2.2.Cluster Point Processes - Replicating and Displacing Points
2.2.3.Cox Point Processes - Mixing Poisson Distributions
2.2.4.Gibbs and Hard-Core Point Processes
2.2.5.Determinantal and Permanental Point Process
2.3.Clustering Comparison Methods
2.3.1.Second-order Statistics
2.3.2.Moment Measures
2.3.3.Void Probabilities
2.3.4.Positive and Negative Association
2.3.5.Directionally Convex Ordering
2.4.Some Applications
2.4.1.Non-trivial Phase Transition in Percolation Models
2.4.2.U-Statistics of Point Processes
2.4.3.Random Geometric Complexes
2.4.4.Coverage in the Germ-Grain Model
2.5.Outlook

3.Random Tessellations and their Application to the Modelling of Cellular Materials / Andre Liebscher

3.1.Introduction
3.2.Random Tessellations
3.2.1.Definitions
3.2.2.Tessellation Models
3.3.Estimation of Geometric Characteristics of Open Foams from Image Data
3.3.1.Characteristics based on Intrinsic Volumes
3.3.2.Model-based Mean Value Analysis
3.3.3.Cell Reconstruction
3.4.Stochastic Modelling of Cellular Materials
3.4.1.Modelling of the Cell System
3.4.2.Modelling of Open Foams

4.Stochastic 3D Models for the Micro-structure of Advanced Functional Materials / Ole Stenzel

4.1.Introduction
4.2.Point Process Models and Time Series
4.2.1.Point Processes
4.2.2.Multivariate Time Series
4.3.Stochastic 3D Model for Organic Solar Cells
4.3.1.Data and Functionality
4.3.2.Data Preprocessing
4.3.3.Description of Stochastic Model
4.3.4.Model Fitting
4.3.5.Model Validation
4.3.6.Scenario Analysis
4.4.Stochastic 3D Modeling of Non-woven GDL
4.4.1.Data and Functionality
4.4.2.Data Preprocessing
4.4.3.Description of Stochastic Model
4.4.4.Model Fitting
4.4.5.Model Validation
4.5.Stochastic 3D Model for Uncompressed Graphite Electrodes in Li-Ion Batteries
4.5.1.Data and Functionality
4.5.2.Data Preprocessing
4.5.3.Description of Stochastic Model
4.5.4.Model Fitting
4.5.5.Model Validation
4.5.6.Further Numerical Results
4.6.Conclusion

5.Boolean Random Functions / Dominique Jeulin

5.1.Introduction
5.2.Some Preliminaries
5.2.1.Random Closed Sets
5.2.2.Upper Semi-Continuous Random Functions
5.2.3.Principle of Random Sets and of Random Function Modeling
5.3.Boolean Random Functions
5.3.1.Construction of the BRF
5.3.2.BRF and Boolean Model of Random Sets
5.3.3.Choquet Capacity of the BRF
5.3.4.Supremum Stability and Infinite Divisibility
5.3.5.Characteristics of the Primary Functions
5.3.6.Some Stereological Aspects of BRF
5.3.7.BRF and Counting
5.3.8.Identification of a BRF Model
5.3.9.Tests of BRF
5.3.10.BRF and Random Tessellations
5.4.Multiscale Boolean Random Functions
5.5.Application of BRF to Modeling of Rough Surfaces
5.5.1.Simulation of the Evolution of Surfaces and of Stresses during Shot Peening
5.5.2.Simulation of the Roughness Transfer on Steel Sheets
5.5.3.Modeling of Electro Discharge Textures (EDT)

6.Random Marked Sets and Dimension Reduction / Ondrej Sedivy

6.1.Preliminaries
6.1.1.Random Measures and Random Marked Closed Sets
6.1.2.Second-order Characteristics
6.2.Statistical Methods for RMCS
6.2.1.Random-field Model Test
6.2.2.Estimation of Characteristics
6.3.Modeling of RMCS; Simulation Results
6.3.1.Tessellation Models
6.3.2.Fibre Process Based on Diffusion
6.4.Real Data Analyses
6.4.1.Random-field Model Test in a Neurophysiological Experiment
6.4.2.Second-order RMCS Analysis of Granular Materials
6.5.Dimension Reduction
6.6.Theoretical Results
6.6.1.Investigation of S1
6.6.2.Investigation of S2
6.7.Statistical Methods
6.7.1.Estimation
6.7.2.Statistical Testing
6.8.Simulation Studies
6.8.1.Description of the Simulation
6.8.2.Numerical Results

7.Space-Time Models in Stochastic Geometry / Marketa Zikmundova

7.1.Discrete-Time Modelling
7.1.1.State-space Models and Sequential Monte Carlo
7.1.2.The PMMH Method
7.2.Application: Firing Activity of Nerve Cells
7.2.1.Space-Time Model and Particle Filter
7.2.2.Model Checking
7.3.Systems of Interacting Discs
7.3.1.Background in Space
7.3.2.Space-Time Model
7.3.3.Model Checking
7.3.4.Simulation Study
7.4.Continuous-Time Modelling
7.4.1.Space-Time Point Processes
7.4.2.Space-Time Levy-Driven Cox Point Processes
7.5.Separability and Space-Time Point Processes
7.5.1.Separability
7.5.2.Spatial and Temporal Projection Processes
7.6.Ambit Sets and Nonseparable Kernels
7.7.Estimation Procedures
7.7.1.Estimation of the Space-Time Intensity Function
7.7.2.Estimation by Means of the Space-Time K-Function
7.7.3.Estimation by Means of Projection Processes

8.Rotational Integral Geometry and Local Stereology - with a View to Image Analysis / Allan Rasmusson

8.1.Rotational Integral Geometry
8.1.1.Rotational Integrals of Intrinsic Volumes
8.1.2.Intrinsic Volumes as Rotational Integrals
8.1.3.Rotational Integral Geometry of Minkowski Tensors
8.1.4.A Principal Rotational Formula
8.2.Local Stereology
8.3.Variance Reduction Techniques
8.4.Computational Stereology
9.An Introduction to Functional Data Analysis / Marta Zampiceni
9.1.Some Fundamental Tools
9.1.1.Basic Statistics
9.1.2.Smoothing Techniques
9.2.Principal Components Analysis (PCA)
9.2.1.PCA for Multivariate Data
9.2.2.PCA for Functional Data
9.3.Statistical Models for Functional Data
9.3.1.Linear Regression
Contents note continued: 9.3.2.Estimation, a Second Look
9.3.3.More General Regression Models

10.Some Statistical Methods in Genetics / Alexander Bulinski

10.1.Introduction
10.2.The MDR method and its Modifications
10.2.1.Implementation of the MDR Method
10.2.2.Prediction Algorithm
10.2.3.Estimated Prediction Error
10.2.4.Dimensionality Reduction
10.2.5.Central Limit Theorem
10.2.6.Modifications of the MDR Method
10.3.Logic Regression and Simulated Annealing
10.3.1.Logic Trees
10.3.2.Logic Regression
10.3.3.Operations on Logic Trees
10.3.4.Simulated Annealing
10.4.Models Involving Random Fields
10.5.Concluding Remarks

11.Extrapolation of Stationary Random Fields / Stefan Roth

11.1.Introduction
11.2.Basics of Random Fields
11.2.1.Random Fields with Invariance Properties
11.2.2.Elements of Correlation Theory for Square Integrable Random Fields
11.2.3.Stable Random Fields
11.2.4.Dependence Measures for Stable Random Fields
11.2.5.Examples of Stable Processes and Fields
11.3.Extrapolation of Stationary Random Fields
11.3.1.Kriging Methods for Square Integrable Random Fields
11.3.2.Simple Kriging
11.3.3.Ordinary Kriging
11.4.Extrapolation of Stable Random Fields
11.4.1.Least Scale Linear Predictor
11.4.2.Covariation Orthogonal Predictor
11.4.3.Maximization of Covariation
11.4.4.The Case α E (0, 1]
11.4.5.Numerical Examples
11.5.Open problems

12.Spatial Process Simulation / Zdravko I. Botev

12.1.Introduction
12.2.Gaussian Markov Random Fields
12.2.1.Gaussian Property
12.2.2.Generating Stationary Processes via Circulant Embedding
12.2.3.Markov Property
12.3.Point Processes
12.3.1.Poisson Process
12.3.2.Marked Point Processes
12.3.3.Cluster Process
12.3.4.Cox Process
12.3.5.Point Process Densities
12.4.Wiener Surfaces
12.4.1.Wiener Process
12.4.2.Fractional Brownian Motion
12.4.3.Fractional Wiener Sheet in R2
12.4.4.Fractional Brownian Field
12.5.Spatial Levy Processes
12.5.1.Levy Process.

This volume is an attempt to provide a graduate level introduction to various aspects of stochastic geometry, spatial statistics and random fields, with special emphasis placed on fundamental classes of models and algorithms as well as on their applications, e.g. in materials science, biology and genetics. This book has a strong focus on simulations and includes extensive codes in Matlab and R which are widely used in the mathematical community. It can be seen as a continuation of the recent volume 2068 of Lecture Notes in Mathematics, where other issues of stochastic geometry, spatial statistics and random fields were considered with a focus on asymptotic methods.

(http://www.springer.com/in/book/9783319100630)