Scientific programme

Overview

We plan to have the following 6 cycles of introductory lectures (3 x 75 min. each):

1. Dominique Jeulin: Introduction to fundamental classes of models in stochastic geometry
2. Jesper Møller: Introduction to spatial point processes and simulation-based inference
3. Wilfrid Kendall: Perfect simulation with applications in stochastic geometry
4. Evgeny Spodarev: Introduction to the extrapolation of stationary random fields
5. Ulrich Stadtmueller: Introduction to functional data analysis and its applications
6. Alexander Bulinski: Introduction to stochastic genetics and its applications

Moreover, we plan to have the following 8 cycles of lectures on more specific topics (2 x 75 min. each):

7. Viktor Benes: Random marked sets and space-time models
8. Eva Vedel Jensen: Applications of automatic image analysis in stereology and spatial statistics
9. Claudia Redenbach: Random tessellations and their application to the modeling of cellular materials
10. Lorenz Holzer: Quantitative microstructure analysis in materials science with special emphasis on 3D-aspects and FIB-imaging
    
11. Volker Schmidt: Stochastic segmentation and 3D modeling of microstructures
12. Bartek Blaszczyszyn: Clustering, percolation and directionally convex ordering of point processes
13. Dirk P. Kroese: Stochastic process generation
14. Zakhar Kabluchko: Max-stable random fields and modeling of extremal spatial phenomena

1) Dominique Jeulin (MINES ParisTech, France)

Introduction to fundamental classes of models in stochastic geometry

Abstract

The aim of these lectures is to give an introduction to some theroretical models developed in mathematical morphology and to simulation of random sets and functions (scalar and multivariate). These models are useful in many physical situations with heterogeneous media, for which a probabilistic approach is required. We can mention for instance problems of fracture statistics of materials, the composition of permeabilities in porous media, scanning or transmission electron microscopy images (including multispectral images), rough surfaces or multicomponent composites, but also some biological textures. We will detail the construction and probabilistic properties of Boolean random function models, originally proposed to simulate random rough surfaces, and various types of dead leaves models, well suited to the simulation of multi-component random sets, as well as to simulate perspective views.

Table of Contents

• Lecture 1. Introduction to random sets and functions. The Boolean random function
  • Section 1.1. Random closed sets and the Choquet capacity
  • Section 1.2. Semi-continuous random functions
  • Section 1.3. The Boolean random function
• Lecture 2. Dead leaves random sets
  • Section 2.1. The dead leaves random tesselation
  • Section 2.2. Color dead leaves: construction and main probabilistic properties
  • Section 2.3. Color dead leaves: properties and simulations
• Lecture 3. Dead leaves random functions
  • Section 3.1. Construction and general properties
  • Section 3.2. Bivariate distribution and distribution of the minimum on a compact

2) Jesper Møller (Aalborg University, Denmark)

Introduction to spatial point processes and simulation-based inference

Abstract

Spatial point pattern data occur frequently in a wide variety of scientific disciplines, including seismology, ecology, forestry, geography, spatial epidemiology and material science. In recent years, fast computers and advances in computational statistics, particularly Markov chain Monte Carlo (MCMC) methods, have had a major impact on the development of statistics for spatial point processes. The focus has now changed to likelihood-based inference for flexible parametric models, often depending on covariates, and liberated from restrictive assumptions of stationarity. We summarize and discuss the current state of spatial point process theory and directions for future research, making an analogy with generalized linear models and random effect models, and illustrating the theory with various examples of applications. In particular, we consider Poisson, Gibbs and Cox process models, diagnostic tools and model checking, MCMC algorithms, computational methods for likelihood-based inference, and quick non-likelihood approaches to inference.

Table of Contents

• Lecture 1. Fundamental concepts and tools
  • Section 1.1. Introduction
  • Section 1.2. Poisson processes
  • Section 1.3. Summary statistics
• Lecture 2. Two main classes of spatial point process models
  • Section 2.1. Cox processes
  • Section 2.2. Markov point processes
• Lecture 3. Simulation and inference procedures
  • Section 3.1. Markov chain Monte Carlo methods
  • Section 3.2. Likelihood and moment-based estimation procedures
  • Section 3.3. Overview and concluding remarks

3) Wilfrid Kendall (University of Warwick, United Kingdom)

Perfect simulation with applications in stochastic geometry

Abstract

The aim of these lectures is to present a clear introduction to the notion of perfect simulation and related mathematical ideas, with a strong emphasis on applications to stochastic geometry. Lecture 1 begins with the original finite-state-space context of Propp and Wilson (1996), and illustrate it by developing a coupling-from-the-past (CFTP) algorithm for a simple problem in elementary image analysis. Lecture 2 continues by describing ways in which one can generalize the CFTP algorithm to suitable infinite-state-space contexts (for example as in Kendall and Moeller, 2000), together with work on how far one might be able to go in this direction (Kendall 2004, Connor and Kendall 2007a,b). This will be illustrated by discussion of CFTP for point processes and other object processes. Finally Lecture 3 discusses the use of perfect simulation ideas in other contexts, such as in dealing with boundary effects.

Table of Contents

• Lecture 1. Classic coupling from the past
  • Section 1.1. Simplest possible instances: flip-flops, simple random walks, Ising models
  • Section 1.2. Mathematical precursors and general theorems
  • Section 1.3. Case studies: seeking out monotonicity
• Lecture 2. Dominated coupling from the past
  • Section 2.1. Birth-death processes and point processes
  • Section 2.2. Theoretical limits
  • Section 2.3. Case studies: perpetuation and interaction
• Lecture 3. Complements
  • Section 3.1. Dealing with boundaries
  • Section 3.2. Further refinements
  • Section 3.3. Other variants of perfect simulation

4) Evgeny Spodarev (Ulm University, Germany)

Introduction to the extrapolation of stationary random fields

Abstract

For stationary random fields with finite second moments, the theory of mean square optimal linear extrapolation exists since 1960s due to D. Krige and G. Matheron. In practice however (e.g. in storm insurance) there is also a need to extrapolate heavy-tailed spatial data without finite second moments. These data can often be reasonably modelled by stable random fields given as integrals of a non-random kernel function with respect to a random stable noise. In our lectures, we give an overview of classical and new results in this area. We study the existence and uniqueness properties of the extrapolation methods which are realised as (non-) linear optimisation problems with constraints.

Table of Contents

• Lecture 1. Extrapolation of second order stationary random fields
  • Section 1.1. Stationary random fields, covariance, and variogram
  • Section 1.2. Estimation of the covariance structure
  • Section 1.3. Ordinary kriging and its properties, kriging with drift
• Lecture 2. Stable random laws
  • Section 2.1. Normalisation: Box–Cox transform, stable distributions
  • Section 2.2. Random measures, stochastic integration
  • Section 2.3. Stable random fields and their properties, stability and association
• Lecture 3. Extrapolation of stable random fields
  • Section 3.1. Predictors and their properties
  • Section 3.2. Maximization of covariation
  • Section 3.3. Examples, open problems

5) Ulrich Stadtmueller (Ulm University, Germany)

Introduction to functional data analysis and its applications

Abstract

In many modern statistical questions the data are not just copies of a random variable or a random vector but copies of ininitely dimensional objects as random processes or random fields. If we want to do statistical analysis based on finitely many data and do not have some parametric model in mind, we have to use approaches using a dimension reduction. These lectures give an introduction into the topics of functional data analysis including functional principal component analysis providing appropriate tools. The lectures will include basic modeling, tools and methods together with some theoretical results. Exemplary applications will illustrate the methods.

Table of Contents

• Lecture 1. Introduction and basics
  • Section 1.1. Examples of functional data
  • Section 1.2. Basic tools
  • Section 1.3. Representing functional data
• Lecture 2. Functional principle component analysis
  • Section 2.1. Classical principle component analysis
  • Section 2.2. Functional PCA
  • Section 2.3. Consisteny results for functional PCA
• Lecture 3. Linear models with functional data
  • Section 3.1. Scalar response
  • Section 3.2. Functional response
  • Section 3.3. Classification

6) Alexander Bulinski (Moscow State University, Russia)

Introduction to stochastic genetics and its applications

Abstract

The goal of these lectures is to give an introduction to modern stochastic models in genetics. Such models are used in studying the complex risks of diseases. The challenging problem here is to identify the collection of single nucleotide polymorphisms (SNP) increasing the risk of diseases. There exist various approaches to this problem involving different probabilistic techniques. We will employ ideas of random trees and forests, spatial Markov processes and Bayesian networks. New statistical methods will be also discussed, including exemplary results of data analysis.

Table of Contents

• Lecture 1. Biological background
  • Section 1.1. DNA, genes, SNP: Heredity theory; gene-gene and gene-environment interactions
  • Section 1.2. Complex diseases and the risk-factors; identification of interactions
  • Section 1.3. Measures of importance of interactions
• Lecture 2. Model selection and genome-wide association study (GWAS)
  • Section 2.1. Multifactor dimensionality reduction (MDR) method and its modifications
  • Section 2.2. Logic regression and simulated annealing
  • Section 2.3. Random trees and forests
• Lecture 3. Statistical inference and spatial analysis of DNA
  • Section 3.1. Spatial Markov processes
  • Section 3.2. Bayesian networks
  • Section 3.3. Simulation techniques

7) Viktor Benes (Charles University Prague, Czech Republic)

Random marked sets and space-time models

Abstract

Random marked sets present generalizations of marked point processes. We are interested mostly in marked fibre and surface processes and show their relation to weighted random measures. Geostatistical characteristics as well as moment measures and distance methods are available for a statistical evaluation. Random-field models can be tested in various ways. For multivariate marks we consider the problem of dimension reduction. Results of simulations are presented for the 3D Poisson-Voronoi tessellation and for a curve in a bounded planar domain being a solution of a SDE, both marked by a Gaussian random field. In applications the 3D grain microstructure of metals with crystallographic orientations of cells induces a marked surface process of cell faces with disorientation marks of neighbouring cells. In a neurophysiological experiment the track (a random fibre) of a rat is monitored together with occurrences of action potentials of a hippocampal neuron, the planar intensity function of spiking activity being a mark. A track of the rat elongates in time, therefore it can be viewed as a spatio-temporal random set. Parametric models are used to describe the development of firing fields. Filtering enables to quantify the evolution of parameters in discrete time, sequential Monte Carlo algorithms are used to speed up the calculations and relax traditional Gaussian assumptions on the posterior. Another model presented is a spatio-temporal random set based on the union of interacting discs simulated by MCMC. Besides growth models here all quermassintegrals may have a temporal trend. The particle filter is compared with the maximum-likelihood estimation of parameters.

Table of Contents

• Lecture 1. Random marked sets
  • Section 1.1. Models of integer Hausdorff dimension
  • Section 1.2. Statistical characteristics and simulations
  • Section 1.3. Applications in materials research and neurophysiology
• Lecture 2. Stochastic geometry in space and time
  • Section 2.1. From spatial to spatio-temporal random sets
  • Section 2.2. Sequential Monte-Carlo
  • Section 2.3. Filtering and statistical methods

8) Eva Vedel Jensen (Aarhus University, Denmark)

Applications of automatic image analysis in stereology and spatial statistics

Abstract

The aim of these lectures is to give a number of recent examples where computerized image analysis has been combined with non-uniform sampling to increase the efficiency of estimators in stereology and spatial statistics. In Lecture 1, we focus on estimation of intensities of point processes based on observation in non-uniformly placed sampling windows. The distribution of the windows is determined by a rough estimate of the realized point pattern. For this reason, the technique is called image-based empirical importance sampling. Methods of constructing optimal estimators of the intensity of a point process based on such data are discussed. Lecture 2 deals primarily with applications of automatic image analysis in local stereology. It will be shown that measurements along lines and in planes can be combined in an intelligent way to increase the efficiency of local stereological estimators in common use.

Table of Contents

• Lecture 1. Image-based empirical importance sampling
  • Section 1.1. Importance sampling
  • Section 1.2. Efficiency for homogeneous point processes
  • Section 1.3. Design- and model-based inference
• Lecture 2. Automatic image analysis in local stereology
  • Section 2.1. Local stereological estimators
  • Section 2.2. Efficiency of estimators of particle size
  • Section 2.3. A comparative study

9) Claudia Redenbach (University of Kaiserslautern, Germany)

Random tessellations and their application to the modeling of cellular materials

Abstract

Cellular materials are nowadays used in a wide range of application areas including shock absorption, thermal insulation or filtration. The macroscopic properties of a material, e.g., elasticity, thermal conductivity or permeability, are highly affected by its microstructure. Models from stochastic geometry combined with finite element simulations are important tools for studying these complex structure-property relations. Random tessellations, i.e., space-filling systems of convex, non-overlapping polytopes, are widely used as models for cellular or polycrystalline materials. Foam structures, for instance, are modelled by the systems of edges (open-cell foam) or facets (closed-cell foam) of the tessellation. Random Laguerre tessellations (a weighted form of the well-known Voronoi tessellations) generated by systems of non-overlapping spheres are particularly promising models for foam structures. In our lectures, we will introduce random tessellation models, in particular Voronoi and Laguerre tessellations, together with their geometric characteristics. We will discuss how to estimate these characteristics from 3D images of cellular materials obtained by microcomputed tomography. Using these characteristics, we will fit random tessellation models to the materials. These models are then used to study the dependence of macroscopic properties on the geometric microstructure.

Table of Contents

• Lecture 1. Geometric characterisation of cellular materials
  • Section 1.1. Cellular materials: Properties and applications
  • Section 1.2. Random tessellations, in particular tessellations of Laguerre type
  • Section 1.3. Characteristics of tessellation and their estimation from image data
• Lecture 2. Stochastic modeling of cellular materials
  • Section 2.1. Model fitting procedure
  • Section 2.2. Examples of application
  • Section 2.3. Prediction of macroscopic properties

10) Lorenz Holzer (Zurich University for Applied Sciences, Switzerland)

Quantitative microstructure analysis in materials science with special emphasis on 3D-aspects and FIB-imaging

Abstract

This lecture intends to highlight the materials scientists approach in elaborating the relationships between microstructure parameters and materials properties. Based on the fundamental understanding of these relationships it will be possible to improve specific material properties such as permeability, effective conductivity, electrochemical activity or durability. The typical methodologies which are used in the work flow for microstructure analysis are discussed and illustrated. These methodologies include image acquisition, basic image processing procedures, extraction of quantitative parameters and finally the elaboration of relationships with experimental results.  Practical applications of these methods will be illustrated with some examples which include a) hydration kinetics (crystallization/precipitation) in cement suspensions, b) transport properties of porous media (electrolysis membranes, concrete, clay stones), c) degradation kinetics of nickel based electrodes and d) microstructure-property relationships (i.e. electrochemical activity) of fuel cell electrodes.

Table of Contents

• Lecture 1: Work flow from image acquisition to microstructure quantification
  • Section 1.1. Image acquisition: Overview of 2D- and 3D-methods
  • Section 1.2. Practical aspects of image processing: Correction of image imperfections, stack alignment, feature recognition, boundary truncation problem
  • Section 1.3. Two different concepts for size distributions: Discrete and continuous PSDs
• Lecture 2: Discussion of some microstructure-property relationships
  • Section 2.1. Basic concepts of microstructure - property relationships in materials science
  • Section 2.2. Porous media: Discussion of transport properties with a special focus on tortuosity
  • Section 2.3. Microstructure effects on the performance of solid oxide fuel cells: degradation kinetics and electrode activity

11) Volker Schmidt (Ulm University, Germany)

Stochastic segmentation and 3D modeling of microstructures

Abstract

We will discuss various techniques of stochastic segmentation and modeling of 3D images, which show complex microstructures of composite and porous materials reconstructed by electron and synchrotron tomography. Using a multiscale approach, it is possible to decompose complex microstructures into several (less complex) components. In particular, a macroscale component is determined by morphological smoothing. It can be represented by unions of overlapping spheres, where a certain maximum-likelihood principle is used. This leads to an enormous reduction of complexity and allows us to model the macroscale component by random marked point processes, which is one of the most fundamental classes of models in stochastic geometry. On the other hand, by the morphological smoothing, a small fraction of voxels is misspecified. The set of these misspecified voxels is interpreted as the microscale component of the microstructure. It is modeled separately, using random particle systems of Cox type. Furthermore, stochastic network models are considered which describe the 3D morphology of percolation paths in composite or porous materials. They have the form of random geometric graphs, where the vertex set is modelled by random point processes and the edges are put using tools from graph theory and MCMC simulation. The network models can be applied to analyse transport processes of charges, gases, or fluids, e.g., in polymer solar cells, Li-ion batteries, and fuel cells.

Table of Contents

• Lecture 1. Multi-scale approach to stochastic 3D modeling
  • Section 1.1. Representation by unions of overlapping spheres
  • Section 1.2. Random particle systems of Cox type
  • Section 1.3. Model fitting, model validation, and virtual material design
• Lecture 2. Modeling the 3D morphology of percolation paths
  • Section 2.1. Stochastic network models via random geometric graphs
  • Section 2.2. Model fitting based on tools from graph theory and MCMC simulation
  • Section 2.3. Applications to polymer solar cells, Li-ion batteries, and fuel cells

12) Bartek Blaszczyszyn (INRIA & ENS, Paris, France)

Clustering, percolation and directionally convex ordering of point processes

Abstract

Heuristics indicate that point processes exhibiting clustering of points have larger critical radius for the percolation of their continuum percolation models than spatially more homogeneous point processes. It has already been shown, and we reaffirm it in this presentation, that the dcx-ordering of point processes is suitable to compare their clustering tendencies. Hence, it is tempting to conjecture that the critical percolation radius is increasing in dcx-order. Some numerical evidences support this conjecture for a special class of point processes, called perturbed lattices, which are "toy models" for determinantal and permanental point processes. However the conjecture is not true in full generality, since one can construct a Cox point process with degenerate critical radius being equal to zero, that is dcx-larger than a given homogeneous Poisson point process, for which this radius is known to be strictly positive. Nevertheless, the aforementioned monotonicity in dcx-order can be proved, for a nonstandard critical radius, related to Peierls argument. Furthermore, we show the reverse monotonicity for another nonstandard critical radius. This gives uniform lower and upper bounds on the standard critical percolation radius for all point processes being dcx-smaller than some given point process.  Moreover, we show that point processes which are dcx-smaller than a homogeneous Poisson point process admit uniformly non-degenerate lower and upper bounds on the standard critical percolation radius. In fact, all the above results hold under weaker assumption on the ordering of void probabilities or factorial moment measures only. Examples of point processes comparable to Poisson point processes in this weaker sense include determinantal and permanental point processes with trace-class integral kernels. More generally, we show that point processes dcx-smaller than a homogeneous Poisson point processes exhibit phase transitions in certain percolation models based on the level-sets of additive shot-noise fields of these point process. Examples of such models are k-percolation and SINR-percolation.

Table of Contents

• Lecture 1. Directional ordering and clustering
  • Section 1.1. Definitons, examples and properties
  • Section 1.2. Statistical descriptors of spatial (non-) homogeneity
  • Section 1.3. Examples (Super- and Sub-Poisson, perturbed-lattice, determinantal and permanental point processes)
• Lecture 2. Directional ordering and continuum percolation
  • Section 2.1. Upper and lower critical percolation radius
  • Section 2.2. Sandwich inequality and phase transition
  • Section 2.3. Phase transitions in sub-Poisson models

13) Dirk P. Kroese (The University of Queensland, Brisbane, Australia)

Stochastic process generation

Abstract

Many numerical problems in science, engineering, finance, and statistics are solved nowadays through Monte Carlo methods; that is, through random experiments on a computer. The most basic use of Monte Carlo simulation is to generate realisations of a random (or stochastic) process from a given probabilistic model, in order to observe its behaviour. In these lectures we discuss how to efficiently generate a range of well-known one- and multi-dimensional random processes, including those used in spatial statistics. The theory will be illustrated with working code in Matlab, which can be downloaded from www.montecarlohandbook.org

Table of Contents

• Lecture 1. Fundamental stochastic processes
  • Section 1.1. Gaussian Processes
  • Section 1.2. Markov Processes and Markov Random Fields
  • Section 1.3. Random Point Processes and Poisson Random Measures
• Lecture 2. Other stochastic processes
  • Section 2.1. Stochastic Differential Equations
  • Section 2.2. Fractional Brownian Motion and the Brownian Sheet
  • Section 2.3. Levy Processes

14) Zakhar Kabluchko (Ulm University, Germany)

Max-stable random fields and modeling of extremal spatial phenomena

Abstract

Max-stable random fields are natural models for extremal spatial random phenomena like extreme rainfalls, extreme floods, large insurance losses, etc. Max-stable random fields appear as limits of pointwise maxima of independent copies of a random field as the number of copies goes to infinity. Starting with the classical one-dimensional extreme-value theory we will proceed to multivariate extreme-value distributions and then to max-stable random fields. We will discuss basic properties of max-stable fields, their representations, simulation, define basic classes of max-stable fields and discuss their statistical estimation.

Table of Contents

• Lecture 1. Classical extreme-value theory
  • Section 1.1. Motivating examples; the classical extreme-value theorem; max-stable distributions and their domains of attraction
  • Section 1.2. Multivariate extreme-value theory; spectral representations
  • Section 1.3. Example: Hüsler-Reiss distributions
• Lecture 2. Max-stable random fields
  • Section 2.1. Definition of max-stable random fields and their basic properties; spectral representations of max-stable random fields
  • Section 2.2. Basic classes: mixed moving maxima, positive recurrent max-stable fields, Brown-Resnick fields; simulation of max-stable random fields
  • Section 2.3. Dependence measures and statistical estimation