Book of abstracts - 17th Workshop on Stochastic Geometry

17th Workshop on Stochastic Geometry,Stereology and Image Analysis

Toruń, 11–14 June 2013

in conjunction with

Tomasz Schreiber’s Memorial SessionToruń, 10 June 2013

Under the Honorary Patronage of Mr. Michał Zaleski,the President of Toruń

Programmeand

Book of Abstracts

Toruń 2013

Edited by

BARTOSZ ZIEMKIEWICZ

Cover design

JACEK OWCZARZ, Studio Red Krasnal

c© Copyright by Wydział Matematyki i Informatyki UMK, Toruń 2013

Wydanie pierwsze. Nakład 100 egz.Skład i łamanie: Bartosz ZiemkiewiczWydział Matematyki i Informatyki UMKul. Chopina, 12/18, 87-100 Toruń

International Year of Statistics(Statistics2013)

The International Year of Statistics (Statistics2013) is a worldwidecelebration and recognition of the contributions of statistical science.Through the combined energies of organizations worldwide, Statistics2013 will promote the importance of Statistics to the broader scien-tific community, business and government data users, the media, policymakers, employers, students, and the general public. The goals of theInternational Year of Statistics include:

• increasing public awareness of the power and impact of Statisticson all aspects of society;

• nurturing Statistics as a profession, especially among young peo-ple; and

• promoting creativity and development in the sciences of Proba-bility and Statistics.

You are invited to visit websites www.statistics2013.org andwww.bs2013.org for information about all activities and registrationof your organization in support.

3

17th Workshop on Stochastic Geometry, Stereology and ImageAnalysis has been sponsored by the following institutions:

The Nicolaus Copernicus University

Prezydent Miasta Torunia

Wydział Matematyki i InformatykiUniwersytetu Mikołaja Kopernika

Polska Akademia Nauk

Bernoulli Society

Their support is kindly acknowledged.

4

Scientific Programme Committee

François Baccelli (Inria/ENS, Paris, UT Austin)Victor Beneš (Charles University, Prague)Bartłomiej Błaszczyszyn (Inria/ENS, Paris) – ChairPierre Calka (University of Rouen)Günter Last (KIT, Karlsruhe)Volker Schmidt (University of Ulm)Ryszard Szekli (University of Wroclaw)

Local Organizing Committee

Adam Jakubowski (Toruń) – ChairJoanna Karłowska-Pik (Toruń) – SecretaryBartosz Ziemkiewicz (Toruń)

5

Contents

General Schedule 7

Abstracts 11Invited Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Contributed Talks . . . . . . . . . . . . . . . . . . . . . . . . 23Poster Session . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

List of participants 47

6


in conjunction with

Tomasz Schreiber’s Memorial Session

GENERAL SCHEDULE


Monday, 10 June 2013

Chair: Joseph Yukich

09:00–09:30 Opening09:30–10:20 Lecture I (E.B. Vedel-Jensen)10:20–10:40 Coffee break10:40–11:30 Lecture II (P. Calka)11:30–12:20 Lecture III (R. Fernández)12:20–13:30 Lunch break13:30–14:20 Lecture IV (G. Last)14:20–15:10 Lecture V (M. Penrose)15:10–15:30 Coffee break15:30–16:20 Lecture VI (M.-C. van Lieshout)16:20–17:10 Lecture VII (H.-O. Georgii)19:30 Concert


Tuesday, 11 June 2013

08:45–09:00 Opening of the conference09:00–09:45 Invited Talk 1 (J. Møller)09:45–11:00 Contributed Session 111:00–11:30 Coffee break11:30–12:45 Contributed Session 212:45–14:30 Lunch break14:30–15:15 Invited Talk 2 (J. Yukich)15:15–16:30 Contributed Session 316:30–17:00 Coffee break17:00–18:30 Poster Session19:00–24:00 Conference Dinner

9

Wednesday, 12 June 2013

09:00–09:45 Invited Talk 3 (Ch. Thäle)09:45–11:00 Contributed Session 411:00–11:30 Coffee break11:30–12:45 Contributed Session 512:45–14:30 Lunch break14:30–15:15 Invited Talk 4 (S. Zuyev)15:15–16:30 Contributed Session 616:30–17:00 Coffee break17:00–21:00 Tour at the Toruń Centre for Astronomy

in Piwnice with a picnic

Thursday, 13 June 2013

09:00–09:45 Invited Talk 5 (M. Reitzner)09:45–11:00 Contributed Session 711:00–11:30 Coffee break11:30–12:45 Contributed Session 812:45–14:30 Lunch break14:30–15:15 Invited Talk 6 (L. Decreusefond)15:15–16:30 Contributed Session 918:00–20:00 Tour around Toruń with a guide

Friday, 14 June 2013

09:00–09:45 Invited Talk 7 (G. Peccati)09:45–11:00 Contributed Session 1011:00–11:30 Coffee break11:30–12:45 Contributed Session 1112:45–13:00 Closing of the conference13:00–14:30 Lunch

10


in conjunction with


ABSTRACTS

Invited Talks

Limit theorems for random polytopes

Pierre Calka

University of Rouen, Saint-Étienne-du-Rouvray(France)


We construct a random polytope as the convex hull of a random setof points in Rd which is either a binomial or a Poisson point process. Weare interested in the asymptotic behaviour of various geometric char-acteristics, including the number of k-dimensional faces and intrinsicvolumes, when the size of the input goes to infinity. More precisely, weaim at obtaining second-order results with explicit limiting variance.

In the particular case of the convex hull of a binomial or Poissonpoint process in the unit-ball, the key idea is to apply a scaling trans-form to the whole picture and see the boundary of the random polytope(resp. its Voronoi flower) when the intensity of the point process goesto infinity as a so-called paraboloid hull process (resp. paraboloid growthprocess) in a product-space. Consequences of such approach includethe calculation of explicit limit variances for the characteristics men-tioned above as well as certain functional central limit theorems via thetechnique of stabilization. Thanks to an inversion transformation, theseresults have also counterparts for the typical cell of a Poisson-Voronoitessellation with large inradius. Finally, we will discuss extensions ofthe methods to more general settings, in particular when the unit-ballis replaced with a smooth convex body.

This talk will be a review of works due to Tomasz Schreiber &J.E. Yukich and joint works with them. It aims at emphasizing someof Tomasz Schreiber’s most beautiful contributions to the topic.

13

Invited Talks

Some geometrical aspects of wireless networks

Laurent Decreusefond

Télécom ParisTech(France)

In the next future, every electronic devices we know of (phone,tablets but also cars, houses, fridges, etc.) will be connected togetherand to the Internet. Since their communications are radio operated,their relative locations play a crucial as they determine interferenceand path-loss. We show how topological algebra and stochastic geome-try can be mixed together to analyze the geometrical features of thesefuture configurations.

Models with exclusions: Discretization andperfect simulation

Roberto Fernández

Utrecht University(Netherlands)


Models in continuum space are usually simulated by discretizingthe underlying space. The question arises as to how trustable is such aprocedure from a rigorous mathematical viewpoint. Is the discretizeddistribution close to the continuum one if the discretization step is smallenough? More generally, is the phase diagram of the discretized systemsimilar to that of the original continuum system? For systems definedby hard exclusions — e.g. Widom Rowlison models — we are ableto prove closeness between the continuous and discretized statistical-mechanical descriptions in regimes without phase transition. More pre-cisely, the models must satisfy a condition allowing the convergence ofthe so-called “ancestor’s algorithm”. This algorithm — which can beinterpreted as a probabilistic version of cluster expansions — yields, inaddition, a perfect simulation scheme coupling the continuum modelwith its discretizations. Tomasz Schreiber understood this algorithm

14

Invited Talks

better than its authors and exploited it to develop a sophisticated the-ory of polygonal Markov fields. He would have been a natural coauthorof our work.

Branching random tessellations with interaction– a last project of Tomasz Schreiber

Hans-Otto Georgii

Ludwig Maximilian University of Munich(Germany)

Tomasz Schreiber’s Memorial SessionBranching random tessellations (BRTs) are stochastic processes that

transform any initial tessellation of Rd into a finer tessellation by meansof random cell divisions in continuous time. In a draft entrusted to meby Tomasz Schreiber shortly before his death, he started an investi-gation of the ‘thermodynamics’ of these objects. A special case arethe so-called STIT tessellations, for which all cells split up indepen-dently. By way of contrast, the cells of a BRT are allowed to interact,in that the division rule for each cell may depend on the structure andpast of the surrounding tessellation. In addition, the cells are endowedwith an internal property, called their colour. Under a suitable con-dition, the cell interaction of a BRT can be specified by a measurekernel that determines the division rules of all cells and gives rise toa Gibbsian characterisation of BRTs. For translation invariant BRTs,one can introduce an ‘inner’ entropy density relative to a STIT tessel-lation. Together with an inner energy density for a given ‘moderate’division kernel, this leads to a variational principle for BRTs with thisprescribed kernel, and further to an existence result for such BRTs.

Clark-Okone type martingale representationsfor Poisson martingales

Günter Last

Karlsruhe Institute of Technology(Germany)

Tomasz Schreiber’s Memorial Session15

Invited Talks

Martingale representations are fundamental tools of stochastic anal-ysis. In the case of Brownian motion the Clark-Okone representationyields explicit expressions for the integrand in terms of Malliavin deriva-tives. A similar result is known for Poisson and Levy processes. In thistalk we will explain a general version of this representation for Poissonmartingales, taken from [1]. Our first application are short proofs ofthe Poincare- and the FKG-inequality for Poisson processes. A secondapplication is Wu’s [2] elegant proof of a general log-Sobolev inequalityfor Poisson functionals. If time permits we will also discuss minimalvariance hedging or some potential applications in stochastic geometry.

References[1] G. Last, M.D. Penrose, Martingale representation for Poisson processes with

applications to minimal variance hedging, Stochastic Processes and their Appli-cations 121 (2011), 1588–1606.

[2] L. Wu, A new modified logarithmic Sobolev inequality for Poisson point processesand several applications Probability Theory and Related Fields 118 (2000), 427–438.

Determinantal point process models andstatistical inference

Jesper Møller

Aalborg University(Denmark)

Determinantal point processes (DPPs) are largely unexplored instatistics, though they possess a number of appealing properties andhave been studied in mathematical physics, combinatorics, and randommatrix theory. In this talk we consider statistical models and inferencefor DPPs on Rd, with a focus on d = 2.

DPPs are defined by a function C satisfying certain regularity condi-tions; usually C is a continuous covariance function where its spectrumis bounded by one. DPPs possess the following appealing properties:

(a) They are flexible models for repulsive interaction, except in caseswith strong repulsiveness (as e.g. in a hard-core point process).

16

Invited Talks

(b) All orders of moments of a DPP are described by certain deter-minants of matrices with entries given in terms of C.

(c) A DPP restricted to a compact set has a density (with respect toa Poisson process) which is expressible in closed form.

(d) A DPP can easily be simulated, since it is a mixture of ‘determi-nantal projection processes’.

(e) A one-to-one smooth transformation or an independent thinningof a DPP is also a DPP.

In contrast, Gibbs point processes, which constitute the usual classof models for repulsive interaction, do not in general have moments thatare expressible in closed form, the density involves an intractable nor-malizing constant, rather time consuming Markov chain Monte Carlomethods are needed for simulations and approximate likelihood infer-ence, and an independent thinning of a Gibbs point process does notresult in a tractable point process.

In the talk, we discuss the fundamental properties of DPPs, investi-gate how to construct parametric models, and study different inferentialapproaches based on moments or maximum likelihood.

The work has been carried out in collaboration with Ege Rubak,Aalborg University, and Frédéric Lavancier, University of Nantes.

On limits and transfer on the Poisson space

Giovanni Peccati

University of Luxembourg(Luxembourg)

I will illustrate some recent findings concerning quantitative limittheorems on the Poisson space, that are specifically motivated by prob-lems in stochastic geometry. If time permits, I will show that someobjects arising in our study can be used to build a class of counterex-amples to a ’transfer principle’ (recently proved by Kemp, Nourdin,Peccati and Speicher), concerning central limit theorems for infinitelydivisible random measures in a classical and free framework.

Joint works with S. Borguin and R. Lachièze-Rey.17

Invited Talks

Random parking and rubber elasticity

Mathew Penrose

University of Bath(United Kingdom)


Renyi’s random parking process on a domain D in d-space is apoint process with hard-core and no-empty-space properties that aredesirable for modelling materials such as rubber. It is obtained asfollows: particles arrive sequentially at uniform random locations inD, and are rejected if they violate the hard-core constraint, until theaccepted particles saturate D.

We describe how any real-valued functional on this point process,provided it enjoys certain subadditivity properties, satisfies an averag-ing property in the thermodynamic limit. Consequently in this limit,one has a convergence of macoroscopically-defined energy functionalsfor deformations of the point process, to a homogenized limiting energyfunctional. We may also apply the results to derive laws of large num-bers for classical optimization problems such as travelling salesman onthe parking point process.

This is joint work with Antoine Gloria.

Random polytopes and random polyhedra

Matthias Reitzner

University of Osnabrück(Germany)

Random polytopes can be generated in various natural ways. Let Kbe a convex body and choose random points either uniformly from K oraccording to some density from its boundary. The convex hull of theserandom points is a random polytope. In a somehow dual way, chooserandom halfspaces containing K. The intersection of these halfspacesis a random polyhedra.

18

Invited Talks

In this talk we present old and new results on random polytopesand random polyhedra. In particular we are interested in the number offaces of random polytopes and random polyhedra: expectations, highermoments, and limit theorems.

Combinatorial structure of iteration stabletessellations in R3

Christoph Thäle

Ruhr University Bochum(Germany)

Iteration stable tessellations have been introduced by Nagel andWeiß in 2005 and have attracted considerable interest since then. Lo-cally, they arise as outcome of a space-time process of repeated cell di-vision and are not facet-to-facet. In this talk we focus on the particularcase of space dimension three and explore the rich combinatorial struc-ture of this tessellation model. At first, we introduce several classes ofline segments and polygons and calculate an exhaustive array of meanvalues. In the second part of the talk, we present a redined classificationof line segments associated with an iteration stable tessellation, whichis based on a classification of tessellation vertices. As a consequencewe are able to derive mean values for the neighborhood of the typicalvertex, for example.

The material presented is based on joint works with Viola Weiß.

Discrete multi-colour random mosaics with anapplication to network extraction

Marie-Colette Van Lieshout

Centrum Wiskunde & Informatica (CWI), Amsterdam(Netherlands)

Tomasz Schreiber’s Memorial Session19

Invited Talks

We introduce a class of random fields that can be understood asdiscrete versions of multi-colour polygonal fields built on regular lineartessellations. These models generalise the binary fields introduced bySchreiber and Van Lieshout (2010).

We focus first on consistent polygonal fields, for which we showMarkovianity and solvability by means of a dynamic representation.This representation enables us to design new sampling techniques forGibbsian modifications of such fields, a class of models which includesthe classic lattice based random fields that are widely used in imageanalysis. We then illustrate the applicability of our models by using aflux based modification to extract the field tracks network from a SARimage of a rural area.

References[1] T. Schreiber, M.N.M. van Lieshout, Disagreement loop and path cre-

ation/annihilation algorithms for binary planar Markov fields with applicationsto image segmentation, Scand. J. Statist. 37 (2010), 264–285.

[2] M.N.M van Lieshout, Multi-colour random fields with polygonal realisations,ArXiv 1204.2664, November 2012.

A spatio-temporal model for fMRI data – witha view to resting state networks

Eva B. Vedel Jensen

Aarhus University(Denmark)


Functional magnetic resonance imaging (fMRI) is a technique forstudying the active human brain. During the fMRI experiment, a se-quence of MR images is obtained, where the brain is represented as aset of voxels. The data obtained are a realization of a complex spatio-temporal process with many sources of variation, both biological andtechnical. Most current model-based methods of analysis are based ona two-step procedure. The initial step is a voxel-wise analysis of thetemporal changes in the data while the spatial part of the modelling is

20

Invited Talks

done separately as a second step in the analysis. In this talk, a spatio-temporal point process model approach for fMRI data will be presentedwhere both the temporal and spatial activation are modelled simulta-neously. This modelling framework allows for more flexibility in theexperimental design than most standard methods. It is also possible toanalyze other characteristics of the data than just locations of activebrain regions, such as the interaction between the active regions. Inthis talk, we discuss both classical statistical inference and Bayesianinference of the model. We analyze simulated data without repeatedstimuli both for location of the activated regions and for interactionsbetween the activated regions. An example of analysis of fMRI data,using this approach, will be presented.

Surface order scaling in stochastic geometry

Joseph Yukich

Lehigh University, Bethlehem, PA(United States)

Limit theory for functionals in stochastic geometry having inputgiven by an i.i.d. sample often involves scaling by the sample size.However in many situations, this scaling is not natural. We give gen-eral results showing when it is more natural to scale by the surfacearea of an appropriately chosen re-scaled manifold of lower dimension.Using the general results we deduce variance asymptotics and centrallimit theorems for the Poisson-Voronoi volume estimator, the Poisson-Voronoi surface area estimator, and for the number of Pareto extremepoints in an i.i.d. sample in a smooth subset of Rd.

Stabille random measures and stable pointprocesses

Sergei Zuyev

Chalmers University of Technology, Gothenburg(Sweden)

21

Invited Talks

Stable distributions play fundamental role in probability since theynecessarily arise in variety of limit theorems. The definition is basedon two operations: addition and scaling by positive numbers. How-ever, in discrete semigroups scaling by arbitrary positive factor can-not be defined and is replaced by a stochastic operation which givesrise to the corresponding stable random elements. Particular examplesinclude discrete stable non-negative random integers studied by Steu-tel and van Harn (1979). A scaling operation here can be defined bytransforming an integer into the corresponding binomial distributionwith success probability being the scaling factor. We explore a simi-lar (thinning) operation defined on counting measures and characterisethe corresponding discrete stablility property of point processes. It isshown that these processes are exactly Cox (doubly stochastic Poisson)processes with strictly stable random intensity measures. We obtainspectral and LePage representations for strictly stable measures andcharacterises some special cases, e.g. independently scattered measures.An alternative cluster representation for discrete stable processes is alsoderived using the so-called Sibuya point processes that constitute a newfamily of purely random point processes.

22

Contributed Talks

Random marked sets and statistical applications

Viktor Beneš

Charles University in Prague(Czech Republic)

We consider random marked sets with integer dimension smallerthan the dimension of the given Euclidean space. Statistical analysis isdeveloped which involves the random-field model test and estimationof first and second order characteristics. Special models are presentedbased on tessellations and marking by means of Gaussian random fields.Real data analysis from the materials research investigating a grainmicrostructure with disorientation of faces as marks is presented.

This is a joint work with Ondrej Sedivy and Jakub Stanek.

Spectral percolation

Charles Bordenave

CNRS & University of Toulouse(France)

We will consider edge percolation on Zd where each edge is presentindependently with probability p. Look at the spectrum of the adja-cency matrix of a finite box of size length n. As p increases from 0 to 1,is there a phase transition on the regularity of the spectrum? Does itcoincides with the usual transition for existence of an infinite connectedcomponent? We will answer partly this question in dimension 2 andstudy related questions on locally-tree like graphs.

This is joint work with Arnab Sen and Balint Virag.

23

Contributed Talks

Mixing properties and further results forstationary Poisson cylinder processes

Christian Braeu

University of Augsburg(Germany)

Stationary Poisson cylinder processes Ξk,d for k = 1, . . . , d−1 in thed-dimensional Euclidean space are stationary Poisson processes on thespace of cylinders K ⊕ Lk, where Lk is a linear subspace of dimensionk and the base K ⊂ L⊥k is a compact subset. The particular case Ξ0,d

coincides with Poisson-grain processes (Boolean model) in Rd, see [1],[2]and [3].

In this talk we will show that stationary Poisson cylinder processesΞk,d (k = 1, . . . , d − 1) are always weakly mixing and hence ergodic.Further we give a necessary and sufficient condition for Ξk,d to be mix-ing, which only depends on the directional distribution of the cylindersand the dimension k.

The case k = d− 1 and the bases of the cylinders are intervals is ofparticular interest, that means the cylinders are thick hyperplanes. Westudy the stationary point process defined by the intersection points ofthe thick hyperplanes and give formulas for the intensity and second-order characteristics. Furthermore, we consider the closed complementof the union set. It forms a stationary random closed set, which isthe union of disjoint convex polytopes and may be interpreted as atessellation with thick boundaries. We will derive some mean values ofthe typical polytope.

References[1] P. Davy, Stereology — A Statistical Viewpoint. Thesis, Australian National

University, (1976), Canberra.

[2] M. Spiess, E. Spodarev, Anisotropic Poisson processes of cylinders, Methodol.Comput. Appl. Probab. 13 (2011), 801–819.

[3] L. Heinrich, M. Spiess, CLT for the volume of stationary Poisson cylinderprocesses in expanding domains, Adv. App. Prob. 45 (2013), (to appear).

24

Contributed Talks

Variational approach for spatial point processintensity estimation

Jean-François Coeurjolly

Laboratory Jean Kuntzmann, Grenoble(France)

We introduce a new variational estimator for the intensity func-tion of an inhomogeneous spatial point process with points in the d-dimensional Euclidean Rd space and observed within a bounded region.The variational estimator applies in a simple and general setting whenthe intensity function is assumed to be of log-linear form β + θ>z(u)where z is a spatial covariate function and the focus is on estimating θ.The variational estimator is very simple to implement and quicker thanalternative estimation procedures. We establish its strong consistencyand asymptotic normality. We also discuss its finite-sample propertiesin comparison with the maximum first order composite likelihood es-timator when considering various inhomogeneous spatial point processmodels and dimensions as well as settings were z is completely or onlypartially known.

This is a joint work with Jesper Møller (Aalborg University).

Size distributions in random triangles

Daryl Daley

University of Melbourne(Australia)

The random triangles to be discussed are defined by having the di-rections of their sides independent and uniformly distributed on (0, π).To fix the scale one side is assigned unit length; let a and b denotethe lengths of the other two sides. We find the density functions ofa/b, max{a, b}, min{a, b} and of the area of the triangle, the first threeexplicitly and the last as an elliptic integral. The first two density func-tions, with supports in (0,∞) and (half,∞) respectively, are unusualin having an infinite spike at 1 which is interior to their ranges (thetriangle is then isosceles).

25

Contributed Talks

On the topology of Boolean model overstationary point processes

Yogeshwaran Dhandapani

Technion – Israel Institute of Technology, Haifa(Israel)

There has been recent interest in understanding the homology ofBoolean model, studied via random simplicial complexex. Random ge-ometric complexes have points of point processes as vertices and facesare determined by some determinisic geometric rule. I shall try to de-scribe the growth of homology groups as measured via Betti numbers.In particular, i shall focus on the quantitaive differences in the growthof Betti numbers between the Poisson point processes and other pointprocesses which exihibit repulsion such as Ginibre ensemble, Zeros ofGaussian analytic function, perturbed lattices etc. I shall also hint atmore general applications of the proof techniques to analysis of compo-nent counts and Morse critical points.

A Bayesian inference framework for locallyscaled spatial point processes

Eva-Maria Didden

Heidelberg University(Germany)

Pairwise interaction point processes are an important and flexiblemodeling class for spatial point patterns. The unknown normalizingconstant of the likelihood, however, makes inference complicated andrequires elaborate procedures. Beyond that, the consideration of inho-mogeneous point configurations involves additional challenges. Assum-ing that the inhomogeneity follows location-dependent scaling proper-ties, we present an efficient Bayesian inference framework for heteroge-neous Strauss processes. Our approach is based on an auxiliary variablescheme in form of the exchange algorithm and includes coupling fromthe past to yield direct samples from the unnormalized likelihood. In acase-study, we apply the suggested framework to cross-sections through

26

Contributed Talks

maize plants, where the goal is to accurately describe and classify dif-ferent genotypes.

Parameter estimation for inhomogeneousspace-time shot-noise Cox point processes

Jiří Dvořák


Shot-noise Cox point processes constitute an important class ofmodels for spatial or space-time cluster processes. For space-time pointprocesses parameter estimation methodology has not been fully devel-oped yet. In general, for Cox point processes finding maximum likeli-hood estimates is typically computationally too demanding and inho-mogeneous space-time shot-noise Cox processes are no exception. Thus,development of faster, simulation-free methods is desirable. In this con-tribution we will discuss possibilities of parameter estimation for suchprocesses, assuming appropriate non-trivial model structure, and assessthe properties of the resulting estimators.

Moment formulae for general point processes.

Ian Flint

Télécom ParisTech, Paris(France)

We consider a general point process, and show that the relevantquantities to the problem of calculating the moment of stochastic in-tegrals are the so-called Papangelou intensities. Then, we show somegeneral formulae to recover the moment of order n of the stochastic in-tegral of a random process. We will use these extended results to studythe divergence of a general point process. To conclude, we will study arandom transformation of the point process and characterize its law.

27

Contributed Talks

Viral marketing in random networks

Kumar Gaurav

Inria/ENS, Paris(France)

The inspiration for this paper comes from the phenomenon of viralmarketing in social networks. A person after getting acquainted withan advertisement (or a news article or a Gangnam style video, for thatmatter) through one of his “friends”, may decide to share it with rest ofhis friends, some of whom will, in turn, pass it along to their friends,and so on. We model the social network by an enhanced ConfigurationModel where the degree of a vertex would represent the number offriends of a person and the sender degree the number of friends he/sheis able to influence. We give a condition on the degree distributionof the configuration model which if not satisfied would prohibit anyadvertisement etc. to become viral (i.e., the size of the populationreached by the advertisement would be o(n) w.h.p., where n is thesize of the population). Further, we show that if the initial target ofthe advertisement campaign is chosen from a set of “Pioneers” whoinduce the advertisement to go viral, then the core component of thepopulation reached is same regardless of the Pioneer chosen, and weexplicitly evaluate the asymptotic size of this component relative ton. We then introduce a reverse process to trace the initial source ofinfluence of a person and establish a duality relation to identify the largecore component of this process with the set of Pioneers in the originalprocess of advertisement diffusion and thus calculate the asymptoticrelative size of the set of Pioneers. In our study of both the originaland the reverse dual process we use the approach proposed in Janson& Luczak(2008) for the study of the giant component of ConfigurationModel.

This is a joint work with B. Błaszczyszyn.

Improved fitting of point process models byreparametrization

Ute Hahn


28

Contributed Talks

Parameters in point process models often influence both intensityand spatial arrangement of points, which is commonly expressed by sec-ond order statistics such as the K-function. Most point process modelscan however be orthogonally reparameterized, such that intensity andspatial arrangement are determined by different parameters. Togetherwith appropriate versions of second order statistics, minimum contrastfitting becomes less computationally intensive, since the effective di-mension of the parameter space is reduced.

Some inequalities and extremal problems forchord-power integrals of ellipsoids and

parallelotopes

Lothar Heinrich

University of Augsburg(Germany)

First, we introduce the standard definition, several equivalent ex-pressions and some well-known inequalities for chord-power integrals(briefly: CPIs) of order k = 0, 1, ..., d, d + 1 for fixed convex bodies Kin the d-dimensional Euclidean space. Second, we motivate the topic oftalk by discussing some situations, where CPIs occur, for example, asenergy functionals in physical contexts, see [2], or as asymptotic vari-ances in CLTs for motion-invariant Poisson cylinder and hyperplaneprocesses, see [1]. Third, we prove some inequalities of the dth-orderCPI of general ellipsoids and of parallelotopes by comparing them withthe corresponding CPI of d-balls and hyperrectangles, respectively. Fi-nally, we show that the dth-order CPI of parallelotopes with fixed vol-ume is maximized by the dth-order CPI of the d-cube with the samevolume. On the other hand, among all parallelotopes with fixed totaledge length the d-cube possesses the least dth-order CPI. Similar resultscan be shown for ellipsoids which supports an longstanding unprovedconjecture, see [3].

References[1] L. Heinrich, M. Spiess, CLT for the volume of stationary Poisson cylinder

processes in expanding domains, Adv. Appl. Prob. 45 (2013), (to appear).

29

Contributed Talks

[2] J. Hansen, M. Reitzner, Electromagnetic wave propagation and inequalitiesfor moments of chord lengths, Adv. Appl. Prob. 36 (2004), 987–995.

[3] P.J. Davy, Inequalities for moments of secant length, Z. Wahrscheinlichkeits-theorie verw. Geb. 68 (1984), 243–246.

First-passage percolation on random geometricgraphs and an application to shortest-path trees

Christian Hirsch

Ulm University(Germany)

We consider Euclidean first-passage percolation on a large familyof connected fibre processes in the d-dimensional Euclidean space en-compassing various well-known models from stochastic geometry. Inparticular, we establish a strong linear growth property for shortest-path lengths on fibre processes which are generated by point processes.This result comprehends two special cases which are of independentinterest, where we consider the event that the growth of shortest-pathlengths between two (end-) points of the path does not admit a linearupper bound. Our linear growth property implies that the probabil-ity of this event tends to zero sub-exponentially fast if the direct (Eu-clidean) distance between the endpoints tends to infinity. Moreover, theshortest-path length between two points at fixed distance admits a sub-exponential tail. Besides, for a wide class of stationary and isotropicfibre processes, our linear growth property implies a shape theorem forthe Euclidean first-passage model defined by such fibre processes. Fi-nally our shape theorem can be used to investigate a problem whichis considered in structural analysis of fixed-access telecommunicationnetworks, where we determine the limiting distribution of the length ofthe longest branch in the shortest-path tree extracted from a typicalsegment system if the intensity of network stations converges to zero.

The talk is based on joint work with David Neuhaeuser, CatherineGloaguen and Volker Schmidt.

30

Contributed Talks

Minkowski tensor density formulas for Booleanmodels

Julia Hörrmann


A stationary Boolean model is the union set of random compact par-ticles which are attached to the points of a stationary Poisson point pro-cess. For a Boolean model with convex grains we consider a recently de-veloped collection of shape descriptors, the so called Minkowski tensors.By combining spatial and probabilistic averaging we define Minkowskitensor densities of a Boolean model. These densities are global charac-teristics of the union set which can be estimated from observations. Incontrast local characteristics like the mean Minkowski tensor of a singlerandom particle cannot be observed directly, since the particles overlap.We relate the global to the local properties by density formulas for theMinkowski tensors. These density formulas generalize the well knownformulas for intrinsic volume densities and are obtained by applyingresults from translative integral geometry. For an isotropic Booleanmodel we observe that the Minkowski tensor densities are proportionalto the intrinsic volume densities, whereas for a non-isotropic Booleanmodel this is usually not the case. Our results support the idea thatthe degree of anisotropy of a Boolean model may be expressed in termsof the Minkowski tensor densities.

Joint work with Daniel Hug, Michael Klatt and Klaus Mecke.

Second order properties and central limittheorems for geometric functionals of Boolean

models

Daniel Hug


31

Contributed Talks

Let Z be a stationary Boolean model based on a stationary Poissonprocess η of convex particles in Euclidean space Rd. Let W denotea convex observation window. For a large class of functionals ψ, for-mulas for the mean value of ψ(Z ∩W ) are available in the literature.In this talk, we describe the (asymptotic) covariances of ψ(Z ∩ W )for general geometric functionals ψ, which are additive and translationinvariant. For the asymptotic result, it is shown that the rate of con-vergence is determined by the inradius of W . Our argument is basedon the Fock space representation of the Poisson process η. In the im-portant special case of intrinsic volumes the asymptotic covariances canbe explicitly expressed in terms of suitable moments of local curvaturemeasure. Moreover, we provide multivariate central limit theorems in-cluding Berry-Esseen bounds. These are based on a general normalapproximation result obtained by the Malliavin-Stein method.

This is joint work with Günter Last and Matthias Schulte.

Capturing the SINR in cellular networks withPoisson processes

Paul Keeler


The steady rise of user-traffic in wireless cellular networks has re-sulted in the need for developing robust and accurate models of variousperformance metrics. A key metric of these networks is the signal-to-interference-and-noise-ratio (SINR) experienced by a typical user. Fortractability, often the positions of base stations in such networks aremodelled by Poisson point processes whereas actual deployments oftenmore resemble lattices (e.g. hexagonal). Strikingly, under log-normalshadowing it has been observed that the SINR experienced by a typi-cal user is more accurate in a Poisson model than a hexagonal model.In this talk we seek to explain this interesting observation by way ofa convergence result. Furthermore, we present numerically tractable,explicit integral expressions for the distribution of SINR of a cellularnetwork modelled by Poisson process. Our model incorporates a power-law path-loss model with arbitrarily distributed shadowing. The results

32

Contributed Talks

are valid in the whole domain of SINR and, unlike previous methods,do not require the inversion of Laplace transforms.

Joint work with B. Błaszczyszyn and M.K. Karray.

Matérn’s hard core models of types I and IIwith arbitrary compact grains

Markus Kiderlen


Matérn’s classical hard core models can be interpreted as modelsobtained from a stationary marked Poisson process by dependent thin-ning. The marks are balls of fixed radius, and a point is retained whenits associated ball does not hit any other balls (type I) or when itsrandom birth time is strictly larger than the birth times of all ballshitting it (type II). Extending ideas of [Månsson, M. and Rudemo, M.(2002). Random patterns of nonoverlapping convex grains. Adv. inAppl. Probab. 34, 718—38], who considered grains that are isotropicrotations or random scalings of a fixed convex set, we discuss these twomodels in d-dimensional space when the marks are arbitrary randomcompact grains. We determine the intensity and the mark distributionafter thinning. In the case where the distribution of the birth-timesis atom-free and independent of the locations of the points and theirgrains, we find the second order factorial moment density of the groundprocess for model II, and determine the volume density of the associatedprocess of non-overlapping particles. Using Brunn-Minkowski inequal-ity, this volume density turns out to be bounded by 2−d. This boundis sharp. It is attained asymptotically (when the proposal intensitytends to infinity) only when all grains coincide with one deterministicorigin-symmetric compact set. We also discuss how known connectionsof these models with the Stienen model and the process of intact grainsof the dead leaves model leads to analogous results for the latter.

33

Contributed Talks

Stochastic geometry in multiscale electrontomography

Albert F. Lawrence

University of California, San Diego, La Jolla(United States)

Biological processes take place on a wide range of spatial and tem-poral scales, from the molecular to the whole organism. Ongoing workin light and electron microscopy has extend the scale available for imag-ing to the level of large molecules and molecular assemblies. Fluores-cence microscopy, in addition to the structural information availablefrom electron microscopy, has added dynamical information as well asinformation about the location of specific proteins. Modern stainingtechniques also can locate the site of fluorescence within nanometers.We may exploit the opportunities created by these two imaging tech-nologies in several diverse areas.

Correlated microscopies. The first problem is to correlate the infor-mation gained by light and electron microscopy. The immediate task isto overlay images or sections of 3D reconstructions so that identifiablestructures are in geometric correspondence. This task is complicated bythe range of spatial scales and the nature of the images at the variousspatial scales.

Uncertainty quantification. Microscopy is Inherently probabalistic.Binding and staining are the result of random processes, and florescenceitself is subject to quantum uncertainty at the molecular level. Theselead to uncertainty in both position and magnitude or relevant physicalevents. A second source of error is due to uncertainty in the trajectoryof the interrogating radiation. Paths of electrons through samples andlenses are subject to local fluctuations in electrical and magnetic fields,and light paths are affected by varying optical densities across materialinterfaces. A third source of error is in the manual marking of structuresfor alignment. As microscopy becomes more quantitative, in terms ofmeasurement of structure and dynamics of biological processes the needfor identification and quantification of sources of uncertainty becomesmore acute.

Systems biology. Decades of work on the metabolic processes hasresulted in the construction of network models relating the chemical

34

Contributed Talks

interactions and transformations in living cells. There is increasing ev-idence that this chemical activity is compartmentalized and that geo-metric proximity is a large factor determining whether specific reactionstake place. Accordingly it is necessary to map the metabolic network tothe spatial reconstructions available from light and electron microscopy.This would entail the introduction of new mathematical structures intotheoretical biology in order to go beyond the database approaches cur-rently in vogue.

Law of large numbers for matchings, extensionsand applications

Marc Lelarge


The fact that global properties of matchings can be read from localproperties of the underlying graph has been rediscovered many timesin statistical physics, combinatorics, group theory and computer sci-ence. I will present a probabilistic approach allowing to derive law oflarge numbers. I will show how it extends previous results in severaldirections and describe some algorithmic applications.

Joint work with Charles Bordenave, Justin Salez, Mathieu Leconte,Laurent Massoulié, Hang Zhou.

Superlinear expectations of random sets andtheir applications

Ilya Molchanov

University of Bern(Switzerland)

The conventional selection (Aumann) expectation of random sets islinear with respect to the Minkowski addition. The talk presents defini-tions of sub- and superlinear expectations of random sets. The construc-tion of superlinear expectation is based on applying univariate sublinear

35

Contributed Talks

expectations to selections of random sets. The main mathematical re-sults concern the dual representation for the introduced expectationsand the Fatou lemma for superlinear expectations of unbounded randomclosed sets. The introduced concepts are used in finance to constructmultivariate risk measures for models with transaction costs.

New exact methods for envelope testing

Tomáš Mrkvička

University of South Bohemia, České Budějovice(Czech Republic)

The conventional envelope test and the deviation test are the maintools for testing simple spatial hypotheses and goodness-of-fit testingfor spatial point processes. The deviation test is a proper statisticaltest, but it says only a little about the reason of rejection of the nullhypothesis and it does not have a direct visual interpretation that isoften desired by researchers. The conventional envelope test is a not aproper statistical tool while it suffers from multiple testing problem andmay have an unacceptably high type I error. Recently, Grabarnik, Myl-lymäki and Stoyan (2011) proposed the refined envelope test in whichthe type I error is adjusted to reach a reasonable level. This test how-ever does not provide a p-value of the test and it demands adjustingthe number of simulations from the null hypothesis together with eval-uation of the type I error of the test. We introduce new methods whichprovide the exact p-value and have the graphical interpretation that, ifthe data characteristic is not inside the constructed envelope, the nullhypothesis is rejected with a given significance level. The new meth-ods are completetely nonparametric RANK ENVELOPE TEST, thesemiparametric ASYMMETRIC QUANTILE and VARIANCE STA-BILIZED ENVELOPE TESTs and the parametric APPROXIMATENORMAL ENVELOPE TEST for fast inference for models with te-dious simulation. We illustrate the new envelope tests through a dataexample and study their performance by a simulation study.

36

Contributed Talks

STIT tessellation processes and their ergodicproperties

Werner Nagel

Friedrich Schiller University Jena(Germany)

We consider random processes where the states are tessellations ofthe plane or a higher-dimensional Euclidean space. The division of cellsby random hyperplanes as well as the iteration of tessellations can causethe transition of states. Among these models, the STIT tessellationmodel (stochastically STable under the operation of ITeration of tessel-lations) appears to be the most fruitful one and potentially a referencemodel for applications. It was introduced by Nagel and Weiß (2005). Asubstantial progress in the study of its properties was achieved by thework of Tomasz Schreiber and Thäle. In the talk, a survey of importantfeatures is given, and recent results on ergodic properties (joint workwith Martínez) are discussed.

Column tessellations

Nguyen Ngoc Linh

Friedrich Schiller University Jena(Germany)

In this talk a new class of random spatial tessellations is introduced– the so-called column tessellations. Based on a given stationary planartessellation, we consider three constructions of different kinds of columntessellations in R3. All these spatial tessellations are not face-to-face.By using parameters of the planar tessellation, the intensities, meanvalues and some distributions of the column tessellation are given.

Statistics for Poisson models of overlappingspheres

Zbyněk Pawlas


37

Contributed Talks

We consider a stationary d-dimensional Boolean model with spher-ical grains. A family of nonparametric estimators for the radius distri-bution is proposed. These estimators are based on observed distancesand radii, weighted in an appropriate way. They are ratio-unbiasedand asymptotically consistent for growing convex observation window.We establish asymptotic normality under a suitable integrability as-sumption on the weight function. The proof is based on the truncationargument and approximation by m-dependent random fields. We alsodiscuss some related estimators. A simulation study is performed tocompare the behaviour of different estimators.

Joint work with Daniel Hug, Günter Last and Wolfgang Weil.

Moderate deviations for stabilizing functionalsin geometric probability

Martin Raič

University of Ljubljana(Slovenia)

The purpose of this paper, which is a joint work with Peter Eichels-bacher and Tomasz Schreiber, is to establish explicit bounds on moder-ate deviation probabilities for a rather general class of geometric func-tionals enjoying the stabilization property, under the assumption of acertain control over the growth of the moments of the functional and itsradius of stabilization. Our proof techniques rely on cumulant expan-sions and cluster measures and yield completely explicit bounds. Weapply our results to two groups of examples: random packing modelsand nearest neighbor graphs.

Recurrence or transience of random walks onrandom graphs generated by point processes

in Rd

Arnaud Rousselle

University of Rouen, Saint-Étienne-du-Rouvray(France)

38

Contributed Talks

Random walks on random graphs in Rd arise naturaly to describeflows, molecular diffusions, heat conduction or other problems fromsatistical mechanics in random and irregular media. The general ideais to extend known results on the lattice Zd to graphs generated bypoint processes in Rd.

In this talk, we consider conductance random walks on graphs de-pending on the geometry of an infinite random set of points. Moreprecisely, given a realisation of a simple point process in Rd, an infinitelocally finite connected graph G = (V,E) is constructed and equippedwith a conductance C, that is a positive symmetric function on its edgeset. The random walk on G associated with C is the (time homoge-neous) Markov chain (Xn)n with transition probabilities given by:

P[Xn+1 = v

∣∣Xn = u]

=C(u, v)

w(u),

where w(u) :=∑

v∼uC(u, v).We present two general criteria for recurrence or almost sure tran-

sience of such walks. The proofs of these results rely on a well-knownanalogy between random walks and electrical networks, and on a com-parison with random walk on percolation clusters in Zd. Under suitableassumptions on the point process N and the conductance C, we provethat random walks on the Delaunay triangulation, the Gabriel graphor the skeleton of the Voronoi tiling of N are recurrent if d = 2 andtransient if d ≥ 3.

Distances between Poisson k-flats

Matthias Schulte


The flats of a stationary Poisson k-flat process with k < d/2 in Rddo not intersect almost surely so that every pair of flats has a well de-fined distance and a midpoint. For the number of pairs of flats havinga distance less than a given threshold and a midpoint in an increasingobservation window the asymptotic variance is computed and a centrallimit theorem is proven. Moreover, it is shown that the distances be-tween the flats behave after a suitable rescaling asymptotically like a

39

Contributed Talks

Poisson point process on the real line. As a consequence, we obtain thelimiting distribution of the m-th smallest distance between two flats.

This is joint work with C. Thäle.

Bounds for the probability generatingfunctional of a Gibbs point process

Kaspar Stucki


We derive explicit lower and upper bounds for the probability gen-erating functional of a stationary locally stable Gibbs point process,which can be applied to summary statistics like the F function. Forpairwise interaction processes we obtain further estimates for the G andK functions, the intensity and higher order correlation functions. Theproof of the main result is based on Stein’s method for Poisson pointprocess approximation.

Local digital algorithms for grey-scale images

Anne Marie Svane


Local algorithms are widely used for estimating intrinsic volumesbased on black-and-white digital images. In the real world, however,recorded images are usually blurred grey-scale images that are con-verted to black-and-white by thresholding, thus ignoring most of theinformation in the original image.

Local algorithms are typically biased in the design based setting,even in the limit when the resolution tends to infinity. In this talk weintroduce a version of local algorithms that can be applied directly togrey-scale images and we shall see how this in some situations leads toasymptotically unbiased estimators.

40

Contributed Talks

Clique number of random geometric graphs

Anais Vergne

Télécom ParisTech(France)

The clique number C of a graph is the largest clique size in thegraph. For a random geometric graph of n vertices, taken uniformly atrandom, including an edge beween two vertices if their distance, takenwith the uniform norm, is less than a parameter r on a torus ad, wefind the asymptotic behaviour of the clique number. Setting θ = (r/a)d,in the subcritical regime where θ = o(1/n), we exhibit the intervals oftheta where C takes the same value asymptotically almost surely. In thecritical regime, θ ∼ 1/n, we show that C is growing slightly slower thanlnn asymptotically almost surely. Finally, in the supercritical regime,1/n = o(θ), we prove that C grows as n∗θ asymptotically almost surely.We also investigate the behaviour of related graph characteristics: thechromatic number, the maximum vertex degree, and the independencenumber.

The local behaviour of the surface area ofparallel sets.

Steffen Winter


It is well known that the boundary surface area of parallel sets ofan arbitrary bounded set in Rd can be characterized in terms of itsparallel volume, which allows e.g. to extract important informationon the asymptotic behaviour of the boundary measure from studyingMinkowski contents. Using the notion of metrically associated sets, wewill demonstrate that these relations can be localized and that also thelocal behaviour of the surface area can be understood in terms of thelocal parallel volume. The results are based on the localization of atheorem by Stachó. Possible applications include self-similar and self-conformal sets as well as random sets such as the Wiener sausage orrandom fields.

41

Contributed Talks

Local stereology of tensors

Johanna F. Ziegel


Local stereology is widely used in the biosciences. It provides meth-ods to estimate quantitative properties of spatial structures such as vol-ume or surface area from measurements performed on sections passingthrough fixed points. Minkowski tensors are generalisations of intrin-sic volumes. They provide information about size, shape and orienta-tion of spatial structures. We present local stereological estimators forMinkowski tensors. In this general framework, many well-known localstereological estimators of volume and surface area appear as specialcases, but there are also new estimators, for example for the centre ofgravity and tensors of rank two.

This is joint work with Eva B. Vedel Jensen.

42

Poster Session

Estimating the volume of bird braincomponents by fakir probe

Jiří Janáček

Institute of Physiology, Academy of Sciences of the Czech Republic, Prague(Czech Republic)

Brains of extant birds were acquired using MR with high resolution.Volumes and exposed surface areas of main brain components (telen-cephalon, diencephalon, midbrain,cerebellum and pons with medullaoblongata) were estimated using virtual 3D grids of lines superimposedwith the MR image stacks. General formula for variance of the estima-tor is used for optimiyation of the measurements design.

Performance of Bayesian methods forparameters estimation for cluster point

processes

Jiri Kopecky

University of South Bohemia, České Budějovice(Czech Republic)

The preferred point process model to analyze clustered point pat-terns is Cox process. In this project we consider the problem of fittingNeyman-Scott point processes models, which are an important specialcase of Cox processes. Currently there are several methods to estimatethe parameters required for fitting model to real point patterns. Min-imum contrast method, composite likelihood method and Palm likeli-hood method would be a typical example.

However, more demanding methods are more accessible with thedevelopment of computer technology. Examples are Bayesian methodsusing Markov chain Monte Carlo.

43

Poster Session

The aim of the project is to compare Bayesian methods with meth-ods mentioned above. We present results of a simulation study in whichperformance of these estimating methods was compared for Neyman-Scott processes with different types and strength of clustering and inter-point interactions.

Image segmentation by locally specifiedmulti-coloured polygonal Markov fields

Michał Matuszak

Nicolaus Copernicus University in Toruń(Poland)

We present a novel algorithm for multi-colour image segmentationbased on Markovian optimisation dynamics combining the simulatedannealing ideas with those of Chen-style stochastic optimisation, inwhich successive segmentation updates are carried out simultaneouslywith adaptive optimisation of the local activity functions. We extenda class of polygonal Markov fields driven by local activity functionsto their multi-coloured form. The local nature of the field specifica-tion ensures substantial additional flexibility for statistical applicationsin comparison to classical polygonal fields. It has been shown that anumber of simulation algorithms and graphical constructions, as devel-oped in previous work of M.N.M. van Lieshout, R. Kluszczynski and T.Schreiber, carry over to this more general framework.

Dimension reduction in extendedQuermass-interaction process

Katerina Stankova Helisova

Czech Technical University in Prague(Czech Republic)

Consider a planar random set given by a union of randomly scat-tered interacting discs with centres in a bounded window and randomradii. The set is described by a parametrical density (with respect to

44

Poster Session

the probability measure of a stationary random-disc Boolean model)which depends on geometrical characteristics of the whole union. Inorder to estimate the parameters (e.g. by maximum likelihood methodusing MCMC simulations – MCMC MLE), it is useful to reduce thedimension of the geometrical characteristics in the density (and accord-ingly the dimension of the vector of corresponding parameters), sincethe estimating methods are very time-consuming, mainly because oflooking for the best estimate in the space of high dimension. This con-tribution concerns different methods for reduction of dimension of thegeometrical characteristics and consequent statistical inference of thereduced model.

On the use of particle Markov chain MonteCarlo in parameter estimation of space-time

interacting discs

Markéta Zikmundová


Particle Marginal Metropolis Hastings algorithm (PMMH) com-bines classical MCMC methods with sequential Monte Carlo. Based ontime–space parametrical model given by an union of interacting discswe study using PMMH for estimating its parameters.

Consider a random union of discs Uy ([4],[3]) given by density

p(y|x) = c−1x exp (x(A(Uy), L(Uy), χ(Uy)))

with respect to a reference Poisson point process of discs, where cxis an normalising constant, A(Uy), L(Uy), χ(Uy) total area, perimeterand Euler–Poincaré charakteristic of union respectively. Parameter x ∈R3 develops in time k according to a transition density pθ(xk|xk−1),where θ is an auxiliary parameter. Further we consider time dependencefor the configurations of union Uy (how to generate such process isdescribed in [5]).

PMMC algorithm is useful in case of unknown parameter θ. It isan combination of common particle filter ([2]) and Metropolis Hastingsalgorithm. At each iteration of Metropolis Hastings we firstly propose

45

Poster Session

parameter θ and then use common particle filter with this "known"parameter to estimate x. Then Hastings ratio is an combination ofproposal distribution of θ, its aprior and the marginal likelihood of p(y).A couple of proposals (θ,x) is accepted with propability correspondingto this ratio.

References[1] C. Andrieu, A. Doucet, R. Holenstein, Particle Markov Chain Monte

Carlo Methods, JRSS B 72(3) (2010), 269–342.

[2] A. Doucet, N. de Freitas, N. Gordon, Sequential Monte Carlo methods inpractice Springer, New York 2001.

[3] J. Moller, K. Helisová, Power diagrams and interaction process for unionsof discs, Adv Appl Prob 40 (2008), 321–347.

[4] J. Moller, K. Helisová, Likelihood inference for unions of interacting discs,Scand J. Statist 37 (2010), 365–381.

[5] M. Zikmundová, K. Staňková Helisová, V. Beneš, Spatio-temporal modelfor a random set given by a union of interacting discs, Methodology and Com-puting in Applied Probability, DOI.

46


in conjunction with


LIST OF PARTICIPANTS

List of participants

Viktor BENEŠ [email protected]łomiej BŁASZCZYSZYN [email protected] BORDENAVE [email protected] BRAEU [email protected] BUCHTA [email protected] CALKA [email protected] Tang CHRISTENSEN [email protected]çois COEURJOLLY [email protected] DADUNA [email protected] DALEY [email protected] DECREUSEFOND [email protected] DHANDAPANI [email protected] DIDDEN [email protected]ří DVOŘÁK [email protected] EICHELSBACHER [email protected] FERNÁNDEZ [email protected] FERRAZ [email protected] FLINT [email protected] GALLEGO [email protected] GAURAV [email protected] GEORGII [email protected] GORONCY [email protected] HAHN [email protected] HEINRICH [email protected] HIRSCH [email protected] HÖRRMANN [email protected] HUG [email protected]ří JANÁČEK [email protected] JASIŃSKI [email protected] KARŁOWSKA-PIK [email protected] KEELER [email protected] KIDERLEN [email protected] KLIMSIAK [email protected] KOPECKY [email protected] KOUSHOLT [email protected] JAKUBOWSKI [email protected]ünter LAST [email protected] F. LAWRENCE [email protected] LELARGE [email protected]ł MATUSZAK [email protected] MOLCHANOV [email protected] MØLLER [email protected]áš MRKVIČKA [email protected]

49

List of participants

Nguyen NGOC LINH [email protected] NIEMIRO [email protected]ěk PAWLAS [email protected] PECATTII [email protected] PENROSE [email protected] RAIČ [email protected] REITZNER [email protected] REJCHEL [email protected] ROUSSELLE [email protected] ROZKOSZ [email protected] RYCHLIK [email protected] SALEZ [email protected] SCHULTE [email protected] SEREINIG [email protected] SŁOMIŃSKI [email protected] SOJA-KUKIEŁA [email protected] STANKOVA HELISOVA [email protected] STEMESEDER [email protected] STUCKI [email protected] Marie SVANE [email protected] SZEKLI [email protected] S. SZEWCZAK [email protected] THÄLE [email protected] VAN LIESHOUT [email protected] B. VEDEL JENSEN [email protected] VERGNE [email protected] WEIL [email protected] WINTER [email protected] YUKICH [email protected] ZAIGRAJEW [email protected] F. ZIEGEL [email protected] ZIEMKIEWICZ [email protected]éta ZIKMUNDOVÁ [email protected] ZUYEV [email protected]

50

Documents

Book of abstracts - 17th Workshop on Stochastic Geometry