# 2011

Abstracts, see below.

13/1, Dr. Raphaël Lachièze-Rey, University of Lille-1, France, Ergodicity of STIT tessellations.
27/1, Alexey Lindo, Department of Mathematical Sciences, Chalmers University of Technology, A probabilistic analysis of Wagner's k-tree algorithm.
3/2, Sergei Zuyev, Chalmers, Discussion seminar - Optimal design of dilution experiments.
17/2, Peter Gennemark, Mathematical Sciences, Identifying and compensating for systematic errors in a large-scale phenotypic screen.
24/2, Takis Konstantopoulos, Uppsala University, A stochastic ordered graph model.
3/3, Victor Brovkin, Research Group Climate-Biogeosphere Interactions, Max Planck Institute for Meteorology, Hamburg, Germany, Land biosphere models for future climate projections.
10/3, Andrey Lange, Bauman Moscow State Technical University, Discrete stochastic systems with pairwise interaction.
10/3, Graham Jones, Durness, Scotland, Stochastic models for phylogenetic trees and networks.
15/3, Adam Jonsson, Luleå University of Technology, Invariant sets for continuous QMF functions. Abstract
31/3, Stig Larsson, Numerical approximation of stochastic PDEs.
7/4, The annual Mathematical Statistics Division meeting on teaching (lärarmöte)
19/4, Dustin Cartwright, Berkeley: How are SNPs distributed in genes?
3/5, Maria Deijfen, Stockholm University, Scale-free percolation.
5/5, Erik Kristiansson, Next generation DNA sequencing: data analysis and applications
12/5, András Bálint, The critical value function in the divide and colour model.
26/5, Vitali Wachtel, Mathematical Institute, LMU, München, Random walks in Weyl chambers.
31/5, Jenny Jonasson, Discussion seminar - Can we use extreme value theory to analyse data from naturalistic driving studies.
7/6, Chris Glasbey, Biomathematics & Statistics Scotland, Dynamic programming versus graph cut algorithms for fitting non-parametric models to image data.
16/6, Stefan Hoberg and Malin Persson, Optimal design for pharmacokinetic trials (Master thesis presentation).
23/6, Sonny Johansson and Tommy von Brömsen, Modelling Dependent Defaults in Static Credit Portfolios (Master thesis presentation)
16/8, Nitis Mukhopadhyay, University of Connecticut-Storrs, Sequential Fixed-Precision Estimation: A Review. Abstract
1/9, Ilya Molchanov, University of Bern, Switzerland, Partially identified models and random sets.
22/9, Serik Sagitov and Altynay Shaimerdenova, Al-Farabi University, Kazakhstan, Time to extinction for a population model with a high carrying capacity
29/9, Nanny Wermuth, The special features of regression graph models
30/9, Daniel Ahlberg, doctoral thesis, Asymptotics and dynamics in first-passage and continuum percolation
4/10, Mari Myllymäki, Aalto University, Testing of mark independence for marked point patterns.
6/10, Maria Deijfen, Stockholm University, Stable bigamy on the line.
7/10, Emilio Bergroth, licentiate thesis, Topics on Game Theory
13/10, Olle Häggström, A stochastic model for consensus formation in social networks
20/10, Martin S Ridout, University of Kent, UK, Numerical Laplace transform inversion for statisticians.
26/10, Erik Mellegård, master thesis, Obtaining Origin/Destination-matrices from cellular network data.
27/10, Jorge Mateu, Department of Mathematics, University Jaume I, Castellon, Spain, Functional spatial statistics with a focus on geostatistics and point processes.
28/10, Ottmar Cronie, doctoral thesis, Modelling and Inference for Spatio-Temporal Marked Point Processes
17/11, Jeffrey Steif, The behaviour of the lower tail of the distribution of a supercritical branching process at a fixed large time.
1/12, David Belius, ETH Zürich, Fluctuations of certain cover times.
8/12, Marina Axelson-Fisk, Using HMMs to predict driver manoeuvers (discussion seminar)

13/1 Dr. Raphaël Lachièze-Rey, University of Lille-1, France: Ergodicity of STIT tessellations
Abstract:
Random tessellations form a relevant class of models for many natural phenomena in biology, geology, materials science. STIT tessellations (for STable under ITeration), introduced in the 2000's, are characterised by their stability under an operation called "iteration", which confers to them a privileged role in modelling phenomena of cracking or of division in nature. After a clear exposition of the model, we will present its main characteristics, establishing in particular its mixing properties.

27/1 Alexey Lindo, Department of Mathematical Sciences, Chalmers University of Technology: A probabilistic analysis of Wagner's k-tree algorithm
Abstract:
David Wagner introduced an algorithm for solving a k-dimensional generalization of the birthday problem (see [1]). It has wide applications in cryptography and cryptanalysis. A probabilistic model of Wagner's algorithm can be described as follows. Suppose that elements of the input lists are drawn from additive group of integers modulo $n$. Let the random variable W represents the number of solutions found by Wagner's algorithm in the introduced model. We first observe that W is the sum of the dependent indicators. Then using Chen-Stein method we derive Poisson approximation to the distribution of W. An upper bound on a total variation distance given in [2,3] is particularly essential for the proof. The bound allows to estimate the strength of encoding by the algorithm in terms of its parameters.
[1] D. Wagner. A generalized birthday problem. http://www.cs.berkeley.edu/~daw/papers/genbday.html
[2] R. Arratia, L. Goldstein and L. Gordon. Two Moments Suffice for Poisson Approximations: The Chen-Stein Method.
[3] L.H.Y. Chen. Poisson approximation for dependent trials.

3/2 Sergei Zuyev, Chalmers: Discussion seminar - Optimal design of dilution experiments
Abstract:
This is the first in a (hopefully) series of Discussion seminars: more questions than answers are expected, so come open-minded and be ready for discussion!
Stem cells generated a lot of excitement in the last decade: these are cells produced in an embryo and they have capability to turn into specialised tissue cells which potentially opens a way to cure many diseases. In recent stem-cells research the following experiment was conducted. Haematopoietic stem cells (HSC) were extracted from a mouse embryo and then transplanted into adult recipient mice which previously received a doze of potentially deadly radiation doze. After such treatment, all the mice recovered successfully.
The problem, however, is to estimate how many HSCs were extracted from the embryo if we know that even one such cell is sufficient for a mouse to recover? Unfortunately, there is still no way to count this number directly, say, by using a microscope. The answer can be obtained by carefully designing experiment when a differently diluted dozes are transplanted to different irradiated mice, so that inevitably some of the mice do not receive any HSCs. Then the proportion of survived mice as a function of the dilution rate allows to estimate the number of HSC extracted. The main constraint in designing such experiment is that we want to save as much lab mice as possible, but still get sensible quality estimates.

17/2 Peter Gennemark, Mathematical sciences: Identifying and compensating for systematic errors in a large-scale phenotypic screens
Abstract:
We consider statistical questions concerning analysis of yeast growth curves. Each curve is based on measurements of the growth of a cell culture during 48 hours with three measurements per hour. The experimental set-up is large scale and allows 200 cultures to be monitored simultaneously. We study reproducibility in such large-scale experiments using a set of control experiments of only wild-type strains.
It is found that the false-positive rate is under-estimated in current significance tests, partly because of bias from, e.g., spatial plate effects, that affect the tests when high precision measurement techniques are used, and partly because of dependence between repetitions in the current design. By stringent data pre-processing and improved experimental design it is demonstrated that one can counter the effects of systematic errors and increase the accuracy.
Joint work with Olle Nerman (Mathematical sciences) and Anders Blomberg (Cell and Molecular Biology, GU)

24/2 Takis Konstantopoulos, Uppsala University: A stochastic ordered graph model
Abstract:
We consider a stochastic directed graph on the integers whereby a directed edge between $i$ and a larger integer $j$ exists with probability $p_{j-i}$ depending solely on the distance between the two integers. Under broad conditions, we identify a regenerative structure that enables us to prove limit theorems for the maximal path length in a long chunk of the graph. We first discuss background literature of this stochastic model. The model is an extension of a special case of graphs studied by Foss and the speaker. We then consider a similar type of graph but on the `slab' $\Z \times I$, where $I$ is a finite partially ordered set. We extend the techniques introduced in the in the first part of the paper to obtain a central limit theorem for the longest path. When $I$ is linearly ordered, the limiting distribution can be seen to be that of the largest eigenvalue of a $|I| \times |I|$ random matrix in the Gaussian unitary ensemble (GUE). This is joint work with S Foss and D Denisov.

3/3 Victor Brovkin, Max Planck Institute for Meteorology, Hamburg, Germany: Land biosphere models for future climate projections
Abstract:
The Earth System Models (ESMs) are the best tools available for projecting changes in the atmospheric CO2 concentration and climate in the coming decades and centuries. ESMs include models of land biosphere which are based on well established understanding of plant physiology and ecology, but these models typically use very few observations to constrain model parameters. Extensive ground-based measurements of plant biochemistry, physiology, and ecology have led to a much better quantification of ecosystem processes during the last decades. Recent assimilation of many thousands of measurements of species traits in global databases opens a new perspective to specify plant parameters used in ecosystem models which predominantly operate at the level of large-scale plant units such as plant functional types (PFTs). Instead of constraining model parameters using values from a few publications, a novel approach aggregates plant traits from the species level to the PFT level using trait databases. A study to employ two global databases linking plant functional types to decomposition rates of wood and leaf litter to improve future projections of climate and carbon cycle using an intermediate complexity ESM, CLIMBER-LPJ, will be presented.

10/3 Andrey Lange, Bauman Moscow State Technical University: Discrete stochastic systems with pairwise interaction
Abstract:
A model of a system of interacting particles of types T_1, ... , T_n is considered as a continuous-time Markov process on a countable state space. Forward and backward Kolmogorov systems of differential equations are represented in a form of partial differential equations for the generating functions of transition probabilities. We study the limiting behaviour of probability distributions as time tends to infinity for two models of that type.
First model deals with an open system with pairwise interaction. New particles T immigrate either one or two particles at a time, and the interaction T+T leads to the death of either one or both of the interacting particles. The distribution of the number of particles is studied as the time tends to infinity. The exact solutions of the stationary Kolmogorov equations were found in terms of Bessel and hypergeometric functions. The asymptotics for the expectation and variance as well as the asymptotic normality of the stationary distribution were obtained when the intensity of new particles arrival is high.
The second model describes a system with particles T1 and T2. Particles of the two types appear either as the offspring of a particle of type T1 or as a result of interaction T1+T1. The distribution of the final number of particles T2 is considered when the subpopulation of particles T1 becomes extinct. Under certain restrictions on the distribution of the number of appearing particles, the asymptotics for the expectation and variance as well as the asymptotic normality of the final distribution are obtained when the initial number of particles T1 is large.

10/3, Graham Jones, Durness, Scotland: Stochastic models for phylogenetic trees and networks
Abstract:
The Tree of Life does not look as though it was generated by a constant rate birth-death process, since too many nodes show unbalanced splits where one branch leads to only a few tips and the other to very many. Generalizations of the constant rate birth-death process (age-dependent and multitype binary branching processes) can produce trees which look more like real phylogenetic trees. A method for numerical calculation of the probability distributions of these trees will be presented.
Hybridization produces phylogenetic networks instead of trees, and things can get complicated even in the simplest cases. What sequences of evolutionary events (speciations, extinctions and hybridizations) can produce two diploids and one tetraploid from a single diploid ancestor?

31/3, Stig Larsson: Numerical approximation of stochastic PDEs
Abstract:
Together with several co-workers during recent years I have studied numerical approximation of evolution PDEs perturbed by noise. You may consider this as a ''discussion seminar'' where I will review our work and ask for your advice and possible cooperation for future work.

9/4, Dustin Cartwright, Berkeley: How are SNPs distributed in genes?
Abstract:
The organization of genes into 3-base codons has certain consequences for the distribution of bases. Many of these consequences have been known for a long time. I will talk about a particular method for detecting these artefacts if we didn't already know the underlying cause. The central analytical tool will be the notion of the rank of a tensor.

3/5, Maria Deijfen, Stockholm University: Scale-free percolation
Abstract:
I will describe a model for inhomogeneous long-range percolation on Z^d with potential applications in network modeling. Each vertex is independently assigned a non-negative random weight and the probability that there is an edge between two given vertices is then determined by a certain function of their weights and of the distance between them. The results concern the degree distribution in the resulting graph, the percolation properties of the graph and the graph distance between remote pairs of vertices. The model interpolates between long-range percolation and inhomogeneous random graphs, and is shown to inherit the interesting features of both these model classes.

12/5, Andras Balint: The critical value function in the divide and colour model
Abstract:
The divide and colour model is a simple and natural stochastic model for dependent colourings of the vertex set of an infinite graph. This model has two parameters: an edge-parameter p, which determines how strongly the states of different vertices depend on each other, and a colouring parameter r, which is the probability of colouring a given vertex red. For each value of p, there exists a critical colouring value R such that there is almost surely no infinite red cluster for all r infinite red cluster exists with positive probability for all r>R. In this talk, I will discuss some new results, obtained jointly with Vincent Beffara and Vincent Tassion, concerning different properties, such as (non-)continuity and (non-)monotonicity, of the critical colouring value as a function of the edge-parameter, as well as both deterministic and probabilistic bounds on the critical colouring value.

26/5, Vitali Wachtel, Mathematical Institute, LMU, München: Random walks in Weyl chambers
Abstract:
We construct $k$-dimensional random walks conditioned to stay in a Weyl chamber at all times. The chief difficulty is to find a harmonic function for a random walk. It turns out that one needs different approaches under different moment assumptions on unconditioned random walks. We prove also limit theorems for random walks confined to a Weyl chamber.

31/5, Jenny Jonasson: Discussion seminar - Can we use extreme value theory to analyse data from naturalistic driving studies
Abstract:
The idea behind naturalistic driving studies is that ordinary people drives cars that are equipped with a number of measuring devices such as cameras both on the road and on the driver, radars, accelerometers, etc. The main question concerns accident prevention. Although the amount of data is enormous there are still not many accidents in the data sets and therefore near-accidents are also extracted from the data. Our task is to decide if accidents and near-accidents are similar in some sense. Near-accidents and accidents can be thought of as extreme events and hence we use extreme value theory.

7/6, Chris Glasbey, Biomathematics & Statistics Scotland: Dynamic programming versus graph cut algorithms for fitting non-parametric models to image data
Abstract:
Image restoration, segmentation and template matching are generic problems in image processing that can often be formulated as non-parametric model fitting: maximising a penalised likelihood or Bayesian posterior probability for an I-dimensional array of B-dimensional vectors. The global optimum can be found by dynamic programming provided I=1, with no restrictions on B, whereas graph cut algorithms require B=1 and a convex smoothness penalty, but place no restrictions on I. I compare conditions and results for the two algorithms, using restoration of a synthetic aperture radar (SAR) image for illustration.

16/6, Stefan Hoberg and Malin Persson: Optimal design for pharmacokinetic trials (Master thesis presentation)
Abstract:
When performing a pharmacokinetic study one measures the concentration of the drug several times. When, and how many times to do this, is not always easy to determine. Using optimal design theory, this thesis will show a method to find an optimal number of measurements and also the times to conduct them. The robustness of this design will be investigated by shifting the design points to determine if that will have a big effect on the estimations of the parameter values. For the model used in this thesis a design with three different design points was the optimal one. The second and third time points proved to be unaffected by most shifts on the times. If the first design point was moved close to or past the time when the concentration is at its maximum, problems appeared. This resulted in difficulties obtaining estimates for the parameters, and the ones acquired proved to be unreliable.

1/9, Ilya Molchanov, University of Bern, Switzerland: Partially identified models and random sets
Abstract:
A statistical model is partially identified if it does not make possible to come up with a unique estimate of the unknown parameter, even if the sample size grows to infinity. The talk presents several examples of such models related to interval regression, statistical analysis of games and treatment response and explains how tools from the theory of random sets can be used to provide a unified solution to all these problems.

4/10, Mari Myllymäki, Aalto University, Finland: Testing of mark independence for marked point patterns
Abstract:
The talk discusses the testing of independence of marks for marked point patterns. Many researchers use for this purpose the popular envelope test. However, this may lead to unreasonably high type I error probabilities, because in this test spatial correlations are inspected for a range of distances simultaneously. Alternatively, the deviation test can be used, but it says only little about the reason of rejection of the null hypothesis. In this talk, it is demonstrated how the envelope test can be refined so that it becomes both a valuable tool for statistical inference and for understanding the reasons of possible rejections of the independence hypothesis. This is joint work with Pavel Grabarnik and Dietrich Stoyan.

6/10, Maria Deijfen, Stockholm University: Stable bigamy on the line
Abstract:
Consider a vertex set that consists of the points of a Poisson process on R^d. How should one go about to obtain a translation invariant random graph with a prescribed degree distribution on this vertex set? When does the resulting graph percolate? One natural way of constructing the graph is based on the Gale-Shapley stable marriage, and the question of percolation has then turned out to be surprisingly difficult to answer. I will describe some existing results and a number of open problems, with focus on the case d=1 and constant degree 2. (Joint work with Olle Häggström, Alexander Holroyd and Yuval Peres.)

20/10, Martin S Ridout, University of Kent, UK: Numerical Laplace transform inversion for statisticians
Abstract:
We review some methods of inverting Laplace transforms numerically, focusing on methods that can be implemented effectively in statistical packages such as R. We argue that these algorithms are sufficiently fast and reliable to be used within iterative statistical inference procedures. Illustrative examples cover calculation of tail probabilities, random number generation and non-Gaussian AR(1) models.

27/10, Jorge Mateu, Department of Mathematics, University Jaume I, Castellon, Spain: Functional spatial statistics with a focus on geostatistics and point processes
Abstract:
Observing complete functions as a result of random experiments is nowadays possible by the development of real-time measurement instruments and data storage resources. Functional data analysis deals with the statistical description and modeling of samples of random functions. Functional versions for a wide range of statistical tools have been recently developed. Here we are interested in the case of functional data presenting spatial dependence, and the problem is handled from the geostatistical and point process contexts. Functional kriging prediction and clustering are developed. Additionally, we propose functional global and local marked second-order characteristics.

26/10, Erik Mellegård: Obtaining Origin/Destination-matrices from cellular network data
Abstract:
"Mobile devices in America are generating something like 600 billion geo-spatially tagged transactions per day" says Jeff Jonas, chief scientist at IBM. A lot of this data are passing through the mobile operators systems and are collected for billing and networking purposes. This data could be used to obtain valuable information about people's movements, something that is not being done today. The main reason for this is that the operators are afraid of what would happen if someone would mistreat this data and used if to track people. This thesis presents a method for ﬁnding Origin/ Destination-matrices from the mobile network data in a way that keeps the individuals' privacy. Since the operators are reluctant to let us used any real data, the method has been applied to synthetic data and some call data records. The results of this thesis shows that it is feasible to obtain Origin/Destination-matrices from mobile network data.