Seminars
Locatie: Snijderszaal, (LH 01.010), 1st verdieping EWI building, Mekelweg 4, 2628 CD Delft
Tijd: Woensdag 11:00 - 12:00 map
23 mei 2012 Markus Heydenreich (UL)
The incipient infinite cluster - construction, scaling and random walk properties
Understanding critical behavior of spatial systems is an important though challenging task. We consider the incipient infinite cluster, an object closely related to critical percolation, on the high-dimensional lattice. It is a natural example of a spatial object showing fractal properties, self-similarity, and a non-degenerate scaling limit. In particular, the behaviour of random walk on the incipient infinite cluster significantly deviates from Euclidean lattices.
During the talk, I explain various constructions of the incipient infinite cluster, and discuss properties of the corresponding random walk. Subsequently, I present a result about the scaling limit of the backbone of the incipient infinite cluster. This result is achieved through a new lace expansion for percolation.
Joint work with Remco van der Hofstad (Eindhoven), Tim Hulshof (Eindhoven) and Grégory Miermont (Orsay).
16 mei 2012 Rob van den Berg (CWI and VU University)
Disjoint-occurrence inequalities: introduction and recent progress
The BK inequality for product measures (van den Berg and Kesten (1985)) says that the probability that two increasing events `occur disjointly' is smaller than or equal to the product of the two individual probabilities.
This result is often used in percolation and interacting particle systems, but can also be interpreted in operations-research terms (e.g. random distribution of resources). The conjecture that the inequality even holds for all events was proved by Reimer in the mid-nineties.
In spite of Reimer's work several natural, fundamental problems in this area remained open. In this talk I will start with an introduction and general overview, and then discuss recent progress. In particular I will show and explain an extension of the BK inequality to randomly drawn subsets of fixed size (joint work with Johan Jonasson) and more recent extensions for the ferromagnetic Ising model and the antiferromagnetic Curie-Weiss model (joint work with Alberto Gandolfi).
9 mei 2012 Eduard Belitser (TUE)
Optimal two-stage procedures for estimating location and size of maximum of multivariate regression functions
We propose a two-stage procedure for estimating the location and size of the maximum of a smooth d-variate regression function.
In the first stage, a preliminary estimator of the location obtained from a standard nonparametric smoothing method is used.
At the second stage, we ``zoom-in'' near the vicinity of the preliminary estimator and make further observations at some design points in that vicinity.
We fit an appropriate polynomial regression model to estimate the location and size of the maximum. We establish that, under suitable smoothness conditions and appropriate choice of the zooming, the second stage estimators have better convergence rates than the corresponding first stage estimators of the location and size of the maximum. More specifically, for $\alpha$-smooth regression functions, the optimal nonparametric rates $n^{-(\alpha-1)/(2\alpha+d)}$ and $n^{-\alpha/(2\alpha+d)}$ at the first stage can be improved to $n^{-(\alpha-1)/(2\alpha)}$ and $n^{-1/2}$ respectively for $\alpha>1+\sqrt{1+d/2}$.
These rates are the optimal rates in the class of all possible sequential estimators.
Interestingly, the two-stage procedure resolves the curse of dimensionality problem to some extent, as the dimension does not control the second stage convergence rates, provided that the function class is sufficiently smooth. We consider a multi-stage generalization of our procedure that attains the optimal rate for any smoothness level $\alpha>2$ starting with a preliminary estimator with any power-law rate at the first stage. Based on joint work with S.Ghosal and H. van Zanten.
2 mei 2012 Shashi Jain (CWI/TU Delft)
Stochastic grid method with bundling for pricing Bermudan Options
Tilley's bundling algorithm was amongst the first few methods to price Bermudan options using simulation. His method however suffered from some drawbacks especially non trivial extension to multi-asset problems. The talk describes extension of Tilley's bundling algorithm to multi-asset problems using the stochastic grid method, with several improvements to the original algorithm. We give numerical results for multi-asset arithmetic and geometric basket options, and max on several assets.
18 april 2012 Pasquale Cirillo (Bern)
Polya lattice models for cascading failures
A cascading failure is a failure in a system of interconnected parts, in which the breakdown of one element can lead to the subsequent collapse of the others. The aim of this talk is to introduce a class of simple combinatorial models for the study of cascading failures. The new models are called Polya Lattice Models (PLM). In particular, having in mind particle systems and Markov random fields, we take into consideration a network of interacting urns displaced over a lattice. Every urn is Pólya-like and its reinforcement matrix is not only a function of time (time contagion), but also of the behavior of the neighboring urns (spatial contagion), and of a random component, which can either represent fate, or the impact of exogenous factors. In this way a non-trivial dependence structure among the urns is built, and it is used to study default avalanches over the lattice.
Thanks to its flexibility and its interesting probabilistic properties, the given construction may be used to model different phenomena characterized by cascading failures such as financial networks or power grids. During the talk we will also show how PLM can be used from a Bayesian nonparametric point of view, and present a first application to the study of financial fragility and firms dynamics.
28 maart 2012 Florian Völlering (Universiteit Leiden)
Random walks in random environments
Random walks in random environments arise when the transition kernels of simple random walks are modified according to a random environment. This modification is dependent on the position of the walk in the environment, which makes the study of the random walk non-trivial. I will give an overview of the topic, presenting both traditional and new results, for environments which are either static or changing in time.
21 maart 2012 Berend Roorda (Universiteit Twente)
Smarter valuation under weaker time consistency.
One of the central topics in mathematical finance is to determine the value of products, or financial positions, with a risky payoff. In a dynamic setting, most approaches rely on the paradigm of conditional valuation, postulating state-dependent certainty equivalents for the position under consideration. By replacing the position at some horizon date by its certainty equivalent at some earlier date in a backward recursive way, valuation effectively reduces to stepwise analysis on a short time scale, in the spirit of Dynamic Programming. In risk neutral valuation, the standard valuation methods in the field, this is justified by the law of iterated expectations. Also in more general valuation methods based on axiomatic frameworks for risk measures, the prevailing rule of (strong) time consistency precisely imposes this type of backward recursive evaluation (see e.g. Föllmer and Schied, Stochastic Finance – an introduction in discrete time, 3rd edition 2011).
In this talk we indicate some fundamental limitations of this setting, in particular in situations where the law of one price does not hold anymore due to market frictions or model uncertainty. The problem is that appropriate degrees of risk aversion per time step typically aggregate to excessive levels over longer periods. We discuss weaker forms of time consistency (as described in [1]) that bring in a new dimension in valuation, in which there is freedom to choose long term features of pricing operators without affecting its local properties. The observations give rise to rethink the role of certainty equivalents, Bayesian updating, and the Dynamic Programming principle.
Some simple examples will illustrate the main ideas.
[1] B. Roorda & J.M. Schumacher (2010) When Can a Risk Measure Be Updated Consistently? Under revision. An earlier version of this paper has been circulated under the title “Time Consistency of Nonconvex Risk Measures” (Netspar Discussion Paper 01/2009-006).
14 maart 2012 Jeanine Houwing-Duistermaat (LUMC)
Estimation of parameters using information from family and twin studies.
Jeanine J Houwing-Duistermaat and Bruna Balliu
Dept of Medical Statistics and Bioinformatics
Leiden University Medical Center
To enrich for genetic factors, families are often selected on at least one case member. These studies are typically underpowered. We therefore propose to use the joint likelihood and to combine family and twin data.
We have developed a likelihood-based approach allowing for several ascertainment schemes, to accommodate for the outcome-dependent sampling scheme, and a family-specific random term, to take into account the correlation between family members. We estimate the parameters using maximum likelihood based on the combined joint likelihood approach.
Simulations show that the combined approach is more efficient than the retrospective or prospective approach. To illustrate our approach, we use data from a family and a twin study from the United Kingdom on rheumatoid arthritis.
29 februari 2012 Hitoshi Nakada (Keio University, Yokohama, Japan)
On costs of some Euclidean type algorithms over F_q[X]^3.
We consider Euclidean type algorithms of three polynomials of F_q -coefficients. There are three possibilities for such algorithms. We estimate some cost functions to find efficient one among these algorithms.
(joint work with V. Berthe and R. Natsui)
15 februari 2012 Henk Don (TUD)
Improved lower bounds for the critical value in fractal percolation.
The fractal percolation model will be introduced. Then I will discuss a result that connects fractal percolation with site percolation. This result can be used to obtain lower bounds for the critical value p_c in fractal percolation. Actually, we are able to construct a sequence of lower bounds that converges to p_c. The terms in this sequence can in principle be calculated algorithmically, but this is computationally very intensive. To obtain numerical lower bounds for p_c, we developed an algorithm to compute lower bounds for the terms in the sequence
8 februari 2012 Peter Spreij (UVA)
Affine diffusions with non-canonical state space.
Multidimensional affine diffusions have been studied in detail for the case of a canonical state space. We present results for general state spaces and provide a complete characterization of all possible affine diffusions with polyhedral and quadratic state space. We give necessary and sufficient conditions on the behavior of drift and diffusion on the boundary of the state space in order to obtain invariance and strong existence and uniqueness. Joint work with Enno Veerman.
1 februari 2012 Maik Schwarz (Universite catholique de Louvain, Belgium)
Adaptive circular deconvolution by model selection under unknown error distribution.
7 december 2011 Frank Redig (TU Delft)
Path space large deviations and Gibbs-non-Gibbs transitions.
Motivated by the phenomenon of dynamical Gibbs-non-Gibbs transitions, we consider the typical behaviour of nearly deterministic processes conditioned on the future. Depending on how far in the future one conditions, there can be unique or non-unique optimal trajectories. Part of the talk will also be devoted to the recently developed path-space large deviation formalism by Feng and Kurtz.
30 november 2011 Ellen Saada (Paris universite Paris Descartes et CNRS)
A shape theorem for an epidemic model in dimension $d\ge 3$.
We prove a shape theorem for the set of infected individuals in a spatial epidemic model with 3 states (susceptible-infected-recovered) on $\Z^d,d\ge 3$, when there is no extinction of the infection.
For this, we derive percolation estimates (using dynamic renormalization techniques) for a locally dependent random graph in correspondence with the epidemic model.
This is a joint work with E. D. Andjel and N. Chabot.
23 november 2011 Marco Loog (TU Delft)
Challenging Semi-Supervised Learning.
Semi-supervised learning aims learn classification rules from both labeled and, typically more easily obtainable, unlabeled data. Though studied since the late 60s and early 70s, surprisingly little headway has been made with respect to methods that can guarantee, in expectation, to always outperform their supervised counterparts. A principle problem is that current state-of-the-art semi-supervised learning techniques make additional assumptions about the underlying data in an attempt to exploit all unlabeled instances. These assumptions, however, typically do not hold true and, as a result, making them can considerably deteriorate classification performance.
After giving a brief impression of the day-to-day worries of a pattern recognizer, I will present and discuss some of my very preliminary ideas and results concerning the problem of semi-supervision. My basic proposal is to develop semi-supervised learning techniques that do not make assumptions beyond those implicitly or explicitly made by the classification scheme employed. The overarching idea to achieve this is to exploit constraints and prior knowledge intrinsic to the classifiers considered. A simple example using the nearest mean classifier is provided. After my presentation there will hopefully be time for the audience to answer some of the questions I have.
12 oktober 2011 Sandjai Bhulai (VU)
Optimal Allocation of Resources in Adaptive Survey Designs.
Survey nonresponse occurs when members of a sample cannot or will not participate in the survey. It remains a problem despite the development of statistical methods that aim to reduce nonresponse. Instead, we address the problem of resource allocation in survey designs in which the focus is on the quality of the survey results given that there will be nonresponse. Therefore, we propose a novel method in which the optimal allocation of survey resources can be determined.
5 oktober 2011 Tom Kemptom (UU)
Beta Expansions and Bernoulli Convolutions.
Almost every number has a unique expansion to base ten, known as the decimal expansion. Conversely, given a non integer real number beta greater than one, almost every number has uncountably many different expansions to base beta. In this talk we discuss some counting questions relating to beta expansions. We are able to give stronger results in the case that the Bernoulli convolution corresponding to beta is absolutely continuous, and consequently gain a new necessary condition for the absolute continuity of Bernoulli convolutions.
14 september 2011 Anne Fey (TU Delft)
Anisotropic bootstrap percolation in three dimensions.
1 juni 2011 Rik Lopuhaa (TU Delft)
The limit distribution of the supremum distance for Grenander type estimators.
25 mei 2011 Doug Hensley (Texas A&M University (TAMU)
Computing key parameters of continued fraction dynamical systems.
18 mei 2011 Anca Hanea (TU Delft)
Parameter estimation using dynamic non-parametric Bayesian networks.
11 mei 2011 Karl Petersen (University of North Carolina at Chapel Hill, USA)
Invariant measures and combinatorics of some nonstationary adic systems.
We review recent work (much of it joint with Frick or Varchenko) on adic (Bratteli-Vershik) dynamical systems which come from walks or reinforced walks on finite graphs. Identification of the ergodic invariant measures depends on knowing path counts between vertices in the associated diagram, and this leads to interesting combinatorial problems and formulas involving binomial coefficients as well as Eulerian, Stirling, and Delannoy numbers. Among dynamical properties that can be determined are lack of point spectrum, faithful coding by subshifts, topological weak mixing, loosely Bernoulli, and complexity.
27 april 2011 Ronald Meester (VU) Long range percolation on the hierarchical lattice.
We study long-range percolation on the hierarchical lattice of order N, where any edge of length k is present with probability p_k=1-exp(-eta^{-k} alpha),independent of all other edges. For fixed eta, we show that the critical value alpha_c(eta) is non-trivial if and only if N < eta < N^2. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability. The uniqueness problem involves a discussion of the so called Neumann-Kakutani transformation – we will explain the connection.
20 april 2011 Sergey Foss (Herriot Watt University) Convergence in the total variation, directed percolation,last passage percolation, chains with infinite memory, and extended renovation theory.
I will discuss conditions for convergence in the total variation for functionals of a Markov chain (or, more generally, of a stochastic recursion) which may depend on entirely infinite future and/or past.Various examples will be given.
13 april 2011 Tina Nane (TUD) "Shape constrained nonparametric baseline estimators in the Cox proportional hazards model".
Within survival analysis, the Cox proportional hazards model is one of the most acknowledged approaches to model right-censored time to event data in the presence of covariates. Different functionals of the lifetime distribution are commonly investigated. The hazard function is of particular interest, as it represents an important feature of the time course of a process under study, e.g., death or a certain disease.
Numerous survival studies indicate explicit evidence of monotone baseline hazard functions. The main objective is therefore to derive nonparametric baseline hazard estimators under monotonicity constraints and investigate their asymptotic behavior. Through the classical graphical representation, our first approach starts from the maximum likelihood estimator of the baseline cumulative hazard estimator, namely the Breslow (1972) estimator. For a nondecreasing baseline hazard, we define the least-squares (LS) baseline hazard estimator as the left-hand slope of the Greatest Convex Minorant (GCM) of the Breslow estimator. This estimator can be viewed as a least-squares projection on the space of all distributions with nondecreasing baseline hazards.
Succeedingly, a maximum likelihood estimator (MLE) of a nondecreasing baseline hazard has been derived by maximizing the (log)likelihood function over the set of all distributions with nondecreasing baseline hazards. Similarly, a monotone baseline density estimator has been defined and its strong consistency established.
23 maart 2011 Tobias Mueller (CWI) Random geometric graphs.
If we pick n points at random from d-dimensional space (i.i.d. according to some probability measure) and fix an r > 0, then we obtain a random geometric graph by joining two points by an edge whenever their distance is at most r.
I will give a brief overview of some of the main results on random geometric graphs and then describe my own work on Hamilton cycles and the chromatic number of random geometric graphs.
16 maart 2011 Paul Eggermont ( University of Delaware) Moment discretization of ill-posed problems and reproducing kernel Hilbert spaces.
2 maart 2011 Jasper Anderluh (TU Delft en Fondsbeheerder HiQ) Real Options in Nuclear Reactor Valuation.
Real options are opportunities for economic players. Consider a nuclear plant operator who builds a special type of reactor, the so called Fast Reactor, that will give him in 25 years the opportunity to recycle part of the nuclear waste and feed it back into his reactor. In 2035 years he has to decide whether to continue generating waste or he can decide to change his fuel cycle by recycling the waste and feeding it back into the system. As it is his choice to change the fuel cycle, he owns an option to do so, which is called a real option. From an economic point of view, this option should have a value. In this talk we will address the issues of determining the real option value and how it is different from computing the price of a standard financial equity option. The talk will be mainly focussing on the concepts that are needed and a little less on the detailed mathematics. Joint work with Ulrike Lauferts.
23 februari 2011 Erik van Zwet (LUMC) What is Causal Inference?
Researchers often want to know if one thing causes another. Statisticians tend to respond that they are happy to test for association, but that association does not imply causation. Now "causal inference" aims to address causation itself. With its particular notation and terminology, causal inference seems very different from standard statistical inference. Judea Pearl, who wrote a book on causality, even states: "Almost by definition, causal and statistical concepts do not mix". From Pearl's book, I learned that at the heart of causal inference lies a very neat idea from 1960 due to Robert Strotz and Herman Wold. This idea leads us to interesting parameters to estimate. Estimating these parameters from data is, of course, just standard statistics. I should mention that I am not an expert on causal inference. My goal is just to help bridge the gap between causality and mainstream statistics.
16 februari 2011 Wessel van Wieringen (VU) A random effects model for regional co-expression associated with DNA copy number aberrations.
We combine coupled DNA copy number and gene expression high-throughput data in order to study regional co-expression, i.e. the phenomenon of neighborhoods of contiguous genes showing similar expression patterns. Such neighborhoods appear throughout the cancer genome and often coincide with DNA copy number aberrations (CNA). We use a random coefficients model to link DNA copy number data of a genomic region to its genes' expression data. The model facilitates a global analysis of regional co-expression at the level of the region (rather than its genes) to assess whether a) there is a shared CNA effect on expression levels of genes within the region, and b) the CNA effect is identical for all genes. To estimate the parameters from high-throughput data, we optimize estimation wrt computational speed and memory use, while incorporating prior knowledge on the parameters. Two examples illustrate the methodology.
9 februari 2011 Derong Kong (TU Delft) The Markov binomial distribution and a stochastic reactive transport model.
We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal or trimodal. Moreover, we give a closed form expression for the variance of the Markov binomial distribution (MBD), and expressions for the mean and the variance conditioned on the state at time n.
In the second part of our talk we introduce a discrete time microscopic single particle model for kinetic transport. The kinetics is modeled by a two-state Markov chain, the transport by deterministic advection plus a random space step. The position of the particle after n time steps is then given by a random sum of space steps, where the size of the sum is given by the Markov Binomial Distribution. We prove that by letting the length of the time steps and the intensity of the switching between states tend to zero linearly, we obtain a random variable S(t), which is closely connected to a well known deterministicPDE reactive transport model from the engineering literature. Our model explains (via bimodality of the MBD) the well known double peaking behavior of the concentration of solutes in the PDE model. Moreover, we show that (under a restriction on the initial distribution of the Markov chain) the partial densities do exist, and do satisfy the partial differential equations.
2 februari 2011 Rui Castro (Eindhoven University of Technology) Active Learning and sequential experimental design for classification and sparse signal inference.
Many traditional approaches to statistical inference and machine learning are passive, in the sense that all data are collected prior to any analysis. However, in many practical scenarios it is possible to actively use information gleaned from previous observations to sequentially focus the data collection process, closing the loop between data analysis and acquisition. Such scenarios are often denoted as active learning or inference using sequential experimental designs. Despite the potential to dramatically improve inference performance, analysis of such procedures is difficult, due to the complicated data dependencies created by the closed-loop observation process. These difficulties are further exasperated by the presence of measurement uncertainty or noise. This talk will be divided in two parts. First, I'll summarize some results on minimax performance bounds for active learning in non-parametric classification settings.
Second, I'll present a novel adaptive sensing procedure - Distilled Sensing - which is highly effective for detection and estimation of high-dimensional sparse signals in noise. Large-sample analysis shows that the proposed procedure provably outperforms the best possible detection methods based on non-adaptive sensing, allowing for detection and estimation of extremely weak signals, imperceptible without adaptive sensing. Some extensions of these ideas to the compressed sensing framework will also be discussed.
26 januari 2011 Ivan Corwin (Courant Institute of Mathematics, New York) Beyond the Gaussian Universality Class.
The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers, particle systems, matrices and stochastic PDEs, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian.
15 december 2010 Roger Cooke (TU Delft / Resources for the future, USA) Obesity Index and Tail Risk.
This reports on results from a National Science Foundation project on Tail Risk, and illustrates techniques em0loyed at RFF to make economists, insurers and government people aware of mathematical problems in dealing with tail risk. The main mathematical results concern conditions under which tail dependence is amplified by aggregation, and a new measure of tail obesity which does not involve estimating a parameter in a hypothetical distribution, but is suitable for measuring tail obesity in finite samples.
8 december 2010 Mark van de Wiel (VUMC) Comparing predictors in a training-testing setting.
Statistics has a long tradition in developing tests and information criteria for the purpose of model selection. These approaches do usually not apply to prediction models for high-dimensional data. Therefore, one resorts to training-test set approaches, by means of cross-validation, random subsampling or resampling. In the literature, much attention is spent on estimating predictor error in this setting. This talk highlights three different aspects of prediction error: comparative testing, confidence intervals and variability with respect to the training data sets.
We start by motivating why one might be interested in comparative inference rather than simple comparison of estimated prediction errors. A simple testing procedure is introduced that applies simultaneously to multiple splits of the same data set. For each split, both procedures predict the response of the same samples, which results in paired residuals to which a signed-rank test is applied. Hence, multiple splits result in multiple p-values. The median p-value and the mean inverse normal transformed p-value are proposed as summary (test) statistics, for which theoretical bounds on the overall type I error rate under a variety of assumptions are provided.
Next, we shortly discusss the potential to extend the testing approach to confidence intervals. Finally, we focus on another aspect of prediction: variability of the predictions across training data sets. We introduce the notion of a confidence score, which quantifies such variability. We show that the well-known decomposition of the Brier score, a popular prediction error measure, nicely generalizes to inclusion of this variance component. The latter is not true for another popular prediction error measure, area under the curve (AUC). Our methods are illustrated on several (high-dimensional) data sets with binary or survival response.
1 december 2010 Evgeny Verbitskiy (UL) Thermodynamics of a binary symmetric channel.
Binary symmetric channel (BSC) is probably the simplest communication model with noise studied in information theory.I will discuss a very basic question: How does the binary symmetric channel affect the thermodynamic properties of the input process? For example, given that the input is a Markov (or, more generally, a Gibbs) process, will the output process remain Gibbs.
24 november 2010 Sonja Cox (TUD) `Burkholder-Davis-Gundy inequalities in Banach spaces'.
I will sketch how decoupling inequalities like the UMD inequality play a role in proving a Burkholder-Davis-Gundy type inequality for Banach space-valued stochastic integrals. (A small) part of my talk will be based recent work by Mark Veraar and myself.
10 november 2010 Michel Mandjes (UVA) Simulation-based computation of the correlation function in a Levy-driven queue.
In this talk I consider a single-server queue with Levy input, and in particular its workload process $Q(t)$, focusing on its correlation structure. With the correlation function defined as $r(t) := {m Cov}(Q(0), Q(t))/{m Var} Q(0)$ (assuming the workload process is in stationarity at time 0), we first study its transform $int_0^infty r(t)e^{-theta t} dt,$ both for the case that the Levy process has positive jumps, and that it has negative jumps. These expressions allow us to prove that $r(t)$ is positive, decreasing, and convex, relying on the machinery of completely monotone functions. For the light-tailed case, we estimate the behavior of $r(t)$ for $t$ large. We then focus on techniques to estimate $r(t)$ by simulation. Naive simulation techniques require roughly $1/r(t)^2$ runs to obtain an estimate of a given precision, but we develop a coupling technique that leads to substantial variance reduction (required number of runs being roughly $1/r(t)$). If this is augmented with importance sampling, it even leads to a logarithmically efficient algorithm.
(This talk is based on joint work with P.W.G. Glynn, Stanford, and will appear in Adv. Appl. Prob. later this year)
3 november 2010 Marianne Jonker (VU) A frailty model for censored family survival data, applied to the age at onset of mental problems.
Family survival data are often used in genetic research to estimate genetic and environmental contributions to the age at onset of a disease or of a specific event in life. The survival data can be modeled with a correlated (gamma) frailty model. We use such a model to estimate the degree of heredity (heritability), environmental effects, and twin effects on the age at which people contact social service for the first time, to test whether these terms differ for males and females, and to investigate whether the survival functions differ for twins and non-twins. Our data come from an ongoing study on health, lifestyle, and personality. Longitudinal data were collected of Dutch monozygotic and dizygotic twins and of their siblings at five time points between 1991 and 2002. At every of these timepoints it is observed whether an individual ever contacted social service; so the age at which an individual contacted social service for the first time is interval censored. The frailty variable in the model is decomposed as a linear combination of four independent gamma distributed random variables which represent the genetic contribution to the age at onset, contributions by the common environment of all siblings (twins and non-twins), a twin-effect and the contribution by individual specific, unshared alleles and environment. The simultaneous survival function is expressed in terms of the marginal survival function and several hypotheses are tested with a likelihood ratio test.
27 oktober 2010 Mia Deijfen (Stockholm University) Preferential attachment models and general branching processes.
A much studied type of models for growing networks is based on so called preferential attachement: vertices are succesively added to the network and are attached to existing vertices with probability proportional to degree. This mechanism has been shown to lead to power law degree distributions, which is in agreement with empirical studies on many types of real networks. I shall describe how general branching processes can be used to derive results on the degree distribution in preferential attachment models and also in an extensions of the model where vertices are not just added to the network but may also be removed.
20 oktober 2010 Peter D. Sozou (RWTH Aachen Univeristy) A model of copying with delay.
Consider an individual choosing between alternative resources, e.g. a person choosing a restaurant or an animal choosing between foraging patches. The individual may have some information about which is likely to be the best choice, but this information is not perfect. Should the individual choose according to her own information or should she instead seek to copy the actions of another individual who may have better information? We consider the following specific problem. Two individuals must each choose between two resources. They know that one resource is better than the other. Neither knows with certainty which is the better resource; each has her own private signal about which is likely to be better. The strength of a signal, which determines the probability that the resource which appears to be the better one really is the better one, is drawn from some known distribution. Each individual knows the strength of her own signal but not that of the other’s signal. The decision problem proceeds in discrete time steps. In each time step, each individual can either choose a resource according to her own private signal, or wait with a view to seeing if the other animal chooses a resource and then copying that choice on the next step. There is, however, a cost to delaying, modelled by means of a constant discount factor. We derive equilibrium strategies, such that each individual’s strategy is a best response to that of the other. The main result is that each individual has a signal threshold above which she should go with her own signal and below which she should wait. This threshold decreases with successive steps; it is possible for both individuals to wait for several time steps before one of them takes the plunge. Some further general results will be presented.
This is joint work with Steve Alpern.
13 oktober 2010 Jelle Goeman (LUMC) Cherry-picking: multiple testing for exploratory research.
Motivated by the practice of exploratory research, we formulate an approach to multiple testing that reverses the traditional roles of the user and the multiple testing procedure. Rather then to let the user choose the error criterion, and the procedure the resulting rejected set, we propose to let the user choose the rejected set freely, and to let the multiple testing procedure return a confidence statement on the number of false rejections incurred. In our approach, such confidence statements are simultaneous for all choices of the rejected set, so that post hoc selection of the rejected set does not compromise their validity. As a tool to achieve this reversal of roles we use the familiar closed testing procedure, but focus on the non-consonant rejections that this procedure makes. We suggest several shortcuts to avoid the computational problems associated with closed testing.
6 oktober 2010 Frank den Hollander (UL) Random walk in dynamic random environment.
We consider an interacting particle system on the integer lattice in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied sites has a local drift to the right but on vacant sites has a local drift to the left. We describe some recent results for the empirical speed of walk: law of large numbers, central limit theorem, and large deviation principle. We compare these results with what is known for static random environments, and list some key open problems. This is joint work with Luca Avena and Frank Redig.
29 september 2010 Rob van den Berg (CWI/VU) Sublinearity of the travel-time variance for dependent first passage percolation.
Suppose we assign to each edge e of the d-dimensional cubic lattice a non-negative value t(e). The passage time of a path in the lattice is then defined as the sum of the t-values of the edges in the path. The passage time from a vertex v to a vertex w is defined as the infimum of the passage times of all paths from v to w. Benjamini, Kalai and Schramm proved (in a paper in Ann. Probab. (2003)) that if the dimension d is at least 2 and the t(e)'s are i.i.d. two-valued random variables, the variance of the passage time from the vertex 0 to a vertex v is sublinear in the distance from 0 to v. (Note that if the dimension d = 1, the variance is of course linear in the distance). A few years ago, this result was extended to a large class of independent, continuously distributed t-variables by Bena"im and Rossignol.
We extend the result by Benjamini, Kalai and Schramm in a very different direction, namely to a large class of models where the t(e)'s are dependent. This class includes, among other interesting cases, a model studied by Higuchi and Zhang, where the passage time corresponds with the minimal number of sign changes in a subcritical `Ising landscape'.
This is joint work with Demeter Kiss.
15 september 2010 Jan van Neerven (TU Delft) Approximating the coefficients in parabolic stochastic partial differential equations.
In this joint work with Markus Kunze, we investigate the continuous dependence on the data A, F, G and _ of mild solutions of abstract parabolic stochastic partial di_erential equations of the form
dX(t) = [AX(t) + F(t;X(t))] dt + G(t;X(t))dW(t); X(0) = _;
where W is a (cylindrical) Brownian motion. We provide su_cient conditions for continuous dependence of the compensated solutions X(t) ��etA_ in the norms Lp(;C_([0; T];E)). The results are applied to a concrete class of semilinear parabolic SPDEs with _nite-dimensional multiplicative noise.
8 september 2010 Zhe Guo (TU Delft) The Hammersley process on the circle.
In this talk, I’ll introduce the HP(Hammersley’s Process) on the circle, and some basic speed theorems of the particles and second class particles which start from the Uniform distribution in this model; Moveover, as the main conclusion in this talk, one result of the longest way will be shown by constructing two related interacting particle systems namely L-system and M-system.
26 mei 2010: Shankar Bhamidi (University of North Carolina, Chapel Hill) Flows, rst passage percolation and random disorder in networks.
19 mei 2010: Markus Haase (TU Delft) Renewal Sequences and Convergence Rates in Ergodic Theorems.
12 mei 2010: Mike Keane (Wesleyan University) Ergodicity of Adic Transformations.
An adic transformation is a very general object, defined on the unit interval by a so-called cutting and stacking procedure. The name arose because of the example given by the dyadic odometer, or equivalently, the classical rotation by 1 of the 2-adic integers. Nowadays it has become common to describe such transformations by Bratteli diagrams, and in this guise any measure preserving transformation of a Lebesgue space can be found. In this lecture we begin with a presentation of the binomial transformation, explain the relationship of it with the binary odometer, and present a simple proof which I found several years ago of its ergodicity. Next, I discuss the Euler transformation, or perhaps more descriptively the "rise and fall transformation", and explain how we (joint work with Sarah Bailey, Karl Petersen, and Ibrahim Salama in Math.Proc.Camb.Phil.Soc.(2006), Vol. 141, 231-238) use a similar idea to prove its ergodicity. Finally, I'd like to present the in my opinion most interesting open problem in this area, for which we currently have no idea for a solution: Is the binomial transformation weakly (or strongly) mixing? The lecture is designed to be accessible for faculty, graduate students, and advanced undergraduates with some knowledge of measure theory and probability.
28 april 2010: Radboud Duintjer-Tebbens (TU Delft)
The Role of System Dynamics Models in the Debate of Control vs. Eradication of Polio.
The global polio eradication program missed its original target date of 2000 due to a number of challenges, including financial shortfalls. In 2006, a number of prominent public health leaders suggested abandoning the eradication objective in favor of a policy of "control". The ensuing policy debate was informed by a mathematical model and ultimately led to a renewed commitment to finish global polio eradication. System dynamics concepts not only helped build the model but also helped identify the heuristics working against the eradication objective. I will present the context for the polio debate and give a crash course in system dynamics modeling concepts that helped build the model and communicate the insights to a policy audience.
21 april 2010: Peter Grunwald (CWI/UL)
The Catch-Up Phenomenon in Model Selection and Model Averaging.
We partially resolve a long-standing debate in statistics, known as the AIC-BIC dilemma: model selection/averaging methods like BIC, the Bayes factor, and MDL are consistent (they eventually infer the correct model) but, when used for prediction or adaptive estimation, the rate at which predictions improve can be suboptimal. Methods like AIC and leave-one-out cross-validation are inconsistent but typically converge at the optimal rate. We give a novel analysis of the slow convergence of the Bayesian-type methods. Based on this analysis, we propose the switching method, a modification of Bayesian model averaging that achieves both consistency and minimax optimal convergence rates. Experiments with nonparametric density estimation confirm that our large-sample theoretical results also hold in practice in small samples. We also discuss how our results can coexist with those of Yang (2005), who proved that the strengths of AIC and BIC cannot always be shared. Joint work with T. van Erven (CWI) and S. de Rooij (Cambridge)
24 maart 2010: Piet Groeneboom (emeritus hgl TU Delft) Monotone hazards and life and death.
About forty years ago, at the start of my career, a well-known statistician told me that isotonic regression was a dead subject. Twenty years ago, another well-known statistician told me that the bootstrap was dead. Around the same time Apple computer was declared dead by the Microsoft following community. So, somewhat appropriately, I recently used my Apple computer to resurrect isotonic regression and the bootstrap from their graves to perform a danse macabre. Perhaps Apple computer, the bootstrap and isotonic regression aren't as dead as some people want us to believe.
10 maart2010: Gerard Hooghiemstra (TU Delft) The Poisson-Dirichlet distribution and first passage percolation on random graphs.
The Poisson-Dirichlet distribution is a {it random} probability distribution on the positive integers. More specifically, let $E_1,E_2,ldots$, be an i.i.d. sequence of exponentially distributed random variables with mean 1 and define $Gamma_i=E_1+ldots E_i$. Then for $alphain (0,1)$, the Poisson-Dirichlet probabilitiesare given by ${P_i}_{igeq 1}$, where $$P_i= (Gamma_i)^{-1/alpha}/(sum_{j=1}^infty (Gamma_j)^{-1/alpha} ).$$ These random probabilities will play an important role in first passage percolation on certain random graphs.
3 maart 2010: Ernst Wit (Rijksuniversiteit Groningen) Sparse inference en differential geometry -- with applications in genomics.
The advent of high-dimensional datasets has presented a challenge to traditional statistical inference. The n>p paradigm turned out to be too restrictive and statisticians seemed to be for a while in high seas. However, they found their (wet) feet again, when they realized the connections between high-dimensional inference on the one hand and model choice and penalized methods on the other. L_1 penalized inference had the additional advantage of also resulting in sparse solutions. We give a background to L_1 penalized inference and consider some extensions to other types of "path estimators". In particular, we will consider how to use differential geometry to extend sparse inference for non-linear models. We look at an application of penalized inference in a genomic network context.
17 februari 2010: Dorota Kurowicka (TU Delft) Regular vines / new developments.
Copulae (distributions on unit hypercube with uniform margins) have become very popular in dependence modeling in financial as well as other engineering contexts. Bivariate copulae are well studied, understood and applied. Multivariate copulae, however, are often limited in range of correlation structures and other properties as e.g. tail dependence that they can handle. A new graphical model introduced in 1997, called regular vines, allows specification of a joint distribution on n variables with given margins by specifying n- choose-2 bivariate copulas and conditional copulas. Estimating parameters of copulae on a vine using the maximum likelihood principle, named Pair Copula Construction (PCC) is performed sequentially starting form the first tree. This landmark advance in associating bivariate copulae to a vine and estimating copula parameters from data demonstrated the superiority of vines and opened large areas of application in mathematical finance, risk analysis and uncertainty modeling in engineering. Regular vines were studied by few researchers from this very department. It took time before the community of researchers interested in this model grew sufficiently. Vines’ ‘fan club’ contains now members form e.g. Norway, Germany and Canada. The rapid growth of the vine community in the last few years bodes well for the pursuit of this research agenda. In this talk we introduce the graphical model vines and present briefly its basic properties. Moreover some new results and open research questions concerning vines will be discussed.
10 februari 2010: Eduard Belitser (Universiteit Utrecht) On oracle projection posterior rate and model selection.
We apply the Bayes approach to the problem of projection estimation of the signal observed in Gaussian white noise model and we study the rate at which the posterior distribution concentrates around the true signal from the space $ell_2$, as the information in the observations tends to infinity. A benchmark is the rate of the so called oracle projection risk, i.e. the smallest risk of the unknown true signal over all projection estimators. Under appropriate hierarchical prior, we study the performance of the resulting (appropriately adjusted: shifted, rescaled or empirical Bayes) posterior distribution and establish that the posterior concentrates around the true signal with the oracle projection convergence rate.
The results are nonasymptotic and uniform over $ell_2$. Another important feature of our approach is that our results on the oracle projection posterior rate are always stronger than any result about posterior convergence with the minimax rate over all nonparametric classes for which the corresponding projection oracle estimator is minimax over this class. Based on posterior, we construct a Bayes adaptive estimator and show that it satisfies an oracle inequality. We also study implications for the model selection problem, namely we propose a Bayes model selector and assess its quality in terms of the so called false selection probability.
3 februari 2010: Yanick Heurteaux (Université Blaise Pascal) Measures and the law of the iterated logarithm.
Le $m$ be a unidimensional probability measure with dimension $d$. A natural question is to ask if the measure $m$ is comparable with the Hausdorff measure (or the packing measure) in dimension $d$. We give an answer (which is in general negative) in several situations including self-similar measures and quasi-Bernoulli measures. The law of the iterated logarithm or estimations of the Lq-spectrum in a neighborhood of 1 are the tools to obtain such estimations.
9 december 2009: Frank Redig (Universiteit Leiden) Duality and hidden symmetries in interacting particle systems.
We consider a model of heat conduction, the so-called Brownian momentum process. We show that this model is dual to a particle system which is a natural analogue of the symmetric exclusion process, with attractive instead of repulsive interaction. We further show that this model is self-dual, and give some applications of these duality relations. We also show in a general context how to obtain duality functions from symmetries of the generator. The talk is based on joint work with C. Giardina, J. Kurchan and K. Vafayi.
2 december 2009: Shota Gugushvili (Eurandom / TU Eindhoven) Nonparametric inference for discretely sampled L’evy processes.
Given a sample from a discretely observed L'evy process $X=(X_t)_{tgeq 0}$ of the finite jump activity, we study the problem of nonparametric estimation of the L'evy density $ho$ corresponding to the process $X.$ We propose an estimator of $ho$ that is based on a suitable inversion of the L'evy-Khintchine formula and a plug-in device. Our main result deals with an upper bound on the mean square error of the estimator of $ho$ at a fixed point $x.$ We also show that the estimator attains the minimax convergence rate over a suitable class of L'evy densities.
25 november 2009: Roberto Fernandez (Universiteit Utrecht) Loss of gibbssianness in un-quenching evolutions.
If an Ising ferrmognet at low temperature is subjected to a high-temperature spin-flip dynamics, the evolved measure may cease to be Gibbsian after a finite time. In this talk I will discuss such phenomenon. The talk will include a survey of the notion of Gibbs and non-Gibbsian measures and a discussion of what is known an expected regarding the dynamical loss of Gibsianness.
11 november 2009: Rik Lopuhaa (TU Delft) Central limit theorem and influence function for the MCD estimators at general multivariate distributions.
The minimum covariance determinant (MCD) estimators of multivariate location and scatter are robust alternatives to the ordinary sample mean and sample covariance matrix. Nowadays they are used to determine robust Mahalanobis distances in a reweighting procedure, and used as robust plug-ins in all sorts of multivariate statistical techniques which need a location and/or covariance estimate, such as principal component analysis, factor analysis, discriminant analysis and linear multivariate regression. For this reason, the distributional and the robustness properties of the MCD estimators are essential for conducting inference and perform robust estimation in several statistical models. Butler, Davies and Jhun (1993) show asymptotic normality only for the MCD location estimator, whereas the MCD covariance estimator is only shown to be consistent. Croux and Haesbroeck (1999) give the expression for the influence function of the MCD covariance functional and use this to compute limiting variances of the MCD covariance estimator. However, the expression is obtained under the assumption of existence, continuity and differentiability of the MCD-functionals at perturbed distributions, which is not proven. Moreover, the computation of the limiting variances relies on the von Mises expansion of the estimator, which has not been established. In this presentation we define the MCD functional by means of trimming functions which are in a wide class of measurable functions. The class is very flexible and allows a uniform treatment at general probability measures, including empirical measures and perturbed measures. We prove existence of the MCD functional for any multivariate distribution P and provide a separating ellipsoid property for the functional. Furthermore, we prove continuity of the functional, which also yields strong consistency of the MCD estimators. Finally, we derive an asymptotic expansion of the functional, from which we rigorously derive the influence function, and establish a central limit theorem for both MCD-estimators. All results are obtained under very mild conditions on P and essentially all conditions are automatically satisfied for distributions with a density. For distributions with an elliptically contoured density that is unimodal we do not need any extra condition and one may recover the results in Butler, Davies and Jhun (1993) and Croux
28 oktober 2009: Nelly Litvak (Universiteit Twente) The power law behaviour of the PageRank distribution.
The PageRank algorithm is designed by Google to rank Web pages according to their importance. According to this algorithm, the importance scores of pages depend on the quantity and the quality of incoming links. We study and explain the properties of PageRank scores in complex information networks characterized by power laws. It is a well-known fact that in the Web, the distribution of the PageRank and In-degree follows a power law distribution with the same exponent. We explain this similarity by presenting a PageRank distribution as a solution of a stochastic equation. Using this model, we apply analytical methods to derive the asymptotic behavior of the PageRank distribution. The obtained results are in good agreement with experimental data. Next, we suggest to measure the dependencies between power law parameters using the notion of angular measure developed within extreme value theory. This technique reveals that the WWW, the Wikipedia and the Growing Network graphs have a completely different dependence structure. In our stochastic model, we can also derive the angular measure analytically, which allows to quantify the proportion of pages that receive a high ranking due to a large in-degree. This is a joint work with Yana Volkovich, Debora Donato, Werner Scheinhardt and Bert Zwart.
21 oktober 2009: Judith Timmer (Universiteit Twente) Cooperation in Tandem Lines.
We consider a number of servers in a tandem line. The servers may improve the efficiency of the system by redistributing their service capacities. This improvement is due to the reduction in the steady-state mean total number of customers in the tandem line. We investigate how the cost of the system after redistribution should be divided among the servers. For this we use tools from cooperative game theory.
14 oktober2009: Noel van Erp & Pieter van Gelder (TU Delft ) Finding Proper Non-informative Priors for Regression Coefficients.
It is a known fact that in problems of Bayesian model selection improper priors may lead to biased conclusions. In this presentation we first give a short introduction to the procedure of Bayesian model selection. We then demonstrate for a simple model selection problem, involving two regression models, how improper uniform priors for the regression coefficients will exclude automatically the model with the most regression coefficients. Having established the problematic nature of improper priors for this particular case we proceed to derive a parsimoneous proper uniform prior for univariate regression models, firstly, and then generalize this result to multivariate regression models, secondly.
7 oktober2009: Sasha Gnedin (Universiteit Utrecht) Quasi-exchangeability and generalizations of de Finetti's theorem.
A random sequence is quasi-exchangeable if its distribution is quasi-invariant under permutations. We discuss generalizations of de Finetti's theorem for this setting, and make connections to a boundary problem for lattice random walks. Major attention is given to the q-exchangeability, for which it is shown that all ergodic sequences are obtainable from a single measure on the space of infinite permutations.
30 september 2009: Charles Berger (Nederlands Forensisch Instituut) Modern forensic methodology.
In this talk we will introduce the modern forensic methodology, where the aim is to find objective evidence using the likelihood ratio as a measure for the strength of that evidence given a set of hypotheses. We will illustrate this approach with some example projects, such as the inference of identity of source for varied traces such as ballpoint ink, speech, paper structure and fingermarks.
23 september 2009: Dierk Schleicher (Jacobs-Universitaet Bremen) Dynamics of transcendental entire functions and a dimension paradox.
16 september 2009: Paul Vitanyi (CWI) om 4 uur Similarity by Compression.
We survey a new area of parameter-free similarity distance measures useful in data-mining, pattern recognition, learning and automatic semantics extraction. Given a family of distances on a set of objects, a distance is universal up to a certain precision for that family if it minorizes every distance in the family between every two objects in the set, up to the stated precision (we do not require the universal distance to be an element of the family). We consider similarity distances for two types of objects: literal objects that as such contain all of their meaning, like genomes or books, and names for objects. The latter may have literal embodyments like the first type, but may also be abstract like ``red'' or ``christianity.'' For the first type we consider a family of computable distance measures corresponding to parameters expressing similarity according to particular features between pairs of literal objects. For the second type we consider similarity distances generated by web users corresponding to particular semantic relations between the (names for) the designated objects. For both families we give universal similarity distance measures, incorporating all particular distance measures in the family. In the first case the universal distance is based on compression and in the second case it is based on Google page counts related to search terms. In both cases experiments on a massive scale give evidence of the viability of the approaches
9 september 2009: Anne Fey (TU Delft) A splitting headache.
We consider the following game. Start with a pile of mass n at the origin of the rectangular grid, and mass h<1 at every other site. Mass may be moved by `splitting piles', that is, one may take all the mass from one site, and divide it evenly among the neighbors. There are restrictions though: One may only split piles of mass at least 1, and one may not stop before all the piles have mass less than 1. We call T the set of sites where at least one split was performed. Will the mass spread over the whole grid, or will T be finite? In the first case, how does it spread? In the last case, what is the size and shape of T, depending on h and n? How many splits are needed? Do any of these answers depend on the order of splitting? This game is related to the abelian sandpile growth model, for which a number of limiting shape results are known. The splitting game however is more difficult to analyse because it is not abelian. Nevertheless, we found several limiting shape results for the splitting game, some of which have no counterpart in the abelian sandpile growth model.
17 augustus 2009: Schmuel Gal (University of Haifa) Coordinated Linear Search.
17 juni 2009: Ian Melbourne (University of Surrey ) Statistical properties of Lorentz gases.
10 juni 2009: Henk Bruin (TU Delft and University of Surrey) Li-Yorke chaos and Cantor attractors for interval maps.
Whereas most unimodal interval maps are either chaotic in any mathematical sense of the word, or have periodic attractors attracting almost every point, there are unimodal maps with more interesting attractors. Such attractors are Cantor sets, and the dynamics on them is less chaotic: e.g. entropy and Lyapunov exponents are 0.In this talk, I want to explain that regarding the existence of Li-Yorke pairs (i.e., points x,y such that 0 = liminf |f^n(x)-f^n(y)| limsup |f^n(x)-f^n(y)| ), these attractors can still be quite interesting.(This work is joint with Victor Lopez-Jimenez, Murcia)
3 juni 2009: Guus Balkema (UvA en ETH) ZAAL F (different location!) Level sets and dependence.
The behaviour of multivariate extremes is determined by the tail behaviour of the marginals and by the (asymptotic) dependence structure of the distribution. Multivariate dfs tell us little about the distribution of the probability mass and the structure of large sample clouds. In this talk we focus on densities, which are assumed to be continuous and unimodal, and to have level sets which are of the same shape asymptotically. This gives a good impression of what sample clouds from the distribution look like. I will discuss recent work with Natalia Lysenko and Paul Embrechts (both ETH, Zurich) on the relation between the shape of level sets and asymptotic dependency.
27 mei 2009: Bert van Es (UvA) Two dimensional uniform kernel deconvolution.
Bert van Es Korteweg-de Vries Instituut, Universiteit van Amsterdam email: a.j.vanes@uva.nl
In a general deconvolution model we have a sample of n independent Xi which are equal to the sum of independent and unknown Yi and Zi. So Xi=Yi+Zi. We assume that the Zi have a known distribution. The aim is to estimate the probability density f of the Yi from this sample of Xi. Since the density of the observed Xi is equal to the convolution of the densities of the Yi and Zi one can derive a density estimator of f by Fourier inversion and kernel estimation of the density of the observations. This approach has proven to be useful in many deconvolution models, i.e. different known distributions of the Zi. However, it fails in the model where the known density of the Zi is uniform. This model is usually called uniform deconvolution. We will present an alternative method based on kernel density estimation and a different, non Fourier, type of inversion of the convolution operator in this model. Following earlier work for the one dimensional model, cf Van Es 2002, we will use the same approach in the two dimensional model where the Xi, Yi and Zi are two dimensional random vectors and where the distribution of the Zi is uniform on the unit square. We will derive expansions for the bias and variance and present some simulated examples.
Reference
[1] B. van Es. (2002) Combining kernel estimators in the uniform deconvolution model , ArXiv:math.PR/0211079.
20 mei 2009: Ruud van Ommen (TU Delft) Monitoring of complex chemical systems - a chemical engineer's view on applying non-linear signal analysis.
In the chemical process industry, various pieces of equipment show complex behaviour. A good example is a fluidized bed: a vessel filled with powder, through which a gas is blown upward at such a velocity that the particles (the grains of the powder) become fluidized (i.e., they are 'floating' on the gas stream). These particles are constantly colliding with each other and with the vessel wall, and form density patterns on a scale much larger than the particle diameter. In various industrial operations (e.g, burning biomass for "green electricity" or production of polymers) the particle can become sticky and form large lumps of materials. This can eventually lead to an unscheduled shut-down of the bed. Currently, the process industry measures typically just average properties such a pressure or temperature. These measurements often do not give an early warning for the operational problem. We propose to base the analysis on dynamic signal (typically pressure measurements at ~ 200 Hz) to obtain more information from the system. We reconstruct an 'attractor' in the state space, a technique borrowed from chaos analysis. By following this attractor in time, it can be observed if this attractor - and thus the monitored system - is changing, which is an indication of particles start to stick together. The above described technique of 'attractor comparison' is very sensitive, but we would like to improve the selectivity of the monitoring: unimportant events in the system should not be detected. To this end, we recently developed a screening methodology to find the most selective combination of filtering method and analysis method. Some practical results from this methodology will be shown.
13 mei 2009: Michael Schroeder (VU Amsterdam en Universität Mannheim) A perspective on the integral of geometric Brownian motion, with applications to finance.
The talk will concentrate on exponential functionals of Brownian motion, with the values of Asian options in the BS model as typical examples from finance. We survey how their structure theory emerges by way of establishing interconnections between stochastics and complex analysis on the upper half plane, taking work of Yor's and Dufresne's as starting points. Explicit valuation of Asian options furnishes the points of reference for measuring the progress here; we demonstrate how this is given expression to by at least 2 "arbitrary precision" methods whose potential we illustrate by way of numerical examples."
15 april 2009: Eric Cator (TU Delft) Hammersley process with random weights: stationary measures and Busemann functions. (joint work with Leandro Pimentel)
In this talk I will discuss the Hammersley process with random weights, both as a longest path percolation and as an interacting fluid process. We will show how we can use the Busemann function, as defined in geometry for metric spaces, to find mixing stationary measures for the Hammersley process. This gives us insight in the Busemann function for the classical Hammersley process, and a new description of the multi-class process. If time permits, we will discuss how these methods could be used to prove the cube-root behavior of the Hammersley process with random weights.
1 april 2009: Christophe Croux (K.U.Leuven) Classification Efficiencies for Robust Discriminant Analysis.
Linear discriminant analysis is typically carried out using Fisher's method. This method relies on the sample averages and covariance matrices computed from the different groups constituting the training sample. Since sample averages and covariance matrices are not robust, it has been proposed to use robust estimators of location and covariance instead, yielding a robust version of Fisher's method. In this paper relative classification efficiencies of the robust procedures with respect to the classical method are computed. Second order influence functions appear to be useful for computing these classification efficiencies. It turns out that, when using an appropriate robust estimator, the loss in classification efficiency at the normal model remains limited.
11 maart 2009: Ronald Geskus (Academisch Medisch Centrum) Three Equivalent Forms of the Cause-Specific Cumulative Incidence Estimator and the Fine and Gray Model.
The standard estimator for the cause specific cumulative incidence in a competing risks setting with left truncated and/or right censored data can be written in two alternative forms. One is as an inverse probability weighted estimator and another as a product-limit estimator. The product-limit estimator is based on a simple estimator of the subdistribution hazard, i.e. the hazard that corresponds to the cause specific cumulative incidence curve. As a consequence, estimation of the cause-specific cumulative incidence and regression on the subdistribution hazard can be performed using standard software for survival analysis if the software allows for inclusion of time dependent weights.
4 maart 2009: Bas Kleijn (UvA) Differentiability and the semiparametric Bernstein-Von Mises theorem.
The Bernstein-Von Mises theorem provides a detailed relation between frequentist and Bayesian statistical methods in smooth, parametric models. It states that the posterior distribution converges to a normal distibution centred on the maximum-likelihood estimator with covariance proportional to the Fisher information. In this talk we consider conditions under which the analogous assertion holds for the marginal posterior of a parameter of interest in smooth semiparametric estimation problems. Central is a no-bias condition, requiring that the nuisance posterior converges at a rate fast enough to suppress the bias. (Joint work with P. Bickel.)
25 februari 2009: Kees Oosterlee (TU Delft / CWI)
On Efficient Methods for Pricing Options with and Without Early Exercise
In this presentation we will discuss option pricing techniques in the context of numerical integration. Based on a Fourier-cosine expansion of the density function we can efficiently obtain European option prices and corresponding hedge parameters. Moreover a whole vector of strike prices can be valued in one computation. This technique can speed up calibration to plain vanilla options substantially. This pricing method, called the COS method, is generalized to pricing Bermudan and barrier options. We explain this and present a calibration study based on CDS spreads (if time permits).
18 februari 2009: Mark Veraar (TU Delft)
Probability and moment estimates for centered random variables
In this talk we discuss some sharp inequalities for centered random variables and their applications. In particular we give sharp lower bounds for P(X>0) in terms of moment estimates on X. Here X is a nonzero centered random variable.
11 februari 2009: Ionica Smeets (UL)
The LLL-algorithm: how it originated and how we can use it as a multidimensional continued fraction algorithm
Hendrik Lenstra, Arjen Lenstra and László Lovász published their famous LLL-algorithm for basis reduction in 1982. In 2007 the 25th birthday of this algorithm was celebrated in Caen, France with a three- day conference. Lenstra, Lenstra, Lovász and close bystander Peter van Emde Boas started the conference by telling how the algorithm emerged from misunderstandings, errors and coincidences. Ionica Smeets wrote down these memories for an upcoming book from Springer about the LLL- algorithm. The first part of her talk will be this nice historic story. In the second part she will talk about her own research on continued fractions and explain how you can iterate the LLL-algorithm to find a series of multidimensional continued fractions.
4 februari 2009: Gerard Hooghiemstra (TU Delft)
First passage percolation on random graphs with finite mean degreesjoint work with: Shankar Bhamidi and Remco van der Hofstad
This talk is about first passage percolation on the configuration model. Assuming that each edge of the graph has an independent exponentially distributed edge weight, we derive distributional asymptotics for the minimum weight between two randomly chosen vertices in the network, as well as for the number of edges on the least weight path. 2008
17 december 2008: Sebastian van Strien (University of Warwick en Universiteit Leiden)
On some questions of Fatou, Milnor and Palis on iterations of polynomial maps
This talk is about iterations of polynomials acting on the complex plane and their associated Julia, Fatou and Mandelbrot sets. I will give a survey of some recent results in this area.
10 december 2008: Ruud Hendrickx (UvT) In several jurisdictions, commercially exploiting a game of chance (rather than skill) is subject to a licensing regime. It is obvious that roulette is a game of chance and chess a game of skill, but the law does not provide a precise description of where the boundary between the two categories is drawn. We provide a framework of determining the relative skill level of a game and discuss some computational aspects. We apply this theory to various variants of poker. Confronting the computed relative skill levels with jurisprudence on the Dutch Gaming Act, we conclude that poker should be classified as games of skill.
26 november 2008: Michel Vellekoop (UT)
Dividends and Discontinuities: the Dirty Little Secret of Mathematical Finance
Standard option models usually pay no or little attention to the inclusion of dividends in the model for the underlying asset prices. In this talk we show that option pricing is only possible in practice if dividends are explicitly included, and we provide a general semimartingale framework to do so. As a first apllication, we show how this allows us to extend integral representations for the early exercise premium in American options to the case where dividends are paid. A second application leads to the surprising result that future price processes need not be risk neutral martingales on discontinuous filtrations.
12 november 2008: Frank van der Meulen (TU-Delft)
Bayesian nonparametric estimation for diffusions
Diffusions can be obtained as solutions of stochastic diffeential equations. As such, they are characterized by their drift and diffusion coefficient. In this talk I will discuss Bayesian estimation of these coefficients using either continuous or discrete time observations. If we observe a sample path of a diffusion continuously in time, we only need to estimate the drift parameter. I will present general conditions from which the posterior rate of convergence (the rate at which the posterior contracts around the "true" parameter) for estimating this parameter can be deduced. Then I will move to the discrete time setting. For this case I will show how the posterior can be computed by Bayesian data augmentation. Joint work with Harry van Zanten and Aad van der Vaart (Vrije Universiteit).
8 oktober 2008: Peter Harremoës Centrum voor Wiskunde voor Informatica (CWI)
The Law of Thin Numbers
It is wellknown that binomial distributions and other Bernoulli sums can be approximated by Poisson distributions, which is sometimes called the law of Small Numbers. It is less known that this can be view as a case of entropy maximization. Inspired by ideas from information theory we shall develop a new framework to describe Poisson approximation. One of the ideas is the definition of thinning of a random variable that allow us to formulate a Law of Small Numbers based on iid sequences rather than on triangular arrays. We also get a closer link to the Central Limit Theorem and get a new lower bound on the rate of convergence in the Central Limit Theorem. We also get very tight bounds the total variation distance between a binomial and the Poisson distribution with the same mean.
1 oktober 2008: Sicco Verwer (TUD, informatica)
An efficient algorithm for learning timed processes
We describe an efficient algorithm for learning deterministic real- time automata (DRTA) from positive data. A DRTA is an intuitive model for many real-time systems. The data can be obtained from observations of some process. We assume this process to be stationary. The algorithm uses statistical tests in order to learn an DRTA model that describes this stationary process. This model can be used to reason and gain knowledge about real-time systems such as network protocols, business processes, reactive systems, etc.
24 september 2008: Leandro Pimentel (TU Delft)
Greedy Polyominoes and first-passage times on random Voronoi tilings
Let N be distributed as a Poisson random set on R^d with intensity comparable to the Lebesgue measure. Consider the Voronoi tiling of R^d, { C_v : v in N }, where C_v is composed by points x in R^d that are closer to v in N than to any other v' in N. A polyomino P of size n is a connected union (in the R^d topological sense) of n tiles, and we denote by Pi_n the collection of all polyominos P of size n containing the origin. Assume that the weight of a Voronoi tile C_v is given by F(C_v), where F is a nonnegative functional on Voronoi tiles. In this paper we investigate the tail behavior of the maximal weight among polyominoes in Pi_n: F_n=F_n(N):=max{ sum_{v in P} F(C_v) : P in Pi_n }. As the main application we show that first passage percolation has at most linear variance.
18 september 2008: Richard Gill (UL)
Careless statistics costs lives
I will explain the Snapinn (1992) rule for early stopping of a randomized clinical trial. This very cunning protocol preserves the standard analysis at the end of a not-early-terminated trial, by balancing the chances (under the null-hypothesis) of abandoning the trial early for expected futily when actually the final result would have been significant, and abandoning the trial early for expected signficance when actually the final result would not have been significant. Further cunning features allows the protocol to be extended from the theoretical setting of testing a normal mean (known variance) to the general setting of, for instance, comparing two unknown Bernoulli probabilities. The Snapinn rule was built into the protocol of the now famous PROPATRIA trial of probiotics treatment in acute pancreatits. It appears now that this trial was allowed to run to completion because of a confusion between one-sided and two-sided testing. This confusion together with the fact that the monitoring committee was blinded to the actual treatments given to the two treatment groups made it possible for them to continue the trial, effectively because there was still a good chance of finally obtaining a significant *harmful* effect of the treatment, when, according to their own protocol, they should have stopped it, because there was almost no chance any more of finally obtaining a significant *beneficial* effect of the treatment. I will give recommendations for precautions which should be built into the design of RCT's in the future, in order to prevent this kind of mistake. www.math.leidenuniv.nl/~gill/probiotica.pdf (slides of talk) arxiv.org/abs/0804.2522 (discussion paper)
10 september 2008: Charlene Kalle (UU)
Beta-expansions with arbitrary digits
Beta-expansions with arbitrary digits are generalizations of the well-understood classical beta-expansions which use the integers 0 up to the floor of beta as digit set. After a short review on the classical beta-expansions, we will introduce two transformations that generate expansions with arbitrary digits, the greedy and lazy transformation, and give some of their measure-theoretical properties. We will then consider a random transformation that generates all possible beta-expansions for a given beta and arbitrary digit set.
28 mei 2008: Wioletta Ruszel (Groningen)
What it takes to be Gibbsian for planar rotors
We study the Gibbsian character of time-evolved planar rotor systems on $mathbb{Z}^d$, $dgeq2$ , in the transient regime, evolving with stochastic dynamics and starting from an initial Gibbs measure $
u$. We model the system by interacting Brownian diffusions $(X_i(t))_{i in mathbb{Z}^d, t geq 0}$ moving on circles. We prove that for small times t and both arbitrary initial Gibbs measures $
u$ and arbitrary temperature dynamics, or for long times and both high- or infinite-temperature initial measure and dynamics, the evolved measure $
u^t$ stays Gibbsian. Furthermore, we show that for a low-temperature initial measures evolving under infinite- temperature dynamics there is a time interval such that $
u^t$ fails to be Gibbsian
14 en 21 mei 2008: Mike Keane (Wesleyan)
Once Reinforced Random Walks on Lines and Ladders
In these lectures we shall treat the recurrence (or possible transience; there are open questions here) of once reinforced random walks on the integers and on products of integers with finite segments of integers, called ladders. The first lecture will deal with once reinforcement on the integers, where we can prove that no matter what the strength of the reinforcement (or weakening) is, such random walks are recurrent. In this lecture we also introduce the martingale approach to the recurrence problem. In the second lecture, we shall treat once reinforcement on the ladders. If there are only two copies of the integers, then Sellke has proved that once reinforced random walk is recurrent for any positive reinforcement, and together with Feiden we have now a proof that this remains true for negative reinforcement (i.e. weakening). Both questions are still open for ladders of widths greater than two, although there are positive results for some values of positive reinforcement due to Sellke (low values of positive reinforcement) and Vervoort (high values of positive reinforcement). We sketch some of the proofs and explain the current state of affairs. Of course, it is expected that for any width and any reinforcement, positive or negative, random walk is recurrent, and even if we consider the case of two dimensions, i.e. infinite ladders we expect recurrence. However, the latter problem seems to be well beyond reach using current techniques.
7 mei 2008: Karma Dajani (Utrecht)
Beta-expansions revisited We give an overview of some of the old and new results describing the ergodic and arithmetic properties of algorithms generating expansions to non-integer base.
23 april 2008: Anne Fey-den Boer (TU Eindhoven)
Quasi-units in Zhang's sandpile modelZhang's model is a non-abelian sandpile model. Numerical simulations of this model on large grids have indicated that the stationary height distribution per site is sharply peaked at discrete values, resembling that of the abelian sandpile model, despite the fact that in Zhang's model the heights are continuous. Zhang called these values 'quasi-units'. We have defined and analyzed this model rigorously in dimension 1. Our main result concerns the limit of infinite grid size. We find that the stationary height distribution indeed tends to that of the abelian sandpile model, up to a scaling factor. Among other results, we prove uniqueness of the stationary height distribution. Finally, I will outline some future research plans on this model, for example, study phases transitions in an infinite volume version, study the model in higher dimensions, as a growth model, and eventually form a link with neuronal network modeling.
16 april 2008: Ludolf Meester (TU Delft)
Extremal distributions for sums of iid random variables on [0,1] or Should simple problems have simple solutions? Two old "Problems section conjectures", one from Statistica Neerlandica and one from SIAM Review, concern the following question: Let X_1,..., X_n be i.i.d. random variables on [0,1], satisfying E[X_1]=m, 0<m<1. Let S_n=X_1+...+X_n and 0<=t<n. Given n, m and t, which distribution maximizes P(S_n<=t)? From the answer a non-parametric confidence bound (of interest to auditors) could be derived. It would also imply a sharpening of Hoeffding's inequality. The n=1 version of the problem is easily solved by looking for equality in Markov's inequality (you can do this in 5 minutes). In an attempt to solve the general problem I apply Mattner's Lagrange multiplier approach, a method for finding (all kinds of) extremal distributions, which is of interest in itself. For n=2, the resulting Lagrange conditions can be shown to imply that extremal distributions should be discrete with at most three support points, one of which is 0 or 1. Combining this with some elementary optimization, this case is solved. I will present these solutions and their implications for the published conjectures. In addition, I would like to discuss some other insights, conjectures and attempts for n>2, perhaps generating some new ideas in the audience.
9 april 2008: Peter Sozou (London School of Economics)
Courtship as a waiting gameEvolution selects for courtship that maximise Darwinian fitness. Courtship is modelled as an iterative game in which a male sends out a signal, such as a Valentine's card or a dinner invitation, that the female may accept or reject. If the female accepts, then the male gives another signal. This type of waiting game models mating behaviour in arthropods, hermit crabs and humans.
12 maart 2008: Michel Dekking (TU Delft)
Arithmetic differences of random Cantor sets and the lower spectral radius Let C and D be two Cantor sets. When will their difference C-D = {x-y: x from C, y from D} contain an interval? Necessarily we should have that the sum of their Hausdorff dimensions is larger than 1. When is this also sufficient? This question will be answered almost surely for a natural class of random Cantor sets.
5 maart 2008: Vilmos Komornik (Strasbourg)
Univoque expansions
20 februari 2008: Birgit Witte (TU Delft)
Maximum Smoothed Likelihood Estimation in Censoring Problems We study the stochastic behaviour of the time $X$ it takes before a certain event takes place (also called the survival time). In many cases, the variable $X$ is not observed directly due to some sort of censoring and in this talk we consider smooth estimators in two different but related censoring models. The first model is the current status model where we observe a censoring variable $T$ (independent of $X$) and a variable $Delta = 1_{{X le T}}$ indicating whether the event took place before time $T$ or had not taken place yet. The maximum smoothed likelihood estimator (MSLE) based on the approach of Eggermont & LaRiccia (2001) is similarly characterized as the well studied and natural estimator in this model, the nonparametric maximum likelihood estimator (NPMLE), see also Groeneboom & Wellner (1992). Both estimators are consistent, however the asymptotic properties differ. In the second model we are interested in the bivariate distribution function $F_0$ of the pair $(X,Y)$, where $X$ is the survival time and $Y$ a continuous mark variable. As in the current status model we do not observe the variable $X$ directly, instead we observe a censoring variable $T$ and a variable $Delta=1_{{Xleq T}}$. When $X$ lies to the left of $T$, i.e. $Delta=1$, we also observe the variable $Y$, in case $Delta=0$, we do not. The NPMLE in this model is studied by Maathuis & Wellner (2007), who prove that this estimator is inconsistent. We propose an estimator in the spirit of Eggermont & LaRiccia (2001).
This is joint work with Geurt Jongbloed and Piet Groeneboom
13 februari 2008: Steve Alpern (London School of Economics)
Rotational (and Other) Representations of Stochastic Matrices Joel E. Cohen (1981) conjectured that any stochastic matrix P could be represented by some circle rotation f in the followingsense: for some partition Si of the circle into sets consisting of finite unions of arcs, we have that the entries pij of the matrix P are weights of intersection (*) pij = μ (f (Si ) ∩ Sj ) / μ(Si ), where μ denotes arc length. In this paper we show how cycle decomposition techniques originally used (Alpern, 1983) to establish Cohen's conjecture can be extended to give a short simple proof of the Coding Theorem, that any mixing (that is, P^N > 0 for some N) stochastic matrix P can be represented (in the sense of * but with Si merely measurable) by any aperiodic measure preserving bijection (automorphism) of a Lesbesgue probability space. Representations bypointwise and setwise periodic automorphisms are also established. Based on a joint paper with Raj Prasad
2007
19 december 2007: Gerald Lucassen (Philips Research) Multivariate Data Analysis Applied in Molecular Diagnostics.
In the Healthcare program at Philips Research, various methods are under investigation for diagnostic tests based on specific protein or DNA detection. A common trend in the field of molecular diagnostics of infectious diseases is the need for rapid detection of multiple analytes. Implementation of these so-called multiplex methods requires chemical analysis techniques that allow simultaneous collection of multiple analyte data, and data analysis techniques that allow unscrambling of the compound signals into those of the individual constituents. In the presentation we will highlight an optical method that uses surface-enhanced resonance Raman spectroscopy (SERRS) for the analysis of DNA from clinical samples. The SERRS method is suitable for simultaneous measurement of multiple analytes in solution. As multiplex methods involve observation and analysis of more than one statistical variable at a time, multivariate statistical methods are required, both in experimentation and in data analysis. In the talk we will present the experimental design and multivariate data analysis techniques that we have used to reach clinically relevant results from measured SERRS data.
12 december 2007: Wouter Kager (Eurandom) Scaling limits of the two-dimensional critical Ising model.
I will address recent progress in the mathematical understanding of scaling limits of the critical Ising model in two dimensions, through the work of Wendelin Werner, Stas Smirnov and others. The presentation does not require specialized knowledge of the Ising model, scaling limits or Schramm-Loewner Evolution (SLE). The main aim is to explain that there exists just a one-parameter family of possible scaling limits for models such as the Ising model, namely the so-called Conformal Loop Ensembles (CLEs). To explain this, I will focus on the critical Ising model and argue that its scaling limit, if it exists, should be a CLE. Then, I will present an independent construction of CLEs through the Brownian loop soup introduced by Lawler and Werner. Finally, we will conclude that the scaling limit of the critical Ising model can only be one of these CLEs.
5 december 2007: Frank Redig (UL) Concentration, coupling and stochastic dynamics.
We present a new coupling approach to obtain concentration inequalities for random fields such as lattice-spin systems. In the uniform coupling regime, we obtain Gaussian concentration bounds, in the non-uniform regime weaker inequalities such as moment bounds. We also illustrate how to obtain upper bounds for the speed of relaxation to equilibrium for interacting particle systems using concentration inequalities and coupling.
28 november 2007: Cees Diks (UvA) Weighted Likelihood Ratio Scores for Evaluating Density Forecasts in Tails.
We propose and evaluate several new scoring rules based on likelihood ratios, for comparing density forecasts in the context of VaR modelling and expected loss estimation. Ou approach is motivated by the observation that some existing weighted scoring rules tend to favour fat-tailed predictive densities over thin-tailed predictive densities. Rather than restricting the weight functions, we impose some restrictions on the score functions. Our benchmark case has fixed weights, equal to one in the left tail and zero elsewhere. Two different scoring rules based on partial likelihood are proposed for this zero-one case. After generalizing the new scoring rules to smooth weight functions, their properties are investigated numerically and illustrated by an empirical application.
7 november, 2007: Rie Natsui (Tokyo) Non-monotonicity of the entropy of the alpha-continued fraction transformations
17 oktober 2007: Michiel van Lambalgen (UvA) Logic in the study of autism: reasoning with rules and exceptions.
Autism is a psychiatric disorder characterised by (sometimes very severe) impairments in verbal and social communication, and in executive tasks. The aetiology of autism is controversial. A prominent theory holds that autism is a consequence of a `theory of mind' deficit, as manifested in significantly worse than normal performance in `false belief' tasks. An apparently very different theory maintains that autism is a consequence of defects in executive function, and again there are (non-verbal) tasks which support this interpretation. Clearly the aetiology has implications for treatment. A logical analysis of theories and diagnostic tasks has turned out to be very helpful here. It so happens that both in the false belief tasks and in the tasks supporting executive dysfunction closed world reasoning (CWR) plays a large role. CWR counsels to take those propositions as false which one has no reason to suppose true. Once it has been observed that CWR is important in existing tasks, it becomes possible to devise an experimental paradigm in which CWR is tested more directly. In fact an existing task in the psychology of the reasoning (in healthy adults), the `suppression task', filled the bill. The logical analysis alluded to predicted that autists would behave very differently on this task compared to normals. An experiment on 28 autists and 28 matched controls indeed showed this to be the case. On a more theoretical level, what is interesting is that CWR seems to correspond to a neural architecture in which inhibitory interneurons play an important role; and recent neurological evidence indicates that these are compromised in autism.
10 oktober 2007: Sébastien Blachère (Eurandom) The use of the Green metric for random walks on hyperbolic groups.
Using the Green function of a random walk on a discrete group, we may define a metric which happens to have some nice geometric properties and is also useful to describe, in some cases, asymptotic behavior of the random walk itself. We will first define this Green metric and explain some finite range properties (volume of the associated balls, comparison with the usual graph metric). Then we will focus on random walks on hyperbolic groups. In that setting, the Green metric provides a useful tool to describe the harmonic measure of the random walk on the boundary of the group. It allows to see this harmonic measure as a quasi-conformal one when the boundary is equipped with the limit of the Green metric. Then we can compute the Hausdorff dimension of the harmonic measure. Another consequence is a characterization of the random walks that satisfy the so-called fundamental inequality between the asymptotic entropy and the product of the rate of escape and the logarithmic volume growth.
3 oktober 2007: Peter Sonneveld (TUD) Convergence behaviour of IDR(s) and the use of random testvectors in a Krylov-Galerkin method.
5 oktober, 2007: Intercity Seminar on Number Theory
special topic 'Ergodic Theory and Numbers'.
26 september 2007: Johan van Leeuwaarden (TUE) The maximum of the Gaussian random walk.
We consider the Gaussian random walk (one-dimensional random walk with normally distributed increments) with negative drift, and in particular its all-time maximum M. We derive explicit expressions for all moments of M in terms of Taylor series with coefficients that involve the Riemann zeta function. We build upon the work of Chang and Peres (1997) on P(M=0) and Bateman's formulas on Lerch's transcendent. Our result for E(M) extends Kingman's (1965) first order approximation for a small drift. We show how our results find application for queues in heavy traffic and for the equidistant sampling of Brownian motion.
19 september 2007: Eric Cator (TUD) Adaptivity of the monotone least squares estimator.
In this talk we will consider the estimation of a monotone regression (or density) function in a fixed point by the least squares (Grenander) estimator. We will show that this estimator is fully adaptive, in the sense that the attained rate is given by a functional relation using the underlying function f_0, and not by some smoothness parameter, and that this rate is optimal when considering the class of all monotone functions, in the sense that there exists a sequence of alternative monotone functions f_1, such that no other estimator can attain a better rate for both f_0 and f_1. When defining the rate, we do not look at the expectation of some convex loss function, but rather we bound the probability that the difference between the estimator and the true value is larger than the given rate (probabilistic error).
12 september 2007: Gerard Hooghiemstra (TUD) Some diameter results for a preferential attachment model . (joint work with Remco van der Hofstad)
In this talk I will define the preferential attachment model in question and derive its degree sequence. Then I will present 3 bounds on the diameter of this graph and if time permits sketch a proof of the third result.
