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Aalto Stochastics and Statistics Seminar is organized by Kalle Kytölä, Lasse Leskelä, Pauliina Ilmonen, and Christian Webb. Feel free to contact one of us if you are interested in giving a talk. You may also earn credit points by active participation.
The seminar is intended for epidemiological modellers. I will present a set SEIR models used by THL to model the COVID outbreak in Finland. I will analyse particular model features, discussing whatever they there useful for inference. I will describe different questions which we tried to answer with these models, and wherever modelling was able to provide useful insight. For any questions, contact Mikhail (Mikhail.shubin@thl.fi https://teams.microsoft.com/l/meetup-join/19%3ameeting_NjE4MDVkYWUtOGQ5MS00YWIxLWIwNDYtMzUxZjQyNmExOTE5%40thread.v2/0?context=%7b%22Tid%22%3a%2281d1db0d-c492-4225-a66c-3c3a9e4b8360%22%2c%22Oid%22%3a%221373e033-bd9e-4fa9-898a-a07b58e1f61e%22%7d
The six-vertex model is a planar random model for the crystalline structure of water ice. It has recently given important insights to the connection of Conformal field theory and critical 2D random models, due to its natural representation as a random field, called the height function, and due to couplings to several other important random models (e.g. FK cluster model, Ising and Potts models, dimers, random graph homomorphisms) We prove that the six-vertex height function has a localization/delocalization phase transition. Delocalization means roughly speaking that the model is not sensitive to a boundary condition far away; indeed our result for instance implies that there exists a unique whole-plane six-vertex model in the delocalized phase. The main tools of the proof are an explicit solution of the free energy of the model, and RSW and FKG inequalities similar as in the study of various percolation models. (Based on ongoing work with Hugo Duminil-Copin, Ioan Manolescu, and Mendes Oulamara.) Zoom-link: https://aalto.zoom.us/j/66281157229
The traditional methods for detecting community structure in a network are based on selecting dense subgraphs inside the network. Here we propose to use the methods of coalitional game theory that highlight not only the link density but also the mechanisms of cluster formation. Specifically, we propose an approach which is based on hedonic coalitional games. This approach allows to find clusters with various resolution. Furthermore, the modularity-based approach and its generalizations as well as ratio cut and normalized cut methods can be viewed as particular cases of the hedonic games. Finally, for methods based on potential hedonic games we suggest a very efficient computational scheme using Gibbs sampling. Bio: Konstantin Avrachenkov received the masters degree in control theory from St. Petersburg State Polytechnic University in 1996, the Ph.D. degree in mathematics from the University of South Australia in 2000, and the Habilitation (Doctor of Science) degree from the University of Nice Sophia Antipolis in 2010. Currently, K. Avrachenkov is Director of Research at Inria Sophia Antipolis. His main research interests are Markov chains, Markov decision processes, stochastic games and singular perturbations. He applies these methodological tools to the modelling and control of telecommunication systems and to design data mining and machine learning algorithms. He has won 5 best paper awards. He is an Associate Editor of the International Journal of Performance Evaluation, Probability in the Engineering and InformationalSciences, ACM TOMPECS and Stochastic Models.
Typically, in the functional context, data depth approaches heavily emphasize the location of the functions in the distribution, therefore often missing important shape or roughness features. Commonly, these depth approaches either integrate pointwise depth values to achieve a global value, or measure the expected distance from a function to the distribution. In this talk, we introduce a new class of functional depths, based on the distribution of depth values along the domain, and discuss their properties. We study the asymptotic properties of these $J$th order $k$th moment integrated depths, and illustrate their usefulness in supervised functional classification. In particular, we demonstrate the importance of receptivity to shape variations, and show that, similarly to existing depth notions, the new class of depth functions takes into account the variation in location, while remaining receptive to variations in shape and roughness.
Prostate cancer incidence rate is extremely high and on the rise, counting over 1.2 million new cases annually and causing 350 000 deaths in 2018. While the prognosis is typically good, approximately 20% of the new cases classifies as high-risk prostate cancer with dire consequences. Moreover, the initial prostate cancer diagnosis always reflects as worry and quality of life impairment. The initial prostate cancer determination is based on prostate specific antigen (PSA) measure, which cannot distinguish between low-risk and high-risk cases. After the PSA determination, the tumor state is characterized with various invasive methods such as Gleason score and T-stage classification. However, both methods display inaccuracy and puts patient under infection risk. Our take on this challenge is to use inflammation-related gene single nucleotide polymorphisms (SNP) as predictors of high-risk prostate cancer. SNP is a low-cost, non-invasive, and stable biomarker. We have explored inflammation SNP association with high-risk prostate cancer in a genotyped part of Finnish Randomized Screening for Prostate Cancer cohort (n = 2715) and found several statistically significant associations. Furthermore, our validation model using unknown prostate cancer cohort collected during hospital visits (n = 888) is in concordance with our discovery model. Remarkably, few SNPs increase early high-risk prostate cancer detection over PSA alone.
Many real-life networks can be naturally modeled by assuming an underlying community structure on the nodes. When each node can belong to more than one community, we say that the communities overlap. This talk discusses the modeling of such networks with random intersection graphs. We review some of their asymptotic properties, such as subgraph counts, and discuss consistent moment-based parameter estimation in a sparse setting.
A central problem in metric fixed point theory is to understand when a nonexpansive (i.e. Lipschitz with constant 1) self-map of a metric space has a fixed point. Even in the case where the metric space is a finite dimensional normed space, this is a subtle problem, as the map need not be a Lipschitz contraction and the space is not bounded, so neither the contraction mapping theorem nor the Brouwer fixed point theorem applies. In this talk I will give necessary and sufficient conditions for a nonexpansive map on a finite dimension normed space to have a bounded non-empty fixed point set. Moreover, we will provide a procedure that can detect fixed points of such maps using sets that illuminate the unit ball of the normed space. We will see how horofunctions play a role in this problem. Time permitting I will also discuss some applications to stochastic games.
The halfspace depth is a tool of non-parametric statistics, whose main aim is a reasonable generalisation of quantiles to multivariate data. It was first proposed in 1975; its rigorous investigation starts in the 1990s, and still an abundance of open problems stimulates the research in the area. We present interesting links of the halfspace depth, and some well-studied concepts from geometry. Using these relations we resolve several open problems concerning the depth, and outline perspectives for future research not only in non-parametric statistics, but also in certain areas of convex geometry. The talk is intended to be largely self-contained; no particular knowledge of probability and statistics is necessary.
This project concentrates on simulating the momentum density of annihilating electron-positron pairs. We use the CASINO simulation program in order to optimize the wave function of a system to simulate the the momentum density using Quantum Monte Carlo methods. The simulations involve diamond, silicon and germanium FCC-lattices.
We discuss how discrete and continuous time multivariate stationary processes can be characterized by an AR(1) type of equation and Langevin equation, respectively. Under the assumption of finite second moments, this leads to quadratic matrix equations for the model parameter matrix that are known as continuous time Riccati equations (CAREs). Based on the equations, we define an estimator for the parameter that inherits consistency and the rate of convergence from autocovariance estimators of the (observed) stationary process. Furthermore, the limiting distribution is given by a linear function of the limit random variable of the autocovariance estimators.
A mixed graph is a graph consisting of both undirected edges and directed edges.This talk discusses first passage percolation on a connected mixed random graph with a given degree sequence, where an undirected edge is formed between type-1 nodes and a directed edge between type-1 and type-2 nodes. Weights on edges are assumed to be independent and exponentially distributed. We analyze a flooding time, which is the minimum time that a uniformly chosen node reaches all other nodes. We derive an asymptotic formula for the flooding time as the number of nodes tend to infinity. As an application, we discuss continuous time information spreading on a random regular graph, where we also take into account the impact of passive nodes. Type-1 nodes can be interpreted as active message spreaders and type-2 nodes can be interpreted as passive receivers which may only receive the message. In this setting we derive an asymptotic formula for the flooding time which is also called the broadcast time in the literature
In several fields of science, a generic problem consists of separating useful signals from uninteresting noise and interference. The problem can be approached by implementing latent variable models. In our approach, we aim to find latent processes, when only linear mixtures of them are observable. In this context, we provide an estimation procedure for complex-valued stochastic processes. Furthermore, we study the asymptotic behavior of the so-called unmixing estimators. We provide novel asymptotic theory for scenarios, when the estimators are not root-n consistent and the limiting distributions are not Gaussian.
One of the central objectives of modern risk management is to find a set of risks where the probability of multiple simultaneous catastrophic events is negligible. That is, risks are taken only when their joint behavior seems sufficiently independent. Our objective is to provide additional tools for describing dependence structures of multiple risks when the individual risks can obtain very large values. The study is performed in the setting of multivariate regular variation. We show how asymptotic independence is connected to properties of the support of the angular measure and present an asymptotically consistent estimator of the support. The estimator generalizes to any dimension greater than or equal to two and requires no prior knowledge of the support. The validity of the support estimate can be rigorously tested under mild assumptions by an asymptotically normal test statistic.
It is known that Schramm Loewner evolution (SLE) is coupled with Gaussian free field (GFF) to give a solution to the flow line problem for an imaginary surface. I will overview our recent work where we extended this coupling to the case of multiple SLE. There, we found that the SLE partition function that defines a multiple SLE and the boundary perturbation for GFF are determined essentially uniquely so that the associated multiple SLE and GFF are coupled with each other.
We present a physics approach to conformal field theory in two dimensions: the bootstrap approach. In this approach, one directly imposes conditions on correlation functions inspired by conformal symmetries. Within this framework, we give emphasis to the fusion rules of degenerate representations. Using fusion rules, we build a well-known set of simple models called Minimal Models. Finally, we propose an extension of them at central charge c=0.
Partial differential equations, PDEs, describe many real life phenomena, and they are a subject of active research. Recently a growing attention have been paid to stochastic versions of PDEs - stochastic partial differential equations, or SPDEs for short. Such equations arise naturally as a random shock may represent some external random force affecting the system, or possibly some measurement errors. However, in the study of SPDEs many classical approaches breaks down completely. Indeed, even the concept of differential is subtle - the solution being typically only Hölder continuous. Moreover, as there is a random force affecting the system, the solution is also a random object. Typically, analysing this randomness is very complicated. In this talk, we discuss d-dimensional stochastic heat equations driven by a Gaussian noise which is white in time and has a spatial covariance given by the Riesz kernel. Basic theory and properties of the solutions are discussed. As a main result, we present a quantitative central limit theorem stating that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the radius of the ball tends to infinity. Our central limit theorem is described in the total variation distance, using Malliavin calculus and Stein's method. We also provide a functional central limit theorem and analogous results in the case of space-time white noise. Extensions and further open questions are discussed.
For a standard Gaussian random variable N, integration by parts gives the Stein equation E(Nf(N)- Df(N))=0 The Stein equation characterizes the distribution and it is the key in proving quantitative limit theorems towards the Gaussian. Here we take the first steps in extending the methodology, and give an algorithm producing all the Stein differential operators with polynomial coefficients for target random variables of the form X= p(N_1, ..., N_d), with Gaussian N and polynomial p. This is a joint work with Ehsan Azmoodeh (Bochum) and Robert Gaunt (Manchester)
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