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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.
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)
We will discuss a construction of correlations of discrete fermions for the two-dimensional critical FK-Ising and Ising models as expectations over geometrical configurations. The observable plays the role of a precursor for the free fermion in the Ising CFT, and it inspires the construction of CFT fields in the continuum case in terms of SLE/CLE measures.
Atherosclerotic cardiovascular disease (ASCVD), also known as coronary artery disease (CAD), is one of the leading causes of death in the world.[1] A consensus has been reached that the main cause of ASCVD are low-density lipoproteins (LDL). [2] ASCVD develops in the innermost layer of the coronary artery wall (intima). Once LDL particles enter the wall, they are retained, modified, and accumulate there. [3] There are several well-known risk factors of ASCVD, among which obesity, smoking, hypertension and LDL-cholesterol concentration in the plasma.[3] A novel approach to assessing the risk of ASCVD however suggests that, not only the concentration, but also the quality of LDL might be associated with ASCVD. [3] It shows that the susceptibility of LDL particles to aggregate (in the presence of the enzyme hrSMase) varies between humans and depends on the composition of the LDL particles. [3] The presence of aggregation-prone LDL in the plasma was found to be associated with future coronary artery disease (CAD) deaths. [3] This makes investigating LDL-aggregation further particularly important. This masters thesis studies LDL aggregation of patients who underwent bariatric surgery - a procedure performed on people with obesity, for the purpose of weight loss. It focuses on four main points: ● Creating a nonlinear mixed-effects model of LDL-aggregation and obtaining a single quantitative measure of LDL-aggregation. ● Investigating whether there is a significant difference in LDL-aggregation in patients before and after bariatric surgery ● Studying correlations between LDL-aggregation and lipids from the LDL-lipidome, as well as correlations with clinical data of bariatric surgery patients. ● Investigating whether there is a significant difference in the LDL-lipidome lipids and clinical parameters in the patients before and after the operation The presentation will discuss the progress made on the project. It will cover the following points. ● Problem Overview: Theory and Data ● Solution Plan ● Step 1: Modelling of LDL-aggregation - Nonlinear Mixed-Effect Models - Modelling using the Bayesian approach - Modelling Problems - Possible Solutions References: [1] George, S. and Johnson, J. (2010). Atherosclerosis: Molecular and Cellular Mechanisms. Weinheim: Wiley-VCH-Verl. [2] Ference, B. et al (2017). Low-density lipoproteins cause atherosclerotic cardiovascular disease. 1. Evidence from genetic, epidemiologic, and clinical studies. A consensus statement from the European Atherosclerosis Society Consensus Panel. European Heart Journal, 38(32), pp.2459-2472. [3] Ruuth, M. et al (2018). Susceptibility of low-density lipoprotein particles to aggregate depends on particle lipidome, is modifiable, and associates with future cardiovascular deaths. European Heart Journal, 39(27),pp.2562-2573.
In semi-supervised graph clustering setting, an expert provides cluster membership of few nodes. This little amount of information allows one to achieve high accuracy clustering using efficient computational procedures. Our main goal is to provide a theoretical justification why the graph-based semi-supervised learning works very well. Specifically, for the Stochastic Block Model in the moderately sparse regime, we prove that popular semi-supervised clustering methods like Label Spreading achieve asymptotically almost exact recovery as long as the fraction of labeled points does not go to zero and the average degree goes to infinity.
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.
The computation of the probability of rare events is the main purpose of large deviations theory. For instance, in a simple case, one can consider the rare event in which a sum of i.i.d. Bernoulli variables attains a value which is larger than its average. A completely different and much more difficult problem, is the computation of large deviations probability of nonlinear functionals of the Bernoulli variables, e.g. cubic polynomials. A case in which such nonlinear problems arise is, for instance, the study of complex networks. In this talk I will present the behavior of the so-called scaled cumulant generating function of the number of triangles of an Erdӧs-Rényi random graph (dense case). The scaled cumulant generating function is strictly connected with the theory of large deviations since, when it is possible to apply the Gärtner-Ellis theorem, it turns out to be the Legendre transform of the rate function. More precisely, the aim of this talk is twofold. On one hand, to describe a modified version of a known Monte Carlo method, called Cloning algorithm, tailored for approximating the scaled cumulant generating function of an additive observable in the framework of random graphs. One the other hand, keeping the focus on the triangle observable, to present the numerical investigation performed in the region where the analytical expression of such function is not known (replica breaking regime).
The dimer model studies random configurations of perfect matchings (dimer covers) of bipartite planar graphs. Through an associated height function such a configuration is encoded in a random surface. These random surfaces (with a fixed boundary) exhibit limit shape formation: a deterministic limit surface emerges in the macroscopic limit. The imposed boundary condition can have dramatic effect: in certain regions the dimers line up in an ordered fashion (form a frozen facet) and do not look random at all. A prime example of this phenomenon is the arctic circle of domino tilings of the Aztec diamond from 1995. We now have, mostly due to Kenyon et al., a general theory which describes these phenomena in unprecedented detail. The limit shape is described by a convex but singular and degenerate variational problem with a gradient constraint. These features are responsible for facet formation and the appearance of arctic curves. In the talk, I will use the lozenge tiling model (dimer model on the hexagonal lattice) to showcase these issues and address how one can analyse the variational problem.
MSc thesis presentation.
Crossing Probabilities of Multiple Ising Interfaces The planar Ising model is one of the most studied lattice models in statistical physics. Exhibiting a continuous phase transition, it enjoys conformal invariance in the scaling limit, as has been verified recently in celebrated works initiated by S. Smirnov. In this talk, I discuss crossing probabilities of multiple interfaces in the critical Ising model with alternating boundary conditions. In the scaling limit, they are conformally invariant expressions given by so-called pure partition functions of multiple SLE(kappa) with kappa=3. I also describe analogous results for critical percolation and the Gaussian free field. Joint work with Hao Wu (Yau Mathematical Sciences Center, Tsinghua University)
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