Research Centre for Mathematical Modelling (RCM²)
- Winter semester 2024/2025
Alexander Hartmann, Oldenburg
Large-deviation algorithms for random processes
Freitag, 17.01.2025, 16:00 c.t., H10
Abstract:
For dynamical random processes or other stochastic models, all measurable quantities are characterized by probability distributions. In the best case, one can calculate analytically these distributions, covering the full range of support, which includes the low-probability tails. Unfortunately, most systems cannot be solved analytically, one has to use numerical simulations. When applying standard algorithms, this means running the system K times on a computer, the smallest empirical rate one can observe is 1/K. In order to access the tails of the distributions, to cover a much large range of the support, i.e., targeting probabilities such as 10^(-100) or much smaller, one can use sophisticated large-deviation algorithms.
A standard approach is based on tilting or biasing the underlying model distribution such that rare events become frequent. For a convenient sampling of such biased distributions, here a Markov chain Monte Carlo method is explained, where the configurations are vectors of originally [0,1] uniformly or Gaussian distributed random numbers. The vectors generated by the Markov chain are used in the simulation to drive the model, while the model output is used to calculate the bias.
An example applications discussed here are properties of random graphs, e.g. the distribution of the size of the largest component for Erdös-Renyi random graphs. Also, fractional Brownian motion is considered, here the distribution of end points for a system with an absorbing boundary as well as the distribution of first-passage areas, where the left tail can be analyzed for probability densities as low as 10^(-190). Also few other models are briefly covered such as convex hulls of random walks, biological sequence alignment, or non-equilibrium work processes.Daniel Merkle, Bielefeld
Analytic Combinatorics and Boltzmann Sampling in (Bio-)Chemistry
Friday, 13.12.2024, 16:00 c.t., H10
Abstract:
This talk presents a generic framework for analytic combinatorics, enabling efficient enumeration and sampling of diverse combinatorial structures. Key features include rapid enumeration via Newton iteration, precise Boltzmann sampling with dynamic arbitrary-precision algorithms, and automated workflows for classes defined by context-free and multiple context-free grammars (MCFGs). While broadly applicable, the presentation will focus on applications in (bio-)chemistry, including the enumeration and sampling of monosubstituted hydrocarbons with stereoinformation and of RNA secondary structures with pseudoknots.
(joined work with Casper Asbjørn Eriksen and Markus Nebel)Sebastian Hummel, Zurich
Modeling the evolution and maintenance of cooperative behavior in gut microbiota
Friday, 08.11.2024, 16:00 c.t., H10
Abstract:
Understanding the potential for cooperation among bacteria in the gut environment is crucial for advancing microbiota engineering and optimization. This talk will first review well-established mechanisms that sustain cooperation and then contextualize these mechanisms within the gut environment. More specifically, biological data related to bacterial clustering in the cecum of mice, obtained within our research group, will be presented, along with our strategies for leveraging this information. To this end, we will introduce ongoing work on a simplified mathematical model of bacterial growth in the mouse cecum and discuss its implications. The talk will focus on modeling challenges and rely solely on elementary mathematics, making it suitable for a broad audience.- Summer semester 2024
Johannes Schmidt-Hieber, Twente
Statistical learning in biological neural networks
Friday, 19.07.2024, 16:00 c.t., H9
Abstract:
Compared to artificial neural networks (ANNs), the brain learns faster, generalizes better to new situations and consumes much less energy. ANNs are motivated by the functioning of the brain but differ in several crucial aspects. For instance, ANNs are deterministic while biological neural networks (BNNs) are stochastic. Moreover, it is biologically implausible that the learning of the brain is based on gradient descent. In this talk we look at biological neural networks as a statistical method for supervised learning. We relate the local updating rule of the connection parameters in BNNs to a zero-order optimization method and derive some first statistical risk bounds.
Annika Hoyer, Bielefeld
Meta-Analysis of Diagnostic Test Accuracy Studies - Statistical Insights and Practical Applications
Friday, 12.07.2024, 16:00 c.t., X-E0-200
Abstract:
While methods for meta-analysis, i.e. the weighted summary of the results of several studies, are meanwhile routinely used in the context of intervention studies, there is still a need for newly developed statistical approaches for meta-analysis of diagnostic test accuracy studies. This is due to the complex underlying data structure with an at least bivariate outcome consisting of sensitivity and specificity. Thereby, sensitivity and specificity refer to the conditional probabilities of a positive or negative test result in the population of diseased and non-diseased, respectively.
A particular challenge arises when individual studies report not only a pair of sensitivity and specificity, but complete receiver operating characteristic (ROC) curves, in which sensitivity and specificity are evaluated at several different diagnostic thresholds. One way to appropriately deal with these data is to adapt statistical methods from survival analysis, such as the application of bivariate time-to-event models for interval- censored data.
In this talk, the general framework of the meta-analysis of diagnostic test accuracy studies will be presented with special respect to statistical challenges and practical applications. Furthermore, newly developed, innovative methods for the complex situation of meta-analysis of ROC curves will be introduced and discussed.Mathias Trabs, Karlsruhe
Towards statistical guarantees for uncertainty quantification in deep learning
Friday, 21.06.2024, 16:00 c.t., X-E0-200
Abstract:
An essential feature in modern data science, especially in machine learning as well as high-dimensional statistics, are large sample sizes and large parameter space dimensions. As a consequence, the design of methods for uncertainty quantification is characterized by a tension between numerically feasible and efficient algorithms and approaches which satisfy theoretically justified statistical properties.
In this talk we discuss a Bayesian MCMC-based method with a stochastic Metropolis-Hastings as a potential solution. By calculating acceptance probabilities on batches, a stochastic Metropolis-Hastings step saves computational costs, but reduces the effective sample size. We show that this obstacle can be avoided by a simple correction term. We study statistical properties of the resulting stationary distribution of the chain if the corrected stochastic Metropolis-Hastings approach is applied to sample from a Gibbs posterior distribution in a nonparametric regression setting. Focusing on deep neural network regression, we prove a PAC-Bayes oracle inequality which yields optimal contraction rates and we analyze the diameter and show high coverage probability of the resulting credible sets.
The talk is based on joint work with Sebastian Bieringer, Gregor Kasieczka and Maximilian F. Steffen.Stefan Rotter, Freiburg
Associative remodeling and repair in self-organizing networks
Friday, 24.05.2024, 16:00 c.t., X-E0-200
Abstract:
Structural plasticity and other types of network remodeling are prominent and well-documented processes during brain development, maturation, learning, and aging. Stunningly high turnover rates of neuronal connectivity have been observed under baseline conditions, and even more so upon stimulation or perturbation. However, these findings raise questions about the underlying biological mechanisms and biological function of this drastic form of brain plasticity. Since direct experiments are notoriously difficult to perform and analyze, formal models are crucial for predicting and understanding the emergent properties of graphs or networks with highly dynamic structure. To this end, we consider ensembles of directed multi-graphs with constrained indegrees and outdegrees, the Directed Configuration Model, as this reflects well the limited material resources of brain networks. Similar to the law of mass action in chemistry, we outline a kinetic theory of random graph formation and structural self-organization. We study the equilibria of these networks and describe the transient effects of perturbation. We apply our theoretical findings to biological neural networks of the brain, where the vertices are neurons and the directed edges are chemical synapses between neurons. In this setting, stimulation leads to perturbation of the network structure, followed by dynamic reconfiguration and convergence to a new structural equilibrium that is different from the previous one, reminiscent of hysteresis in solid state physics. I will describe some emergent properties of this new model, including Hebb's rule (“neurons that fire together wire together”) and self-repair of networks under appropriate conditions. For certain perturbation strategies, this provides us with models for brain maturation during adolescence and for engram formation in learning and memory. I will also discuss possible links to psychological phenomena like classical conditioning and describe new machine learning strategies based on the self-organization of networks that can be derived from our biologically motivated theory.Jürgen Schnack, Bielefeld
The Unreasonable Effectiveness of the Finite Temperature Lanczos Method
Friday, 03.05.2024, 16:00 c.t., D5-153
Abstract:
I will explain an astonishingly accurate, yet pretty simple approximation to evaluate thermodynamic equilibrium functions in quantum mechanics. The method rests on the Lanczos procedure, thus can be understood with matrix-vector procedures. Although originally designed to approximate extreme eigenvalues it could be shown that the method can easily be extended to address problems at finite temperature. A mathematical as well as a physical interpretation will be offered.- Winter semester 2023/2024
Roland Langrock, Bielefeld
Periodic variation in hidden Markov models
Friday, 26.01.2024, 16:00 c.t., V2-213
Abstract:
The class of hidden Markov models (HMMs) is a popular tool for modelling time series driven by underlying states. HMMs can be used for example to relate animal movement processes to underlying behavioural modes, to infer the disease state of a patient from biomarkers, or to predict extreme share returns as a function of the underlying nervousness of the financial market. After a general introduction to this versatile class of time series models, I will focus on one particular extension of HMMs, namely model formulations that attempt to capture periodic variation in the state-switching dynamics (e.g. daily variation in animal behaviour, or seasonality in economic markets). Such periodic variation is commonly modelled using trigonometric functions, but can alternatively and more flexibly be incorporated using cyclic splines. I will showcase these modelling approaches in case studies, in which I will also illustrate a positive consequence of such periodic modelling, namely that for periodic HMMs the distributions of the dwell times in the different system states can deviate substantially from a geometric distribution as it would be implied by a homogeneous HMM.
Vitali Wachtel, Bielefeld
Probabilistic approach to risk processes with level-dependent premium rate
Friday, 12.01.2024, 16:00 c.t., V2-213
Abstract:
In this talk I shall discuss classical risk processes and risk processes with level dependent premium rate. I shall present a purely probabilistic approach to processes with level-dependent rate, which allows one to obtain upper and lower bounds for ruin probabilities.
Carolin Herrmann, Berlin
Statistically proven? - the role of a valid sample size calculation in clinical trials
Friday, 08.12.2023, 16:00 c.t., V2-213
Abstract:
Randomized clinical trials are the preferred study design for answering medical research questions in an evidence-based way. Apart from a research idea, different aspects have to be decided on when planning a clinical trial, e.g. the treatments to compare with, the endpoints and the sample size.
Clinical trials can only be deemed successful if they come along with a valid sample size planning. Sample sizes should be large enough to detect an existing relevant effect with a sufficient probability. At the same time, they are supposed to be as small as possible to keep possible study related risks low and to not postpone a potential market approval unnecessarily.
After a short introduction to clinical trials, we consider the basic concept of sample size planning as well as more advanced statistical methods for updating sample sizes during ongoing trials.Alexander Schönhuth, Bielefeld
Diffusion Models: A Tutorial
Tuesday, 28.11.2023, 16:00 c.t., X-E0-200
Abstract:
Diffusion models have been one of the fundamental building blocks of generative deep learning techniques such as "text-to-image" models. In essence, diffusion models reflect a novel, neural network guided sampling strategy. They apply for heavily complex probability distributions. They are based on the general idea that only artificial intelligence techniques can lead one the way from random number generation to, for example, random images showing beaches with palm trees. I will discuss the mathematical and computational foundations of these models. I will also point out why such models may have the potential in attempts to "generate random life".
Arndt von Haeseler, Vienna
Single-cell RNA sequencing and phylogenetic inference
Friday, 10.11.2023, 16:00 c.t., D2-136
Abstract:
In the talk we will merge the emerging field of scRNA sequencing data with the established field of phylogenomics. We will show that a simple transformation of the complex and high-dimensional scRNA data and the subsequent phylogenetic analysis leads to surprisingly clear visualisations and can thus provide the basis for new biological insights. The talk will be rounded off with some illustrative examples.
This is joint work with Christiane Elgert and Julia Naas.- Summer semester 2023
Mike Steel, Christchurch (New Zealand)
Phylogenetic theory in conservation biology
Monday, 17.07.2023, 16:00 c.t., D2-136
Abstract:
The current rapid extinction of species leads not only to their loss, but also to the disappearance of the pendant and interior branches in the underlying evolutionary tree, along with the unique features (e.g. traits) that have evolved along these branches. In this talk, I describe some recent and new results for quantifying different measures of phylogenetic diversity and for predicting its loss under simple extinction models.Kai-Uwe Bux, Bielefeld
The Greek way of saying "let epsilon be greater than zero"
Friday, 14.07.2023, 16:00 c.t., D2-136
Abstract:
Proving equality of areas, lengths or angles was an important aspect of ancient geometrical thought (that's what the word "metric" indicates). For polygonal areas, the standard method is "cut and rearrange". This method does not apply in all cases for volumes or areas of curved shapes like circles. Archimedes in particular was very fond of integrating curved areas and volumes. We now know that limits are unavoidable in this case. So, our colleagues from antiquity (and among them Archimedes) did run into this problem; and they solved it very much in the same way that Cauchy and Weierstrass did. There is but one difference: the Greeks did neither have the concept of a real number nor did they think about functions. I shall outline the way in which ancient mathematicians dealt with limit arguments in a purely geometric fashion where one only deals with quantities and ratios without identifying those with numbers.
Martin Wahl, Bielefeld University
Principal component analysis in infinite dimensions
Friday, 21.04.2023, 16:00 c.t., D2-136
Abstract:
In high-dimensional settings, principal component analysis (PCA) reveals some unexpected phenomena, ranging from eigenvector inconsistency to eigenvalue (upward) bias. While such high-dimensional phenomena are now well understood in the spiked covariance model, the goal of this talk is to present some extensions for the case of PCA in infinite dimensions. As applications, I will discuss the prediction error of principal component regression (PCR) in the overparametrized regime, as well as nonlinear dimensionality reduction methods such as Laplacian eigenmaps and diffusion maps.