 # Abstracts Research Seminar Winter Term 2022/23

#### Andreas Löhne: On Convex Polyhedron Computations Using Floating Point Arithmetic

Convex polyhedra are not necessarily finite sets but they can be finitely represented. Thus they play an important role for various types of set computations, for instance in set optimization. Most of the computational techniques for polyhedra rely in some sense on vertex enumeration, which means to compute the vertices and extremal directions of a polyhedron which is given by (finitely many) linear inequalities. The inverse problem, which is equivalent by polarity, is called convex hull problem. In practice it is quite common to implement vertex enumeration and convex hull methods by using floating point arithmetic. However, in most situations there is no proof of correctness of the methods when inexact arithmetic is used. In particular, there is no correct practicable floating point algorithm known for polytopes of dimension larger than 3. We demonstrate by examples that inexact computations can produce results which are far away from the correct ones. We present an approximate vertex enumeration method, which is shown to be correct for polytopes of dimension 2 and 3. We discuss why a generalization to any higher dimension is, if possible, not trivial.

#### Johannes Moritz Jirak: Relative Perturbation Bounds for Empirical Covariance Operators

We discuss expansions for empirical eigenvalues and spectral projectors, leading to concentration inequalities and limit theorems. One of the key ingredients is a specific separation measure for population eigenvalues, which we call the relative rank, giving rise to a sharp invariance principle in terms of limit theorems, concentration inequalities and inconsistency results. Our framework is very general, requiring (in principle) only p>4 moments and allows for a huge variety of dependence structures.

#### Erik Schlögl: Term Structure Modelling From the SOFR Perspective

The Secured Overnight Funding Rate (SOFR) has become the main benchmark for US dollar interest rates; thus, models need to be updated to reflect the key features exhibited by the dynamics of SOFR and the forward rates implied by SOFR futures. As an index based on transactions in the Treasury overnight repurchase market, the dynamics of SOFR are closely linked to the dynamics of the Effective Federal Funds Rate (EFFR), which is the interest rate most directly impacted by US monetary policy target rate decisions. Therefore, these rates feature jumps at known times (Federal Open Market Committee meeting dates), and market expectations of these jumps drive the prices for futures contracts written on these rates. On the other hand, forward rates implied by Fed Funds and SOFR futures continue to evolve diffusively. The models presented in this talk reflect these features of SOFR dynamics. In particular, they reconcile diffusive forward rate dynamics with piecewise constant paths of the target short rate. The first, Gaussian version of the model allows us to extract the factor dynamics from market data, informing the stochastic modelling choices in the second version of the model, which is calibrated to market prices for options on SOFR futures.

Joint work with K. Gellert.

#### Sara Svaluto-Ferro: Signature-Based Models: Theory and Calibration

We consider asset price models whose dynamics are described by linear functions of the (time extended) signature of a primary underlying process, which can range from a (market-inferred) Brownian motion to a general multidimensional continuous semimartingale. The framework is universal in the sense that classical models can be approximated arbitrarily well and that the model’s parameters can be learned from all sources of available data by simple methods. We provide conditions guaranteeing absence of arbitrage as well as tractable option pricing formulas for so-called sig-payoffs, exploiting the polynomial nature of generic primary processes. One of our main focus lies on calibration, where we consider both time-series and implied volatility surface data, generated from classical stochastic volatility models and also from S&P 500 index market data. For both tasks the linearity of the model turns out to be the crucial tractability feature which allows to get fast and accurate calibrations results.

Joint work with Christa Cuchiero and Guido Gazzani.

#### Charles Bouveyron: Statistical Learning With Interaction Data: Applications

In this talk, we will focus on the problem of statistical learning with interaction data. This work is motivated by two real-world applications: the modeling and clustering of social networks, on the one hand, and of Pharmacovigilance data, on the other hand. To this end, we developed two model-based approaches. First, we propose the deep latent position model (DeepLPM), an end-to-end generative clustering approach which combines the widely used latent position model (LPM) for network analysis with a graph convolutional network (GCN) encoding strategy. An original estimation algorithm is introduced to integrate the explicit optimization of the posterior clustering probabilities via variational inference and the implicit optimization using stochastic gradient descent for graph reconstruction. Second, for the Pharmacovigilance problem, we introduce a latent block model for the dynamic co-clustering of count data streams with high sparsity. We assume that the observations follow a time and block dependent mixture of zero-inflated Poisson distributions, which combines two independent processes: a dynamic mixture of Poisson distributions and a time-dependent sparsity process. To model and detect abrupt changes in the dynamics of both clusters memberships and data sparsity, the mixing and sparsity proportions are modeled through systems of ordinary differential equations. The model inference relies on an original variational procedure whose maximization step trains recurrent neural networks in order to solve the dynamical systems. Numerical experiments on simulated data sets demonstrate the effectiveness of the proposed methodologies for the two problems.

#### Stefan Thonhauser: PDMP Based Risk Models

Ruin theory aims at describing and analysing insurance risks by means of stochastic models for the evolution of a surplus process. Classical models - such as the Cramér-Lundberg or renewal risk model - suffer from a static parameter choice and non-controllability. These simplifying features can be overcome by the use of more general models from the class of piecewise-deterministic Markov processes. For such processes many objects of interest such as ruin probabilities, penalty functions or expected dividend payments can hardly be explicitely computed and one needs to rely on numerical methods. We will show that the Markovian structure allows for the application of quasi-Monte Carlo integration, but that the derivation of error bounds requires some smoothing. As a complement we discuss PDMP-techniques for the identification of the asymptotic behaviour of ruin probabilities in particular risk models with stochastic intensities.

#### Aliaksandr Hubin: Boosting Performance of Latent Binary Neural Networks With a Local Reparametrization Trick and Normalizing Flows

An artificial neural network (ANN) is a powerful machine learning method that is used in many modern applications such as facial recognition, machine translation and cancer diagnostics. A common issue with ANNs is that they usually have millions or billions of trainable parameters, and therefore tend to overfit to the training data. This is especially problematic in applications where it is important to have reliable uncertainty estimates. Bayesian neural networks (BNN) can improve on this, since they incorporate parameter uncertainty. In addition, latent binary Bayesian neural networks (LBBNN) also take into account model uncertainty, enabling inference in both model and parameter space. In this paper, we will consider two extensions to the LBBNN method: Firstly, by using the local reparametrization trick (LRT) to sample the hidden units directly, we get a more computationally efficient algorithm. Secondly, by using normalizing flows on the variational posterior distribution of the LBBNN parameters, the network learns a more flexible variational posterior distribution than the mean field Gaussian. Experimental results show that this improves significantly on predictive power compared to the LBBNN method, while also obtaining a more sparse network. Additionally, we perform a simulation study where the normalizing flow method performs best at variable selection.

Joint work with Lars Skaaret-Lund and Geir Storvik.

#### Marc-Oliver Pohle: Generalised Covariances and Correlations

We introduce and examine new dependence measures. Generalised covariance and correlation allow to measure dependence between two random variables X and Y around arbitrary statistical functionals just as covariance and Pearson correlation measures dependence around their means. The key idea behind this is to replace the error or deviation from the mean showing up in Pearson correlation by a suitable measure for the deviation from a general statistical functional, where identification functions provide us with such a generalized error. Generalised correlation has favourable theoretical properties and a multitude of practically relevant measures arise from this class, for example quantile correlation. Quantile correlation is akin to the extension of least squares to quantile regression. Quantile and the related threshold correlation make it possible to measure dependence locally, for example to analyse tail dependence. When choosing distribution functions as functionals we arrive at distributional correlations, which are two-dimensional functions lying between -1 and 1. They uncover the full dependence structure between X and Y and are closely related as well as natural complements in statistical analysis to the joint CDF and the copula. To condense the full dependence structure into a single number we finally introduce summary correlations as appropriately normalized integrals over distributional covariances with respect to arbitrary measures. Interesting new measures arise, but also Spearman's rho and an improved version of Pearson correlation as canonical special cases.

Joint work with Tobias Fissler.

#### Gabriele Eichfelder: Multiobjective Replacements for Set Optimization and Robust Multiobjective Optimization

Set-valued optimization using the set approach is a research topic of high interest due to its practical relevance and numerous interdependencies to other fields of optimization. An important example are robust approaches to uncertain multiobjective optimization problems which result in such set optimization problems. However, it is a very difficult task to solve these optimization problems even for specific cases.
In this talk, we present parametric multiobjective optimization problems for which the optimal solutions are strongly related to the optimal solutions of the set optimization problem. This corresponds to the well-known idea of scalarization in multiobjective optimization. We give results on approximation guarantees and we examine particular classes of set-valued mappings for which the corresponding set optimization problem is in fact equivalent to a multiobjective optimization problem, like set-valued mappings with a convex graph.

#### Delia Coculescu: A Non-Markov Approach for the Evolution of Contagious Phenomena

In some classical dynamical contagion models, the contagion process is a multivariate process (with coordinates valued in a binary set e.g. {default, survival} or {infected, not infected} etc.) that is assumed Markov, conditionally on the observation of its stochastic environment, with interacting intensities. This assumption requires that the environment evolves autonomously and is not influenced by the transitions of the contagion process.
From the applications perspective, the Markov property implies that there are “key factors” impacting the intensities, that are considered exogenous. Typical factors are macro-economic variables (interest rates, inflation) in the case of default contagion, or observed behavioral patterns in case of disease spread. The Markov assumption is not very realistic for systemic risks, as defaults may influence the macro-economic variables, which in turn impact the default probabilities; similarly, in the case of a pandemic, rising numbers of infected individuals may modify behavior of non infected individuals and hence affect the future spread a disease. These types of feedback loops cannot be integrated in a conditionally Markov framework. The aim of this talk is to present an extension of the classical theory that allows to model this type of phenomena.

#### Kenneth Benoit: A Better Wordscores: Scaling Text With the Class Affinity Model

Probabilistic methods for classifying text form a rich tradition in machine learning and natural language processing. For many important problems, however, class prediction is uninteresting because the class is known, and instead the focus shifts to estimating latent quantities related to the text, such as affect or ideology. We focus on one such problem of interest, estimating the ideological positions of 55 Irish legislators in the 1991 Dáil confidence vote, a challenge brought by opposition party leaders against the then-governing Fianna Fáil party in response to corruption scandals. In this application, we clearly observe support or opposition from the known positions of party leaders, but have only information from speeches from which to estimate the relative degree of support from other legislators. To solve this scaling problem and others like it, we develop a text modeling framework that allows actors to take latent positions on a “gray” spectrum between “black” and “white” polar opposites. We are able to validate results from this model by measuring the influences exhibited by individual words, and we are able to quantify the uncertainty in the scaling estimates by using a sentence-level block bootstrap. Applying our method to the Dáil debate, we are able to scale the legislators between extreme pro-government and pro-opposition in a way that reveals nuances in their speeches not captured by their votes or party affiliations.

Joint work with Patrick O. Perry.