Memory-Based Reduced Modelling and Data-Based Estimation of Opinion Spreading

We investigate opinion dynamics based on an agent-based model and are interested in predicting the evolution of the percentages of the entire agent population that share an opinion. Since these opinion percentages can be seen as an aggregated observation of the full system state, the individual opinions of each agent, we view this in the framework of the Mori-Zwanzig projection formalism. More specifically, we show how to estimate a nonlinear autoregressive model (NAR) with memory from data given by a time series of opinion percentages, and discuss its prediction capacities for various specific topologies of the agent interaction network. We demonstrate that the inclusion of memory terms significantly improves the prediction quality on examples with different network topologies

Sparse Proteomics Analysis - A compressed sensing-based approach for feature selection and classification of high-dimensional proteomics mass spectrometry data

Background: High-throughput proteomics techniques, such as mass spectrometry (MS)-based approaches, produce very high-dimensional data-sets. In a clinical setting one is often interested in how mass spectra differ between patients of different classes, for example spectra from healthy patients vs. spectra from patients having a particular disease. Machine learning algorithms are needed to (a) identify these discriminating features and (b) classify unknown spectra based on this feature set. Since the acquired data is usually noisy, the algorithms should be robust against noise and outliers, while the identified feature set should be as small as possible. Results: We present a new algorithm, Sparse Proteomics Analysis (SPA), based on the theory of compressed sensing that allows us to identify a minimal discriminating set of features from mass spectrometry data-sets. We show (1) how our method performs on artificial and real-world data-sets, (2) that its performance is competitive with standard (and widely used) algorithms for analyzing proteomics data, and (3) that it is robust against random and systematic noise. We further demonstrate the applicability of our algorithm to two previously published clinical data-sets

Communicated by Michael Meyer

In diesem Artikel wird ein mathematisches Modell entwickelt für die Ausbreitung des Wollschafs unter Hirten im Nahen Osten und in Südosteuropa zwischen 6200 und 4200 v. Chr. In unserem Modell werden Hirten als Agenten betrachtet, deren Bewegungen durch Zufallsprozesse gesteuert werden, sodass sich die Agenten mit größerer Wahrscheinlichkeit in Regionen aufhalten, die attraktiv für die Schafhaltung sind. Das Modell berücksichtigt außerdem soziale Interaktionen zwischen Agenten und erlaubt die Weitergabe der Innovation zwischen Agenten mit einer bestimmten Wahrscheinlichkeit. Die Parameter des agentenbasierten Modells werden an die verfügbaren archäologischen Daten angepasst. Ein Simulationsverfahren für die räumliche und zeitliche Entwicklung des Ausbreitungsprozesses soll es ermöglichen, qualitative Effekte von verschiedenen Aspekten zu studieren, die den Ausbreitungsprozess beeinflussen

Measuring Dependencies between Variables of a Dynamical System Using Fuzzy Affiliations

A statistical, data-driven method is presented that quantifies influences between variables of a dynamical system. The method is based on finding a suitable representation of points by fuzzy affiliations with respect to landmark points using the Scalable Probabilistic Approximation algorithm. This is followed by the construction of a linear mapping between these affiliations for different variables and forward in time. This linear mapping, or matrix, can be directly interpreted in light of unidirectional dependencies, and relevant properties of it are quantified. These quantifications, given by the sum of singular values and the average row variance of the matrix, then serve as measures for the influences between variables of the dynamics. The validity of the method is demonstrated with theoretical results and on several numerical examples, covering deterministic, stochastic, and delayed types of dynamics. Moreover, the method is applied to a non-classical example given by real-world basketball player movement, which exhibits highly random movement and comes without a physical intuition, contrary to many examples from, e.g., life sciences

Measuring Dependencies between Variables of a Dynamical System Using Fuzzy Affiliations

A statistical, data-driven method is presented that quantifies influences between variables of a dynamical system. The method is based on finding a suitable representation of points by fuzzy affiliations with respect to landmark points using the Scalable Probabilistic Approximation algorithm. This is followed by the construction of a linear mapping between these affiliations for different variables and forward in time. This linear mapping, or matrix, can be directly interpreted in light of unidirectional dependencies, and relevant properties of it are quantified. These quantifications, given by the sum of singular values and the average row variance of the matrix, then serve as measures for the influences between variables of the dynamics. The validity of the method is demonstrated with theoretical results and on several numerical examples, covering deterministic, stochastic, and delayed types of dynamics. Moreover, the method is applied to a non-classical example given by real-world basketball player movement, which exhibits highly random movement and comes without a physical intuition, contrary to many examples from, e.g., life sciences

Modellierung von Beobachtungen von dynamischen Systemen mit Gedächtnis

Detecting the governing mathematical rules of a dynamical system from data persists to be a challenge. It becomes particularly difficult when the variables of the system can only be partially observed in the form of a so-called observable function. In this case information about variables that can be vital for the prediction of future states is missing. In order to still formulate the dynamics of the observable, it can be shown that by exploiting its memory terms one can make up for the lost information. This can be placed on a mathematical ground by the delay embedding theorem of Takens and the Mori–Zwanzig formalism (MZ). In this thesis, novel numerical methods for the modelling of the observed dynamics were developed by using Takens and MZ to extend known methods used for memoryless systems. Firstly, the method Sparse Identification of Nonlinear Dynamics (SINDy) was combined with the family of autoregressive (AR) models to define Sparse Identification of Autoregressive Models (SINAR) which seeks a sparse representation of memory-exhibiting dynamics. It was compared to various others theoretically and on examples coming from different fields. Another new numerical method was introduced in which a high-dimensional dynamical system is projected onto a low-dimensional convex polytope using the Scalable Probabilistic Approximation (SPA) algorithm. The projection to the polytope was interpreted as an observable and memory was used to estimate the projected dynamics in a newly introduced method called memory SPA (mSPA). It was shown that mSPA can generate strong prediction accuracy for various dynamical systems while guaranteeing stability by keeping the dynamics inside the polytope. As another contribution of this thesis, the identification of memory-exhibiting dynamics was connected with the field of agent-based modelling. To this end, two new ABMs were defined, the high-dimensional representation of the states of all individual agents was interpreted as the full system state and a low-dimensional statistic as the observable. Then the ABM was translated into the setting introduced before. It was shown in a detailed numerical analysis using SINAR that including memory generally improves the accuracy of the model identification, moreover, that adding a sparsity constraint can improve the model and that the model fitting can strongly depend on the data. The thesis was concluded with a small note on how along the different methods discussed, the seemingly unrelated theoretical perspectives of Takens and Mori–Zwanzig could be connected.Die dominanten mathematischen Regeln eines dynamischen Systems aus Daten zu ermitteln ist weiterhin eine Herausforderung, welche besonders schwierig wird, wenn die Variablen eines Systems nur in Form einer so genannten Observablenfunktion partiell beobachtet werden können. Um trotzdem die Dynamik der Observablen formulieren zu können, ist es möglich, die Gedächtnisterme der Observablen zu benutzen, um das Fehlen der Information über die nicht-beobachtbaren Variablen auszugleichen. Dies kann auf ein mathematisches Fundament gestellt werden durch das (eng.) Delay-Embedding-Theorem von Takens und den Mori–Zwanzig Formalismus (MZ). In dieser Arbeit wurden neue numerische Methoden für die Modellierung der beobachteten Dynamik entwickelt, indem mithilfe von Takens und MZbereits bekannte Methoden für gedächtnislose Dynamiken erweitert wurden. Zunächst wurde die Methode Sparse Identification of Nonlinear Dynamics (SINDy) mit autoregressiven (AR) Modellen kombiniert, um Sparse Identification of Autoregressive Models (SINAR) zu definieren – eine Methode, die eine (eng.) sparse Darstellung einer gedächtniszeigenden Dynamik zu finden versucht. Diese Methoden wurden untereinander und mit anderen auf theoretischer Basis und anhand verschiedener Beispiele verglichen. Eine weitere numerische Methode wurde eingeführt, mit der durch den Scalable Probabilistic Approximation (SPA) Algorithmus eine hochdimensionale Dynamik auf ein niedrigdimensionales Polytop projiziert wird. Die Projektion auf das Polytop wurde als Observable interpretiert und Gedächtnis benutzt, um die projizierte Dynamik mit der neuen Methode memory SPA (mSPA) zu schätzen. Es wurde gezeigt, dass mSPA gute Genauigkeit für verschiedene dynamische Systeme erreichen kann und gleichzeitig Stabilität garantiert, indem die Dynamik innerhalb des Polytops bleibt. Als ein weiterer Beitrag dieser Arbeit wurde die Identifikation von gedächtniszeigender Dynamik mit dem Feld von agentenbasierter Modellierung (ABM) verbunden, wofür zwei neue ABMs definiert wurden. Die hochdimensionale Darstellung der Zustände ihrer einzelnen Agenten wurden als den vollen Systemzustand und eine niedrigdimensionale Statistik als die Observable interpretiert und die ABMs wurden in das zuvor eingeführte Konzept übersetzt. Es wurde mit SINAR u.a. detailliert gezeigt, dass das Hinzunehmen von Gedächtnis im Allgemeinen die Genauigkeit der Modellidentifikation verbessert. Diese Arbeit wurde mit einer Beobachtung dar über abgeschlossen, wie durch die verschiedenen untersuchten Methoden die scheinbar nicht verwandten theoretischen Perspektiven von Takens und Mori–Zwanzig verbunden werden könnten

Deterministic and Stochastic Parameter Estimation for Polymer Reaction Kinetics I: Theory and Simple Examples

Two different approaches to parameter estimation (PE) in the context of polymerization are introduced, refined, combined, and applied. The first is classical PE where one is interested in finding parameters which minimize the distance between the output of a chemical model and experimental data. The second is Bayesian PE allowing for quantifying parameter uncertainty caused by experimental measurement error and model imperfection. Based on detailed descriptions of motivation, theoretical background, and methodological aspects for both approaches, their relation are outlined. The main aim of this article is to show how the two approaches complement each other and can be used together to generate strong information gain regarding the model and its parameters. Both approaches and their interplay in application to polymerization reaction systems are illustrated. This is the first part in a two-article series on parameter estimation for polymer reaction kinetics with a focus on theory and methodology while in the second part a more complex example will be considered

Memory-Based Reduced Modelling and Data-Based Estimation of Opinion Spreading

We investigate opinion dynamics based on an agent-based model and are interested in predicting the evolution of the percentages of the entire agent population that share an opinion. Since these opinion percentages can be seen as an aggregated observation of the full system state, the individual opinions of each agent, we view this in the framework of the Mori–Zwanzig projection formalism. More specifically, we show how to estimate a nonlinear autoregressive model (NAR) with memory from data given by a time series of opinion percentages, and discuss its prediction capacities for various specific topologies of the agent interaction network. We demonstrate that the inclusion of memory terms significantly improves the prediction quality on examples with different network topologies

Data-Powered Positive Deviance during the SARS-CoV-2 Pandemic—An Ecological Pilot Study of German Districts

We introduced the mixed-methods Data-Powered Positive Deviance (DPPD) framework as a potential addition to the set of tools used to search for effective response strategies against the SARS-CoV-2 pandemic. For this purpose, we conducted a DPPD study in the context of the early stages of the German SARS-CoV-2 pandemic. We used a framework of scalable quantitative methods to identify positively deviant German districts that is novel in the scientific literature on DPPD, and subsequently employed qualitative methods to identify factors that might have contributed to their comparatively successful reduction of the forward transmission rate. Our qualitative analysis suggests that quick, proactive, decisive, and flexible/pragmatic actions, the willingness to take risks and deviate from standard procedures, good information flows both in terms of data collection and public communication, alongside the utilization of social network effects were deemed highly important by the interviewed districts. Our study design with its small qualitative sample constitutes an exploratory and illustrative effort and hence does not allow for a clear causal link to be established. Thus, the results cannot necessarily be extrapolated to other districts as is. However, the findings indicate areas for further research to assess these strategies’ effectiveness in a broader study setting. We conclude by stressing DPPD’s strengths regarding replicability, scalability, adaptability, as well as its focus on local solutions, which make it a promising framework to be applied in various contexts, e.g., in the context of the Global South