Quantization Errors of fGn and fBm Signals

In this Letter, we show that under the assumption of high resolution, the quantization errors of fGn and fBm signals with uniform quantizer can be treated as uncorrelated white noises

A Bayesian approach to simultaneously characterize the stochastic and deterministic components of a system

The present work provides a Bayesian approach to learn plausible models capable of characterizing complex time series in which deterministic and stochastic phenomena concur. Two main approaches are actually developed. The first approach, is a simple superposition model grounded on the hypothesis that the interactions between the stochastic and deterministic phenomena are negligible. To enable this model to capture complex dynamics, the stochastic part is assumed to be a fractal signal. Under the assumptions of this model, an analysis method is proposed, enabling the characterization of the fractal stochastic component and the estimation the deterministic part. The second main approach relies on Stochastic Differential Equations (SDEs) to model systems where the stochastic and deterministic part interact. First, a non-parametric estimation method for SDEs is developed, using recent advances from Gaussian processes. Finally, the thesis studies how to overcome the main constraint that the use of SDEs imposes: the Markovianity assumption. To that end, a new structured variational autoencoder with latent SDE dynamics is proposed. All the methods are tested on both synthetic and real signals, demonstrating its ability to capture the behavior of complex systems

Autour de la quantification fonctionnelle de processus gaussiens

Cette thèse a pour objectif principal l'étude de résultats asymptotiques autour de la quantification fonctionnelle. Après les résultats obtenus pour Sagna sur le rayon maximal du quantifier optimal en dimension finie, nous cherchons l'asymptotique du rayon maximal en dimension infinie, spécifiquement pour le mouvement brownien. Nous présentons aussi un nouvel algorithme stochastique en dimension finie pour trouver des quantifiers stationnaires. Nous proposons une nouvelle méthode d'estimation pour le paramètre de Hurst dans des processus gaussiens fractionnaires plus robuste pour le calcul numérique que le maximum de vraisemblance en utilisant la décomposition de Karhunen-Loève des processus gaussiens.The purpose of the present thesis is to study the theory of functional quantization for some Gaussian process. Our goal is to investigate some general asymptotic properties of the quantization error and concepts related as the maximal radius of the optimal quantizer. We also develop a new method based on the Karhunen-Loève expansion of fractional Gaussian process to estimate the Hurst parameter associated to this processes. We derive a new stochastic algorithm mainly based on the Competitive Learning Vector Quantization (CLVQ). We examine the convergence of this method and present some numerical results of it behaviour

Epileptic Seizures and the EEG

A study of epilepsy from an engineering perspective, this volume begins by summarizing the physiology and the fundamental ideas behind the measurement, analysis and modeling of the epileptic brain. It introduces the EEG and provides an explanation of the type of brain activity likely to register in EEG measurements, offering an overview of how these EEG records are and have been analyzed in the past. The book focuses on the problem of seizure detection and surveys the physiologically based dynamic models of brain activity. Finally, it addresses the fundamental question: can seizures be predicted? Based on the authors' extensive research, the book concludes by exploring a range of future possibilities in seizure prediction

Field theories for stochastic processes

This thesis is a collection of collaborative research work which uses field-theoretic techniques to approach three different areas of stochastic dynamics: Branching Processes, First-passage times of processes with are subject to both white and coloured noise, and numerical and analytical aspects of first-passage times in fractional Brownian Motion. Chapter 1 (joint work with Rosalba Garcia Millan, Johannes Pausch, and Gunnar Pruessner, appeared in Phys. Rev. E 98 (6):062107) contains an analysis of non-spatial branching processes with arbitrary offspring distribution. Here our focus lies on the statistics of the number of particles in the system at any given time. We calculate a host of observables using Doi-Peliti field theory and find that close to criticality these observables no longer depend on the details of the offspring distribution, and are thus universal. In Chapter 2 (joint work with Ignacio Bordeu, Saoirse Amarteifio, Rosalba Garcia Millan, Nanxin Wei, and Gunnar Pruessner, appeared in Sci. Rep. 9:15590) we study the number of sites visited by a branching random walk on general graphs. To do so, we introduce a fieldtheoretic tracing mechanism which keeps track of all already visited sites. We find the scaling laws of the moments of the distribution near the critical point. Chapter 3 (joint work with Gunnar Pruessner and Guillaume Salbreux, submitted, arXiv: 2006.00116) provides an analysis of the first-passage time problem for stochastic processes subject to white and coloured noise. By way of a perturbation theory, I give a systematic and controlled expansion of the moment generating function of first-passage times. In Chapter 4, we revise the tracing mechanism found earlier and use it to characterise three different extreme values, first-passage times, running maxima, and mean volume explored. By formulating these in field-theoretic language, we are able to derive new results for a class of non-Markovian stochastic processes. Chapter 5 and 6 are concerned with the first-passage time distribution of fractional Brownian Motion. Chapter 5 (joint work with Kay Wiese, appeared in Phys. Rev. E 101 (4):043312) introduces a new algorithm to sample them efficiently. Chapter 6 (joint work with Maxence Arutkin and Kay Wiese, submitted, arXiv:1908.10801) gives a field-theoretically obtained perturbative result of the first-passage time distribution in the presence of linear and non-linear drift.Open Acces

Epileptic Seizures and the EEG

A study of epilepsy from an engineering perspective, this volume begins by summarizing the physiology and the fundamental ideas behind the measurement, analysis and modeling of the epileptic brain. It introduces the EEG and provides an explanation of the type of brain activity likely to register in EEG measurements, offering an overview of how these EEG records are and have been analyzed in the past. The book focuses on the problem of seizure detection and surveys the physiologically based dynamic models of brain activity. Finally, it addresses the fundamental question: can seizures be predicted? Based on the authors' extensive research, the book concludes by exploring a range of future possibilities in seizure prediction

Symmetry and Complexity

Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry

Mathematical Modelling of Spatially Coherent Transcription

Genetics and epigenetics are widely expected to revolutionise our understanding of health and disease. However any attempt to extract relevant information from noisy data requires a combination of modelling and statistical techniques. Given the number of genes and the complexity involved in the genome, sophisticated methods will be needed to properly capture the information that is contained. Many mechanisms and variables can affect and control the expression of a gene. In this thesis, it is specifically spatially coherent variations in transcription which are investigated. Several different areas were examined, producing a broad set of results. Important findings include the demonstration of spatial coherence as the result of epigenetic effects, the creation and validation of a technique to detect spatial coherence, and the extension of spatial modelling to epigenetic data. Other important results include the detection of spatial coherence variation due to confounding variables (PMI and neuronal concentration) and the development of new spatial modelling techniques. The results indicate that spatial modelling provides a useful approach to investigating unusual and unknown aspects of epigenetic and transcriptional regulation