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

Abstract

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.