We are pleased to announce the public defense of the doctoral thesis by José Alberto Martínez Ordóñez, entitled «Discrete-Time Nonlinear Markov Switching Models: Stability, Inference and Physical Applications». The dissertation has been conducted under the supervision of Professor Joaquín Míguez Arenas.

The defense took place Tuesday, July 15th at 11:00 AM, in Sala 08, Sótano 1, Biblioteca Rey Pastor.

The thesis addresses an important problem in the modeling of time series data, particularly in systems where the dynamics evolve over time and switch between different regimes. While autoregressive Markov switching (ARMS) models have been widely used for this purpose, José Alberto’s research introduces two novel extensions: the Delayed Nonlinear ARMS (DN-ARMS) and the Continuous Delayed Nonlinear ARMS (cDN-ARMS) models. These frameworks incorporate both integer and real (non-integer) delays into the system, providing a more realistic and flexible tool for modeling complex, delay-driven processes such as those found in climatology and engineering.

A significant contribution of the thesis is the proposal of a new stability criterion, termed q-stability, along with rigorous conditions ensuring the stability of these delayed models. In terms of inference, the work presents an adapted space-alternating expectation-maximization (SA-EM) algorithm, capable of efficiently estimating model parameters and transition probabilities with linear computational complexity. To estimate the number of dynamical regimes, the thesis employs penalized maximum likelihood techniques.

The practical relevance of these methodologies is illustrated through applications to the El Niño–Southern Oscillation (ENSO) phenomenon and to solar activity data. In addition to synthetic experiments validating the proposed methods, the thesis compares the predictive performance of ARMS models against deep learning techniques on real-world data.

For cases where likelihoods cannot be evaluated in closed form, José Alberto proposes a Bayesian inference approach based on adaptive importance sampling, using the nonlinear population Monte Carlo (NPMC) algorithm. This strategy not only estimates model parameters but also quantifies uncertainty, enhancing the interpretability and robustness of the inference process.

This defense marks the culmination of an ambitious and technically sophisticated project, with valuable contributions to time series analysis and stochastic modeling.