The problem of change point detection in time series has been widely studied in the last decades due to its importance in different areas such as economics and finance, environment and health sciences. Time series often suffer from changes in structure (e.g. changes in the underlying model parameters) that may arise from e.g. policy changes or critical events, which when not detected may result in misleading analyses and conclusions.

The aim of this study is to perform a comparative study between parametric and non-parametric statistical methods to detect change points in time series based on model residuals. In addition, a simulation study was designed to evaluate the performance of each method in some characteristics, namely, the capacity to identify change points and to establish a false positive rate.

For this purpose, an empirical analysis was carried out with long monthly temperature data from 15 different European cities. The performance of both change point detection methods was evaluated, as for the simulation study, taking into account the influence of the time series size, the magnitude of error’s variance and the presence or absence of a temporal correlation structure based on autoregressive moving average models (ARMA).

For the environmental data, the month August 1987 was detected as a change point that was identified in a significant part of the 15 time series analysed. In addition, the parametric method showed a higher number of change point detections than the non-parametric method. Regarding the simulation scenarios, results revealed that in smaller series, i.e. in series of size 50 and 200, the non-parametric method appears to perform better compared to the parametric method. More relevant changes were found in performance indicators with the increase of one of the following parameters: the autocorrelation parameter, the error’s variance and the moving average parameter.

A lógica modal consiste numa extensão da lógica proposicional clássica que inclui modalidades. Estas modalidades foram introduzidas para conseguirmos lidar com possibilidades, e podem ser usadas para modelar fenómenos relacionados com o tempo ou transição entre estados. Geralmente, modelos deste tipo são representados por um conjunto de círculos (estados) e setas (transições entre estados) que estão rotuladas pela ação que levou à transição. A vantagem de usar lógica para entender estes modelos é a possibilidade de especificar propriedades suas em linguagem formal. Ao considerar uma noção de tempo, algumas lógicas modais que são mais usadas são Computation Tree Logic (CTL) e Propositional Linear Temporal Logic (PLTL). Existem diversas ferramentas computacionais que verificam se determinada fórmula é válida em cada estado. Este tipo de verificação é chamado de model checking. Este trabalho contemplou o estudo de modelos de transição de estado probabilísticos, usados na avaliação de tecnologias de saúde, como medicamentos. Estes modelos são conhecidos como Markov chains. A ferramenta Probabilistic Symbolic Modal Checker (PRISM) é um exemplo de ferramenta computacional que efetua model checking e em que uma versão probabilística das lógicas CTL e PLTL são incluídas. Consequentemente, com o auxilio do PRISM podemos verificar diversas propriedades de modelos probabilísticos como Markov chains 

There are several ways to evaluate the association between two variables. While the Pearson's coefficient ρ is useful for evaluating the correlation between two numerical variables extracted from the same set of individuals, Cramer's V and a recently proposed W, based on the Euclidean distance between probability distributions, are more adequate when the two variables are categorical or ordinal. To assess whether there is a significant association between variables, confidence intervals represent a useful statistical tool which, contrary to significance tests, compute the full range of plausible values of a given parameter. Thus, this work covers several methods for computing confidence intervals not only for these three association coefficients individually but also for the difference ρ1-ρ2. Methods for the construction of confidence intervals for the differences V1-V2 and W1-W2 will also be discussed. Application of these methods to a dataset using existing R packages as well as newly developed functions will be presented. 

The mathematical modeling of COVID-19 was essential to predict the spread of the SARS-CoV-2 virus as well as to understand the transmission and the impact of the subsequential interventions taken to prevent its spread. 

Models for this task often rely on systems of differential equations, that although have good results regarding temporal evolution, generally lack consistency when it comes to other aspects like properties of space and heterogeneous populations. 

The aim of this research is to adopt a new approach to the modeling of COVID-19 in order to overcome the limitations of the classical models, such as the SIR model. 

By adopting an approach considering graph theory and cellular automata, it is expected that some of these limitations, like lack of spatial properties and the assumption of homogeneous populations will be mitigated. 

Clinical trials are research studies that determine if a new form of treatment or prevention, such as a new drug, diet, or medical device, is safe and effective by evaluating their effects on human health outcomes.

The design of a clinical trial describes a sequence and structure of activities aiming to reveal a cause-and-effect relationship, generally defined in the research question. Cross-over designs occur when subjects are given different treatments to study the differences between them. This design applies to bioequivalence studies, which aim to demonstrate that different formulations or regimens of drug products are similar, playing a pivotal role in drug development by ensuring that safety and efficacy will be maintained when a patient switches to a new formulation in the marketplace.

The developed work aimed to understand the necessary aspects of implementing clinical trials with two-treatment, two-sequence, and two-period cross-over designs, but also analyze the results that demonstrate the success or failure of the study. The present work focuses on analyses for continuous outcomes through linear regression and analysis of variance. These models can incorporate other random terms and co-variance structures using linear mixed models. AB/BA crossovers applied to bioequivalence studies were studied and analyzed 

We derive a nonstandard finite difference scheme for Bailey's Susceptible-Infected-Removed continuous model. We prove that our discretized system is dynamically consistent with its continuous counterpart and we derive its exact solution. We end with the analysis of the long-term behavior of susceptible, infected and removed individuals, illustrating our results with numerical simulations. This is a joint work with Sandra Vaz and Delfim F. M. Torres.