The first part of this work has been motivated by a multivariate space weather dataset. When the Sun releases high-energy particles into space due to a solar explosion, we experience magnetic storms, that once reaching Earth can last hours or days. Although solar flares emmited by the Sun cannot cause any harm to humans on Earth, if they are too severe they can damage machinery and techonology, such as satellites and radio communication. Thus, the modelling of extreme solar activity is important so we can be prepared for undesireable extreme events. Extreme value analysis can help professionals to understand the risks that severe geomagnetic field fluctuations can pose to Earth. For example, we can characterise the tail of the distribution of geomagnetic disturbances and the probability of extreme events. Hence, we perform a pairwise analysis for modelling the extremes of multiple bivariate processes of geomagnetic activity considering two copula models. The aim is to model the joint extremal probability and depict the pairwise extremal dependence structure between pairs of sites in two regions in Europe. The results show that the dependence structure differs in Northern and Southern Europe and that the dependence weakens as the distance increases.

The second part of this work proposes a control chart for detecting small shifts in the mean of a double-bounded process, such as fractions or proportions, in the presence of control variables. For this purpose, we consider the cumulative sum control chart applied to different residuals of the beta regression model. We conduct an extensive Monte Carlo simulation study to evaluate and compare the performance of the proposed control chart with two other control charts in the literature in terms of run length analysis. The numerical results show that the proposed control chart is more sensitive to changes in the process than its competitors and that the quantile residual is the most suitable residual to be used in our proposal. Finally, based on the quantile residual, we present and discuss applications to real and simulated data to show the applicability of the proposed control chart.