Fernando Sánchez Lasheras

We propose a heuristic method to solve polynomial matrix equations of the type ∑k=1makXk=B, where ak are scalar coefficients and X and B are square matrices of order n. The method is based on the decomposition of the B matrix as a linear combination of the identity matrix and an idempotent, involutive, or nilpotent matrix. We prove that this decomposition is always possible when n=2. Moreover, in some cases we can compute solutions when we have an infinite number of them (singular solutions). This method has been coded in MATLAB and has been compared to other methods found in the existing literature, such as the diagonalization and the interpolation methods. It turns out that the proposed method is considerably faster than the latter methods. Furthermore, the proposed method can calculate solutions when diagonalization and interpolation methods fail or calculate singular solutions when these methods are not capable of doing so.

Air pollution affects human health and is one of the main problems in the world, including in coastal cities with industrial seaports. In this sense, the city of Gijón (northern Spain) stands out as one of the 20 Spanish cities with the worst air quality. The study aims to identify outliers in air quality observations near the El Musel seaport, resulting from the emissions of six pollutants over an eight-year period (2014–2021). It compares methods based on the functional data analysis (FDA) approach and vector methods to determine the optimal approach for detecting outliers and supporting air quality control. Our approach involves analyzing air pollutant observations as a set of curves rather than vectors. Therefore, in the FDA approach, curves are constructed to provide the best fit to isolated data points, resulting in a collection of continuous functions. These functions capture the behavior of the data in a continuous domain. Two FDA approach methodologies were used here: the functional bagplot and the high-density region (HDR) boxplot. Finally, outlier detection using the FDA approach was found to be more powerful than the vector methods and the functional bagplot method detected more outliers than the HDR boxplot.