Alejandro Arias Gómez.

I carried out my Bachelor’s thesis within the Machine Learning Group at UAM, in collaboration with the scikit-fda development team. The work focuses on the mathematical and statistical foundations of Functional Principal Component Analysis (FPCA) for irregularly sampled data.

Functional Data Analysis (FDA) is a branch of statistics that studies data whose observations are functions, rather than finite-dimensional vectors. In this framework, each observation is modeled as a realization of a random variable, allowing the use of tools from probability theory, linear algebra, and functional analysis.

The dissertation develops the theoretical formulation of the PACE (Principal Component Analysis through Conditional Expectation) algorithm, introduced in Yao et al. This algorithm is an extension of classical FPCA designed for sparse and irregular sampling. It studies the estimation of mean and covariance operators via kernel smoothing, the role of bias correction and measurement noise, and the reconstruction of functional principal component scores through conditional expectation, providing a mathematically robust approach to dimensionality reduction in functional settings.

This Bachelor’s thesis studies the theory of primary decomposition in commutative algebra, with a particular focus on the class of monomial ideals in Noetherian rings. The work addresses the structural decomposition of ideals and the conditions under which such decompositions exist.

After introducing the basic notions of commutative algebra, the dissertation develops the general framework of primary decomposition in Noetherian rings. The focus then shifts to monomial ideals in polynomial rings, where their combinatorial structure allows for a more explicit and constructive treatment of primary decompositions.