Facultad de Matemáticas

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Browsing Facultad de Matemáticas by Subject "09 Industry, innovation and infrastructure"

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    Interfaces between statistical learning and risk management.
    (2020) Rubio Varas, Rodrigo Esteban; Galea Rojas, Manuel Jesús; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    The recent hype on Artificial Intelligence, Data Science, and Machine Learning has been leading to a revolution in the industries of Banking and Finance. Motivated by this revolution, this thesis develops novel statistical methodologies tailored for learning about financial risk in the Big Data era. Specifically, the methodologies proposed in this thesis build over ideas, concepts, and methods that relate to cluster analysis, copulas, and extreme value theory. I start this thesis working on the framework of extreme value theory and propose novel statistical methodologies that identify time series which resemble the most in terms of magnitude and dynamics of their extreme losses. A cluster analysis algorithm is proposed for the setup of heteroscedastic extremes as a way to learn about similarity of extremal features of time series. The proposed method pioneers the development of cluster analysis in a product space between an Euclidean space and a space of functions. In the second contribution of this thesis, I introduce a novel class of distributions—to which we refer to as diagonal distributions. Similarly to the spectral density of a bivariate extreme value distribution, the latter class consists of a mean-constrained univariate distribution function on [0, 1], which summarizes key features on the dependence structure of a random vector. Yet, despite their similarities, spectral and diagonal densities are constructed from very different principles. In particular, diagonal densities extend the concept of marginal distribution—by suitably projecting pseudo-observations on a segment line; diagonal densities also have a direct link with copulas, and their variance has connections with Spearman’s rho. Finally, I close the thesis by proposing a density ratio model for modeling extreme values of non-indentically distributed observations. The proposed model can be regarded as a proportional tails model for multisample settings. A semiparametric specification is devised to link all elements in a family of scedasis densities through a tilt from a baseline scedasis. Inference is conducted by empirical likelihood inference methods.
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    The noncommutative geometry of the Landau Hamiltonian
    (2024) Sandoval, Maximiliano; De Nittis, Giuseppe; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    This dissertation will present three works in the areas of noncommutative geometry, the study of the Landau Hamiltonian, and the study of rational noncommutative tori making use of complex geometry and theta functions. The first two works focus primary on extending Bellissard’s work on the noncommutative geometry of the Hall Effect to the case where the medium is continuous case making use of a novel Dirac Operator, closely related to the isotropic quantum harmonic oscillator. We study the resulting spectral triple and its properties as a noncommutative space. Notably we provide proofs for the first and second Connes formulas for this spectral triple. The third work deals with the study of a pair of dual representations of rational noncommutative tori with rational parameters θ and θ −1, how we can naturally construct a vector bundles on a complex tori from to them, and hint how this duality is a manifestation of the Fourier-Mukai-Nahm transform.