MAT Tesis doctorado

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Browsing MAT Tesis doctorado by Subject "03 Salud y bienestar"

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    Flexible spatio-temporal strategies for modeling mosquito-borne diseases
    (2024) Pavani, Jessica Letícia; Quintana Quintana, Fernando; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    Growing awareness of environmental threats has encouraged researchers to increasingly focus on analyzing spatial and temporal patterns of diseases, including vector-borne diseases. A byproduct of this is the also increased interest in cluster analysis. Over the last few decades, the frequency and magnitude of disease outbreaks caused by insects have increased dramatically. In addition to areas that are recurrently affected, outbreaks are spreading into regions that were previously unaffected. Faced with such a scenario, clustering analysis is essential for recognizing areas and times with high disease incidence, thus aiding in intervention planning. Moreover, the increasing availability of large datasets of high quality has culminated in the emergence of more sophisticated statistical models and methods. In response to this need, we have developed some flexible Bayesian approaches whose main goal is to identify and cluster neighboring regions where the infection behaves similarly, and to evaluate how the spatial clustering pattern changes over time. To begin with, we develop a technique for recognizing and grouping regions that display similar time-based patterns for a specific disease. Our method employs product partition models that take into account the influence of neighboring regions to cluster geographical data. This prior is tied to temporal modeling, as it aligns the classification of regions with their time trends. Consequently, the temporal coefficients are common among areas within the same cluster. Furthermore, we introduce a directed acyclic graph structure to manage the spatial dependencies among these regions. As a contribution to the literature on multivariate data, we extend the first approach to jointly modeling multiple diseases, explicitly accounting for potential space-time correlations between them. In this case, we employ a multivariate directed acyclic graph autoregressive framework to capture both spatial and inter-disease dependencies. In the initial two models, the spatial cluster stays unchanged throughout time. However, the challenge of modeling intensifies when we attempt to examine temporal changes across different spatial partitions. To address this, we introduce a model for time-dependent sequences of spatial random partitions, establishing a prior based on product partition models that correlate spatial configurations. By utilizing random spanning trees as a methodological tool, we ease the exploration of the complex partition search space. We validate the properties of all models through simulation studies, demonstrating its competitive performance against alternative approaches. Furthermore, we apply them to mosquito-borne diseases dataset in the Brazilian Southeast region.
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    Mathematical analysis and applications of neural networks, with applications to image reconstruction
    (2025) Molina Mejía, Juan José; Courdurier, Matías; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    This thesis explores two fundamental aspects of neural networks: their frequency learning behavior and their application to quantitative Magnetic Resonance Imaging (MRI) reconstruction. The first part investigates the phenomenon of frequency bias, the empirical observation that neural networks tend to learn low-frequency components of a target function more rapidly than high-frequency ones. To provide a rigorous understanding of this behavior, we develop a theoretical framework based on Fourier analysis. Specifically, we derive a partial differential equation that governs the evolution of the error spectrum during training in the Neural Tangent Kernel regime, focusing on two-layer neural networks. Our analysis centers on Fourier Feature networks, a class of architectures where the first layer applies sine and cosine activations using pre-defined frequency distributions. We demonstrate that the network's initialization, particularly the initial density distribution of first-layer weights, plays a crucial role in shaping the frequency learning dynamics. This insight provides a principled way to control or even eliminate frequency bias during training. Theoretical predictions are validated through numerical experiments, which further illustrate the impact of initialization on the inductive biases of neural networks.The second part of the thesis applies neural network techniques to the reconstruction of quantitative MRI data. Quantitative MRI enables the estimation of tissue-specific parameters (e.g., T1, T2, and T2*) that are vital for clinical diagnosis and disease monitoring. However, these methods typically require long acquisition times, which are often mitigated through aggressive undersampling of k-space data. Undersampling, in turn, introduces reconstruction artifacts that must be addressed through regularization. To this end, we propose CConnect, a novel iterative reconstruction method that incorporates convolutional neural networks into the regularization term. CConnect connects multiple CNNs through a shared latent space, allowing the model to capture common structures across different image contrasts. This design enables the effective suppression of aliasing artifacts and improves image quality, even in highly undersampled scenarios. We evaluate CConnect on in-vivo brain T2*-weighted MRI data, demonstrating its superiority over classical low-rank and total variation methods, as well as standard deep learning baselines.