Functional modeling techniques for high-dimensional data with complex structures

Recent technological advancements have increased the structural complexity of recorded data across various fields of research. Terms such as high-dimensional and Big Data have become commonplace among statisticians and data scientists. Analyzing such data requires specialized techniques tailored to the specific application. This thesis focuses on studying high-dimensional data using techniques from Functional Data Analysis (FDA). The thesis is structured as a compilation of three independent research articles, each presented in a separate chapter, that nonetheless share the goal of enhancing a particular functional regression model. Chapter 2 introduces our first contribution, stringing via Manifold Learning, ML-stringing for short. Our proposal is framed within a wider class of methods that map high-dimensional observations to the infinite space of functions, allowing the use of FDA. Stringing handles any high-dimensional data vector as scrambled realizations of an unknown stochastic process. Its essential feature is a rearrangement of the observed values. Originally, stringing is based on Unidimensional Scaling (UDS), an unsupervised technique that linearly reduces the dimension of the data vectors by preserving distances. Motivated by the linear nature of UDS, we aim to recover more complex relationships between covariates using Manifold Learning. The chapter includes simulation studies showing that ML-stringing achieves higher-quality orderings than UDS-stringing, improving the data’s functional representation. In the context of scalar-on-function regression, ML-stringing also leads to improvements in the estimated model. The chapter also presents an application to a colon cancer study that deals with high-dimensional gene expression arrays. Chapter 3 addresses the scalar-on-function regression problem using functional partial least squares (FPLS), focusing on functional data defined over complex domains that may have multiple dimensions and non-Euclidean structures. Here we introduce our second contribution, a penalized FPLS approach based on a Rank-1 approximation of the empirical sample covariance matrix between the response and the predictor; R1-FPLS for short. When the domain has a manifold topology, R1- FPLS solves the scalar-on-function regression through Finite Element Analysis, which provides interesting sparsity properties that make the algorithm computationally efficient even in the context of large datasets. The chapter includes simulation studies that compare the performance of the proposed R1-FPLS with other FPLS approaches from the literature, using functional data defined over one-dimensional and two-dimensional planar domains. We also apply our method to brain connectivity maps obtained from task-based functional Magnetic Resonance Images. In this case, the brain is viewed as a three-dimensional domain with a non-Euclidean structure. Our results show that using R1-FPLS to discriminate between schizophrenics and healthy patients based on the connectivity maps outperforms other recently proposed approaches. Chapter 4 introduces our third contribution; a novel penalized Function-on- Function Partial Least-Squares (pFFPLS) that solves the function-on-function linear regression problem. pFFPLS introduces an appropriate finite-dimensional functional space with an associated set of bases on which to represent the data and controls smoothness with a roughness penalty operator. Penalizing the FPLS weights imposes smoothness on the resulting coefficient function, improving its interpretability. The chapter compares pFFPLS with the non-penalized counterpart FFPLS. Through a simulation study, it is shown that pFFPLS provides a higher accuracy when predicting the response and the true coefficient function from which the data were generated. The chapter also includes two case studies involving two well-known datasets from the FDA literature. In the first application, we predict log precipitation curves from the yearly temperature profiles recorded in 35 weather stations in Canada. In the second one, we predict the hip angle profiles during a gait cycle of children from their corresponding knee angle profiles.

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