Functional data usually consist of a sample of functions, with each function observed on a discrete grid. The key idea of functional data analysis is to consider each function as a single, structured object rather than a collection of data points. To represent and investigate functional data, curve registration and functional regression are two important techniques. Curve registration is used to align random curves that display time variations. This procedure, known as functional convex averaging, leads to phase-variance adjusted mean functions. Therefore, compared to a simple averaged mean function, phase-variance adjusted mean function by functional convex averaging is a more accurate representation of the inherent function from which the functional data arise. Several curve registration methods are reviewed in this work, including landmark, self-warping and Bayesian hierarchical curve registration (BHCR). For BHCR, when the number of random curves is large or the sampling grid is intensive, the computational cost increases dramatically. To solve this problem, we introduce an accelerated BHCR algorithm via a predictive process model (PPM), known as PPM-BHCR. Tested by a simulation study and real data, this new method is demonstrated to save large amounts of computing time, without a large sacrifice of accuracy.

Functional regression is used to explore the relationship between the outcome and the predictor, where either or both of them are functional. In this work, several functional regression methods are reviewed according to the function-on-scalar, scalar-on-function and function-on-function categories. Registration is traditionally performed as a data preprocessing step before regression. In this work, we introduce a new method called warped functional regression (WFR), which integrates curve registration and functional regression into one joint model. Therefore, we are able to provide prediction based on an unwarped predictor using this new model. The proposed method is evaluated by simulation studies and demonstrates high accuracy. Several case studies illustrate the key contributions of the proposed method in addressing complex scientific questions.

Many modern biomedical studies record vast amounts of data on individual subjects. The observed data may often be conceptualized as arising from an underlying smooth stochastic process after discretization and contimation with noise. Data in this form may exhibit multidimensionality and complex structural features. For example, electroencephalography (EEG) records electrical activity in the brain over continuous time. Repeated trials of cognitive tasks in EEG experiments induce longitudinal \textit{and} and functional dimensions, complicating estimation and inference.

Regularized estimation and rigorous uncertainty quantification is highly sought after in these settings. In this dissertation I leverage techniques from factor analysis, probabilistic principal components analysis, and Gaussian processes (GPs) in the Bayesian paradigm. These techniques are crucial to achieve simultaneous flexible estimation and adaptive regularization. Model performance and calibration is assessed through a series of numerical experiments. The proposed methods are applied to analyze a wide variety of biomedical data, including cognitive EEG experiments, global age-specific fertility rates, and sleep EEG.

Functional brain imaging technologies produce high dimensional data with structured dependency spanning along multiple dimensions. This dissertation focuses on the specific case of Electroencephalography (EEG), even though most methodological developments are applicable to other imaging modalities. The overarching goal is to provide methodological foundations to inferential problems involving population inference in the setting of cognitive experiments. Specifically, I address important challenges associated with the highly heterogenous nature of brain imaging measurements, by reframing complex inferential questions in the context of familiar analytical techniques involving regression, clustering, functional and longitudinal data analysis. These contributions focus on spatio-temporal modeling of EEG measurements, which characterize both intra- and inter-subjects variation within the contexts of neuronal synchronicity and differential band power dynamics. The methodological developments are used to provide analytical insight in several neuro-cognitive studies based on EEG data.

Until recently, very little research has been conducted to assess the potential human health hazards associated with engineered nanomaterials (ENMs). In-vitro high-throughput screening (HTS) assays for the assessment of engineered nanomaterials provide new opportunities to learn how these particles interact at the cellular level, and may aid in reducing the demand for in-vivo testing. The large number of potential factors that could link nanomaterials to adverse human health impacts, create an imperative need to develop a stronger foundation for quantitative risk assessment in nanotoxicology.

In this dissertation we propose a probability model for the analysis of high-throughput cellular assays. In particular, we develop a method that builds a balance between model complexity and interpretability as a tool to be used by subject-matter specialists for assessing cytotoxicity. The resulting multivariate surface-response model allows for joint inference on dose and time kinetics, and associated classical risk assessment parameters of interest. We illustrate the proposed methodology by profiling a multivariate screening study of eight metal-oxide nanomaterials. Next, we present loss-function-based methods for the hazard ranking of engineered nanomaterials. Specifically, we provide a decision-making tool for prioritizing extensive in-vivo testing of emerging nanomaterials. The proposed framework allows for the aggregation of ranks across different sources of evidence while allowing for differential weighting of this evidence based on its reliability and importance in risk ranking. We illustrate the methodology by ranking particles from a multivariate cytotoxicity screening study of eight metal oxides, conducted in two human cell-lines. Finally, we propose methodology for modeling the relationship between physicochemical properties of ENMs and their observed cytotoxicity, as an initial step in the development of a framework for predictive nanotoxicology. In particular, the proposed approach introduces a new measure of toxicity that is seamlessly integrated into a multi-dimensional model that accounts for dose and duration kinetics jointly using a flexible smooth surface fit. Moreover, the designed approach is appropriate for small sample size, and includes data integration and a framework for advanced dimension reduction through variable selection. The proposed method was applied to a library of 24 engineered nanomaterials.

DNA methylation (DNAm) is commonly used to develop aging biomarkers such as predictors of age, mortality risk, and blood cell counts. Challenges arise due to its high-dimensionality, variability in cytosine-phosphate-guanine (CpG) loci coverage across different arrays, and measuring the most relevant tissue DNAm. This dissertation introduces novel approaches to harness DNAm data for biomarker development, imputation accuracy enhancement, and cross-tissue prediction through three interconnected studies.

Because DNAm data are often high dimensional, they require regularized regression frameworks to construct practical prediction models. In the first arm, I developed DNAm-based biomarkers for fitness parameters, like maximum handgrip strength and VO2max, using regularized linear regression. These biomarkers demonstrate significant associations with physical activity across diverse groups, from individuals with low to intermediate activity levels to elite athletes, showcasing their potential for evaluating the epigenetic impacts of physical fitness.

DNAm data has common missingness challenges, and my second arm presents tools that utilize Copula models to enhance imputation accuracy. Unmeasured loci become problematic when DNAm biomarkers require those methylation levels for their algorithm, however, DNAm data do not commonly meet the underlying normality assumption needed for imputation tools. Therefore, we developed algorithms that can improve DNAm imputation by transforming DNAm into gaussian variables using their inherent distribution. While designed with DNAm in mind, our algorithms extend to any continuous variable needing gaussian structure, offering a versatile tool for all research projects.

The final arm explores Transfer Learning (TL) methodologies to facilitate the prediction of DNAm biomarkers across tissues, addressing the limitation of tissue accessibility in biomarker development and measurement. By enabling the use of saliva DNAm to predict blood DNAm biomarker values, this approach significantly broadens the scope of non-invasive epigenetic studies, providing researchers with robust algorithms for cross-tissue biomarker prediction and tools for development of new biomarkers. In doing so, we demonstrate how information from other tissues' DNAm can enhance biomarker prediction, and provide guidelines for researchers to implement our TL methods.

Collectively, this dissertation uncovers novel strategies for extracting valuable insights from high-dimensional DNAm data, contributing new biomarkers for physical fitness, enhancing DNAm imputation methods, and pioneering cross-tissue prediction algorithms. These offer significant advancements integrating epigenetics with the biostatistics field, facilitating a deeper understanding of DNAm and their implications for human health and aging.