Functional data analysis (FDA) is a class of statistical tools to analyze data defined over time, spatial location, or any continuous domains including both Euclidean space and manifolds. Some examples of functional data include differential equation systems, univariate and multivariate longitudinal trajectories, image and spatial data, matrix-valued functions, dynamic networks, and more. This dissertation proposes new statistical models for functional data which do not fit the existing works. Furthermore, this work focuses on combining machine learning-based strategies with statistical tools on functional data. A representation model for functional data has been proposed to compress information about infinite-dimensional objects into low-dimensional vectors. The model is entirely built with neural networks. Functional time series is an emerging topic in FDA where its methodologies are mostly based on existing models. This work proposes an entirely new machine learning-based approach which is shown to be superior in prediction tasks. We propose a composition of differential equations and functional linear models to characterize the behaviors of neurons. We have built an implementation package for easier application of the proposed methodology. Functional principal component analysis (FPCA) is an essential class of models in FDA, and this work has introduced a new approach for functional data defined on manifolds in contrast to those in Euclidean spaces. Last, we design a strategy from functional data analysis for network embedding on dynamic networks.
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Thesis advisor: Cao, Jiguo
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