Integrated Model Selection and Scalability in Functional Data Analysis through Bayesian Learning

Functional data, including one-dimensional curves and higher-dimensional surfaces, have become increasingly prominent across scientific disciplines. They offer a continuous perspective that captures subtle dynamics and richer structures compared to discrete representations, thereby preserving essential information and facilitating more natural modeling of real-world phenomena, especially in sparse or irregularly sampled settings. A key challenge lies in identifying low-dimensional representations and estimating covariance structures that capture population statistics effectively. We propose a novel Bayesian framework with a nonparametric kernel expansion and a sparse prior, enabling direct modeling of measured data and avoiding the artificial biases from regridding. Our method, Bayesian scalable functional data analysis (BSFDA), automatically selects both subspace dimensionalities and basis functions, reducing computational overhead through an efficient variational optimization strategy. We further propose a faster approximate variant that maintains comparable accuracy but accelerates computations significantly on large-scale datasets. Extensive simulation studies demonstrate that our framework outperforms conventional techniques in covariance estimation and dimensionality selection, showing resilience to high dimensionality and irregular sampling. The proposed methodology proves effective for multidimensional functional data and showcases practical applicability in biomedical and meteorological datasets. Overall, BSFDA offers an adaptive, continuous, and scalable solution for modern functional data analysis across diverse scientific domains.