Nonlinear Function-on-Function Geometric Quantile Regression in the Face of Big Data

Advances in data acquisition have enabled the widespread use of large-scale functional data across fields. In the big data era, functional datasets bring both opportunities and challenges. Being infinite-dimensional, functional data are more complex to analyze than scalar data, with storage and computational demands often exceeding a single machine’s capacity. These challenges reveal the limitations of traditional statistical methods and the need for new frameworks. Functional Data Analysis (FDA) addresses these issues, with Function-on-Function (FoF) regression, modeling both response and covariates as functional data, proving highly versatile. However, FoF regression remains underexplored due to the computational complexity of infinite-dimensional predictors and responses. Most existing methods focus on conditional mean estimation, while attempts at FoF quantile regression rely on pointwise modeling, overlooking the geometric structure of functional data and key concepts like conditional distribution functions and quantile intervals. Furthermore, most work is restricted to linear models, limiting the ability to capture nonlinear relationships. Practical applications are also constrained by the high computational demands of large-scale datasets. This project has two main goals. First, we aim to develop a systematic framework for nonlinear FoF quantile regression using M-quantiles (including geometric quantiles) in infinite-dimensional spaces. This framework generalizes univariate quantile regression, enabling the modeling of complex nonlinear relationships while retaining key properties of conditional quantiles. The Principal Investigator (P.I.) and Co-Investigator (Co-I.) have already conducted preliminary investigations, focusing on geometric quantiles, with plans to expand to the broader M-quantile framework, encompassing both geometric quantiles and expectiles. Additional planned work includes developing a data-driven criterion for selecting truncation orders in the Karhunen-Loève expansion, adapting estimation procedures for discretely observed data, and integrating neural operators to flexibly model nonlinear relationships. Second, to address the computational challenges associated with large-scale functional data, we will extend this framework to a distributed computing setting. The P.I. and Co-I. have proposed a tentative distributed computing algorithm for estimating the basis functions. Building on this foundation, we will thoroughly examine its theoretical properties and associated statistical inference aspects, refine the algorithm to ensure that the resulting estimator attains efficiency comparable to the global estimator, develop communication-efficient techniques, and design iterative methods to improve both scalability and computational efficiency. The requested funding will primarily be used to hire a research assistant to conduct essential empirical analyses for the project, and supporting a postgraduate research student to explore specific research questions within the project.