Title: A general theory for large-scale curve time series via functional stability measure
Authors: Xinghao Qiao - London School of Economics (United Kingdom) [presenting]
Shaojun Guo - Institute of Statistics and Big Data, Renmin Unversity of China (China)
Abstract: Modelling a large bundle of curves arises in a broad spectrum of real applications. However, studies in functional data analysis rely primarily on the critical assumption of independent curve observations. We introduce a general theory for large-scale curve time series, where the number of functional variables, $p$, is large relative to the number of observations, $n$, and the dynamical dependence across observations exists. We propose a functional stability measure for stationary functional processes based on their spectral properties and use it to establish the concentration bounds on the sample covariance matrix function under different functional matrix norms. We also develop concentration results on the relevant estimated terms under the Karhunen-Loeve expansion framework. To illustrate with an example, we consider the vector functional autoregressive models, which characterize the temporal and cross-sectional dependence across multiple curve time series. We develop a regularization approach to estimate the autoregressive coefficient functions under the sparsity constraint. Using our derived concentration inequalities, we investigate the theoretical properties of the regularized estimate in the high-dimensional large $p$, small $n$, regime. Finally, we evaluate the sample performance of the proposed method through simulation studies.