Abstract:
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Modelling a large bundle of curves arises in a broad spectrum of real applications. However, many studies in functional data analysis literature focus primarily on the critical assumption of iid samples of a fixed number of curves. We introduce a measure of functional dependence for stationary functional processes that provides insights into the effect of cross-dependence among high dimensional curve time series. Based on our proposed functional dependence measure, we establish some useful concentration bounds for the relevant estimates when each component of the vector of curve time series is represented through its Karhunen-Loeve expansion. As an example to illustrate, we propose vector functional autoregressive models, which characterize the dynamic dependence across high dimensional curve time series, and develop a regularization approach to estimate autoregressive coefficient functions. We then apply our developed concentration inequalities to derive the non-asymptotic upper bounds for the estimation errors of the regularized estimates. We also show that the proposed method significantly outperforms its potential competitors through both simulations and one real data example.
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