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Abstract:
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We study estimation and forecasting of functional autoregressive (FAR) processes, a statistical tool for modeling functional time series data. Due to the infinite-dimensional nature of FAR processes, the existing literature addresses its inference via dimension reduction and theoretical results therein require the strong assumption of fully observed functional time series. We propose an alternative inference framework based on Reproducing Kernel Hilbert Spaces (RKHS). Specifically, a nuclear norm regularization method is proposed for estimating the transition operators of the FAR process directly from discrete samples of the functional time series. We derive a representer theorem for the FAR process, which enables infinite-dimensional inference without dimension reduction. Optimal theoretical guarantees are established under the more realistic assumption that we only have finite discrete samples of the FAR process. Extensive numerical experiments and a real data application of energy consumption predic- tion are further conducted to illustrate the promising performance of the proposed approach compared to the state-of-the-art methods in the literature.
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