Bootstrap Prediction Bands for Functional Time Series

Abstract

A bootstrap procedure for constructing prediction bands for a stationary functional time series is proposed. The procedure exploits a general vector autoregressive representation of the time-reversed series of Fourier coefficients appearing in the Karhunen–Loève representation of the functional process. It generates backward-in-time functional replicates that adequately mimic the dependence structure of the underlying process in a model-free way and have the same conditionally fixed curves at the end of each functional pseudo-time series. The bootstrap prediction error distribution is then calculated as the difference between the model-free, bootstrap-generated future functional observations and the functional forecasts obtained from the model used for prediction. This allows the estimated prediction error distribution to account for the innovation and estimation errors associated with prediction and the possible errors due to model misspecification. We establish the asymptotic validity of the bootstrap procedure in estimating the conditional prediction error distribution of interest, and we also show that the procedure enables the construction of prediction bands that achieve (asymptotically) the desired coverage. Prediction bands based on a consistent estimation of the conditional distribution of the studentized prediction error process also are introduced. Such bands allow for taking more appropriately into account the local uncertainty of the prediction. Through a simulation study and the analysis of two datasets, we demonstrate the capabilities and the good finite-sample performance of the proposed method.

KEYWORDS:

• Functional prediction
• Prediction error
• Principal components
• Karhunen–Loève expansion

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Bootstrap Prediction Bands for Functional Time Series

Supplementary Material

Supplement: Bootstrap prediction bands for functional time series. The file contains the analysis of a second real-life data example, some auxiliary lemmas and the proofs of all theoretical results presented in this article.

Acknowledgments

The authors are very grateful to the editor, the associate editor and the reviewers for their constructive and insightful comments that lead to an improvement of the article.