Functional time series forecasting: a systematic review

Forecasting functional time series (FTS) has arguably achieved tremendous success in recent years. Time series of curves, or functional time series, exist in many disciplines. Among the numerous existing contributions for forecasting time series, one-step-ahead functional time series forecasting, that is one-step-ahead prediction of a curve-valued time series, has been studied in several practical studies. Predominantly most traditional functional time series studies use functional (Hilbertian) autoregressive models for one-step-ahead forecast, but their application in real-world data remains a pertinent challenge due to a non-stationary behavior. Opposed to such models, several nonparametric approaches have been proposed in the recent literature for forecasting time series of curves. An analysis of the forecasting performances of such nonparametric approaches, validated empirically with a set of real experiments, is presented in this paper. While a complete understanding of these approaches remains elusive, we hope that our perspectives, discussions, and comparisons serve as a stimulus for new statistical research.