Conformal Prediction Bands for Two-Dimensional Functional Time Series

Conformal Prediction (CP) is a versatile nonparametric framework used to quantify uncertainty in prediction problems. In this work, we provide an extension of such method to the case of time series of functions defined on a bivariate domain, by proposing for the first time a distribution-free technique which can be applied to time-evolving surfaces. In order to obtain meaningful and efficient prediction regions, CP must be coupled with an accurate forecasting algorithm, for this reason, we extend the theory of autoregressive processes in Hilbert space in order to allow for functions with a bivariate domain. Given the novelty of the subject, we present estimation techniques for the Functional Autoregressive model (FAR). A simulation study is implemented, in order to investigate how different point predictors affect the resulting prediction bands. Finally, we explore benefits and limits of the proposed approach on a real dataset, collecting daily observations of Sea Level Anomalies of the Black Sea in the last twenty years.