A Conformal Approach for Distribution-free Prediction of Functional Data

Interval prediction has always been a complex problem to solve in the realm of Functional Data Analysis, and the solutions currently proposed to address this very important theoretical and applied issue are not satisfactory. In this contribution we propose a novel approach, based on a non-parametric forecasting approach coming from machine learning, called Conformal Prediction. In the scalar setting, the method is based on simple yet remarkable considerations about sample quantiles. After having stated in a formal way the issue of forecasting for functional data, we develop an algorithm that can be used to generate non-parametric prediction bands for a functional-on-scalar linear regression model. These forecasts are proven to be valid in a statistical sense (i.e., they guarantee a global coverage probability larger or equal to a given threshold) under a very minimal set of assumptions, and thus extremely useful in the statistical practice. The method is then tested on a real world application, namely ensemble emulations for climate economy models, very used in the climate change economics realm.