Linear Processes for Functional Data

TL;DR

This paper reviews the development of linear processes for functional data, highlighting recent advances and future perspectives in modeling continuous-time stochastic processes with applications in prediction.

Contribution

It provides a comprehensive overview of recent theoretical and applied progress in linear processes for functional data, emphasizing new models and research directions.

Findings

Recent advances in linear functional processes are summarized.

Theoretical and applied challenges are identified.

Promising future research perspectives are discussed.

Abstract

Linear processes on functional spaces were born about fifteen years ago. And this original topic went through the same fast development as the other areas of functional data modeling such as PCA or regression. They aim at generalizing to random curves the classical ARMA models widely known in time series analysis. They offer a wide spectrum of models suited to the statistical inference on continuous time stochastic processes within the paradigm of functional data. Essentially designed to improve the quality and the range of prediction, they give birth to challenging theoretical and applied problems. We propose here a state of the art which emphasizes recent advances and we present some promising perspectives based on our experience in this area.