An Innovations Algorithm for the prediction of functional linear processes

TL;DR

This paper introduces a new computationally efficient predictor for functional linear processes, leveraging the Innovations Algorithm, with proven convergence properties depending on eigenvalue decay rates.

Contribution

It develops a novel finite-dimensional predictor for functional linear processes using the Innovations Algorithm, extending applications beyond functional autoregressive models.

Findings

Predictor converges to the optimal predictor as sample size increases.

Convergence rate depends on eigenvalue decay of covariance and spectral density.

Method is computationally tractable for practical applications.

Abstract

When observations are curves over some natural time interval, the field of functional data analysis comes into play. Functional linear processes account for temporal dependence in the data. The prediction problem for functional linear processes has been solved theoretically, but the focus for applications has been on functional autoregressive processes. We propose a new computationally tractable linear predictor for functional linear processes. It is based on an application of the Multivariate Innovations Algorithm to finite-dimensional subprocesses of increasing dimension of the infinite-dimensional functional linear process. We investigate the behavior of the predictor for increasing sample size. We show that, depending on the decay rate of the eigenvalues of the covariance and the spectral density operator, the resulting predictor converges with a certain rate to the theoretically best linear predictor.