Estimating functional time series by moving average model fitting

Abstract

Functional time series have become an integral part of both functional data and time series analysis.Important contributions to methodology, theory and application for the prediction of future trajectories and the estimation of functional time series parameters have been made in the recent past. This paper continues this line of research by proposing a first principled approach to estimate invertible functional time series by fitting functional moving average processes. The idea is to estimate the coefficient operators in a functional linear filter. To do this a functional Innovations Algorithm is utilized as a starting point to estimate the corresponding moving average operators via suitable projections into principal directions. In order to establish consistency of the proposed estimators, asymptotic theory is developed for increasing subspaces of these principal directions. For practical purposes, several strategies to select the number of principal directions to include in the estimation procedure as well as the choice of order of the functional moving average process are discussed. Their empirical performance is evaluated through simulations and an application to vehicle traffic data.

Lagged covariance and cross-covariance operators of processes in Cartesian products of abstract Hilbert spaces

“…A comprehensive study of lag-h-cross-covariance operators of L 2 [0, 1]-valued processes is conducted in Rice & Shum (2019, [46]) who established operator estimates and deduced their limit distribution. Aue & Klepsch (2017, [3]) estimated lagged covariance and cross-covariance operators of processes in Cartesian products of L 2 [0, 1] to deduce asymptotic assertions regarding estimators for operators of linear, invertible processes in L 2 [0, 1]. Enabling processes to have values in Cartesian products was also handy in the study of AR(p) processes with p > 1, see [7].…”

Variable Selection for the Prediction of C[0,1]-Valued Autoregressive Processes using Reproducing Kernel Hilbert Spaces

“…The literature in the field of functional time series analysis is developing quickly. Recent publications include time-domain methods like Hörmann and Kokoszka (2010), where a weak dependence concept is introduced, Aue et al (2015), and , where prediction methodologies based on linear models are developed, and Aue and Klepsch (2017), where an estimator of functional linear processes based on moving average model fitting is derived. Besides, another examples of statistical papers taking advantage of the usefulness of Reproducing Kernel Hilbert Spaces are, among others, Hsing and Eubank (2015); Kadri et al (2015); Berrendero et al (2017Berrendero et al ( , 2018 and Yang et al (2018).…”

“…• Vehicle traffic data (Traffic) presented in Aue and Klepsch (2017). The original data set was provided by the Autobahndirektion Südbayern.…”

Functional Horseshoe Smoothing for Functional Trend Estimation

“…Besse et al, 2000;Hörmann et al, 2013;Gao et al, 2019;Hörmann et al, 2015) and its theoretical properties have been extensively studied (e.g. Bosq, 2000;Aue and Klepsch, 2017;Spangenberg, 2013;Kühnert et al, 2020;Cerovecki et al, 2019). However, in actual data such as GDP, the assumption of stationarity is not satisfied in many situations because the expected value varies from period to period.…”