Hilbertian ARMA model for forecasting functional time series

Abstract

A new forecasting method for functional time series is proposed. This model attempts to generalize the standard scalar ARMA time series model to the $L^2$ Hilbert space in order to forecast functional time series. A functional time series is the realization of a stochastic process where each observation is a continuous function defined in a finite interval $[a,b]$. Forecasting these time series require a model that can operate with continuous functions. The structure of the proposed model is a regression where functional parameters operate on functional variables. The variables can be lagged values of the series (autoregressive terms), past observed errors (moving average terms) or exogenous variables. The functional parameters used are integral operators in the $L^2$ space. In our approach, the kernels of the operators are given as a linear combination of sigmoid functions. The parameters of each sigmoid are estimated using a Quasi-Newton algorithm minimizing the sum of squared errors. This is a novel approach because the iterative algorithm allows estimating the moving average terms. The new model is tested with functional time series obtained from real data of the Spanish electricity market and compared with other functional reference models.