Forecasting functional time series with a new Hilbertian ARMAX model: application to electricity price forecasting

Abstract

Functional time series are the realization of stochastic processes where each observation is a continuous function defined on a finite interval. Forecasting these high dimensional time series requires models that operate with continuous functions. In this paper, a new forecasting method is proposed that attempts to generalize the standard seasonal ARMAX time series model to the L2 Hilbert space in order to forecast functional time series. The structure of the proposed model is a linear regression where functional parameters operate on functional variables. The variables can be lagged values of the series (autoregressive terms), past observed innovations (moving average terms) or exogenous variables. In our approach, the functional parameters used are integral operators in the L2 space and the kernels of the operators are modeled as linear combinations of sigmoid functions. The parameters of each sigmoid are estimated using a Quasi-Newton algorithm for minimizing the sum of squared errors. This novel approach allows estimating the moving average terms in functional time series models. The new model is tested by forecasting the daily price profile of the Spanish electricity market and compared with other functional reference models.