Forecasting Time Series With Complex Seasonal Patterns Using Exponential Smoothing

TLDR

An innovations state‑space modeling framework is introduced to forecast complex seasonal time series with multiple, high‑frequency, non‑integer, and dual‑calendar seasonality. The framework employs Box–Cox transformations, Fourier representations with time‑varying coefficients, ARMA error correction, a computationally efficient maximum‑likelihood estimation, and a trigonometric decomposition that reveals otherwise hidden seasonal components. The model demonstrates versatility across diverse applications, evidenced by three empirical studies, and its trigonometric decomposition successfully extracts hidden seasonal components.

Abstract

An innovations state space modeling framework is introduced for forecasting complex seasonal time series such as those with multiple seasonal periods, high-frequency seasonality, non-integer seasonality, and dual-calendar effects. The new framework incorporates Box–Cox transformations, Fourier representations with time varying coefficients, and ARMA error correction. Likelihood evaluation and analytical expressions for point forecasts and interval predictions under the assumption of Gaussian errors are derived, leading to a simple, comprehensive approach to forecasting complex seasonal time series. A key feature of the framework is that it relies on a new method that greatly reduces the computational burden in the maximum likelihood estimation. The modeling framework is useful for a broad range of applications, its versatility being illustrated in three empirical studies. In addition, the proposed trigonometric formulation is presented as a means of decomposing complex seasonal time series, and it is shown that this decomposition leads to the identification and extraction of seasonal components which are otherwise not apparent in the time series plot itself.