مدل برای بهینه سازی روش تتا و ارتباط آنها با مدل فضای حالت   Models for optimising the theta method and their relationship to state space models

1. Introduction The development of accurate, robust and reliable forecasting methods for univariate time series is very important when large numbers of time series are involved in the modelling and forecasting process. In industrial settings, it is very common to work with large lines of products; thus, efficient sales and operational planning (S&OP) depend heavily on accurate forecasting methods. Despite the advantages of automatic model selection algorithms (Hyndman & Khandakar, 2008; Hyndman, Koehler, Snyder, & Grose, 2002; Poler & Mula, 2011), there is still a need for accurate extrapolation methods. Forecasting competitions have played an important role in moving toward the forecasting of large numbers of time series, with the objective of identifying high-performing methods. The Theta method attracted the attention of researchers by its simplicity and surprisingly good performance (Koning, Franses, Hibon, & Stekler, 2005; Makridakis & Hibon, 2000), and has been one of the benchmarks in more recent forecasting competitions (Athanasopoulos, Hyndman, Song, & Wu, 2011). The Theta method (Assimakopoulos & Nikolopoulos, 2000, hereafter A&N) is applied to non-seasonal or deseasonalised time series, where the deseasonalisation is usually performed via the multiplicative classical decomposition. The method decomposes the original time series into two new lines through the so-called theta coefficients, denoted by θ1 and θ2 for θ1, θ2 ∈ R, which are applied to the second difference of the data. The second differences are reduced when θ < 1, resulting in a better approximation of the long-term behaviour of the series (Assimakopoulos, 1995). If θ is equal to zero, the new line is a straight line. When θ > 1, the local curvatures are increased, magnifying the short-term movements of the time series (A&N). The new lines produced are called theta lines, denoted here by Z(θ1) and Z(θ2). These lines have the same mean value and slope as the original data, but the local curvatures are either filtered out or enhanced, depending on the value of the θ coefficient.