4.4 Chosen forecasting methods
4.4.3 Exponential smoothing methods
In addition to the N1 and N2 methods, also several exponential smoothing (ES) methods are used to see if methods that are more sophisticated and the estimation of the trend component increases the forecasting accuracy. Because a time series may include either an additive or a multiplicative trend, or no trend at all, and the same goes for the seasonal component, there are nine different trend and seasonal component combinations in total. Each of these combinations has their own exponential smoothing method in addition to methods, which either damp the trend and/or the seasonal component. (Gardner 2006) In this research, nine ES methods and three equally weighted method combinations are used. The combinations are equally weighted because they have been argued to usually provide the best results (Blattberg & Hoch 1990, 895). The first ES method used is the simple exponential smoothing method (SES), which assumes no seasonal or trend component in a time series (Makridakis, et al. 1998, 147). The second method is the Holt’s linear method (HLM), which assumes the time series to have a linear trend but no
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seasonality (Holt 2004). If the time series includes a multiplicative seasonality but no trend or a multiplicative seasonality and an additive trend, then the Winter’s multiplicative method (WMM) and the Holt-Winters’ multiplicative method (HWMM) should work well respectively. Finally, for a time series which include an additive trend but no seasonality, the Gardner & McKenzie's damped additive trend method (GMD-AT) should provide good results while the damped multiplicative trend method (DAT-MS) should provide accurate forecasts for a time series with a linear trend and a multiplicative seasonality. The Equations 11–16 below represents these six ES methods respectively. (Gardner 2006, 640; Winters 1960; Gardner & McKenzie 1985)
(11) Lt = αYt + (1 – α)Lt–1 Ft+m = Lt (12) Lt = αYt+(1 – α)(Lt–1 + bt–1) bt = β(Lt – Lt–1)+(1 – β)bt–1 Ft+m = Lt + btm (13) Lt = α(Yt⁄St–p) + (1 – α)Lt–1 St = γ(Yt⁄Lt) + (1 – γ)St–p Ft+m = LtSt–p+m (14) Lt = α(Yt⁄St–p) + (1 – α)(Lt–1 + bt–1) bt = β(Lt – Lt–1) + (1 – β)bt–1 St = γ(Yt⁄Lt)+(1 – γ)St–p Ft+m = (Lt + btm)St–p+m (15) Lt = αYt + (1 – α)(Lt–1 + ϕbt–1) bt = β(Lt – Lt–1) + (1 – β)ϕbt–1 Ft+m = Lt + btm (16) Lt = α(Yt⁄St–p) + (1 – α)(Lt–1 + ϕbt–1) bt = β(Lt – Lt–1) + (1 – β)ϕbt–1 St = γ(Yt⁄Lt) + (1 – γ)St–p Ft+m = (Lt +∑ ϕibt m i=1 ) St–p+m
Where p is the number of periods in the seasonal cycle, α is the smoothing parameter for the level, β for the trend, and γ for the seasonal indices of the series, and ϕ is the damping parameter. Lt is the smoothed level of the series, bt is the slope of the additive trend and St is the multiplicative seasonality at the end of period t. (Gardner 2006, 640–641) As the
reader may have noticed, the seasonal adjustments are made in the forecasting methods multiplicatively. Therefore, it will also be studied if the methods are changed so that they
use pseudo-additive decomposition. Technically, this means that the (Yt/St – p) is changed
to (Yt – TtSt + Tt). This study can be seen to have a high theoretical contribution value
because such studies have not been provided before. Finally, as argued before, the St was
replaced with the seasonal mean because of the stochastic seasonal component (Dekker, et al. 2004; Miller & Williams 2003).
From the smoothing parameters alpha, beta, and gamma may have a value between zero and one while phi can have any positive value. A smaller alpha makes the forecasted time series smoother and the forecasts to lag more. The forecasts with a larger alpha again lag less but the risk them including noise in the forecasts increases. When the alpha is one then the Holt’s linear method becomes the N1 method. (Makridakis, et al. 1998) As it can be seen, the DAT-MS (Equation 15) is the same than HWMM (Equation 13) when the damping parameter ϕ is set to one and WMM (Equation 12) when set to zero. If the value of the phi is close to one, then the trend is expected to be relatively persistent, however, if the data is noisy or the trend is expected to change the value of the phi is lowered towards zero. The method assumes that there is no linear trend whatsoever if the phi value is zero and an exponential trend when it is over one. (McKenzie & Gardner 2010) These ES methods were chosen because the time series forecasted here include the assumed characteristics. In addition, the damped trend methods are used because there are numerous studies where they have shown to outperform traditional ES methods and some of the more complicated methods have been able to provide only slight increases in the forecasting accuracy in certain circumstances. The damped trend methods are robust because they do not consider the trend to be persistent but rather takes uncertainty into account and damps the trend. (Armstrong et al. 2015; Fildes, Nikolopoulos, Crone & Syntetos 2008; Armstrong 2006; Gardner & McKenzie 1985) Even though they have performed well, they are still rather unknown outside of a small group of academics (Gardner 2015). Therefore, by studying its functionality and comparing it with other ES methods, this thesis provides valuable theoretical contribution value.
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