7.6 A taxonomy of exponential smoothing methods

Exponential smoothing methods are not restricted to those we have presented so far. By considering variations in the combination of the trend and seasonal components, fifteen exponential smoothing methods are possible, listed in Table 7.7. Each method is labelled by a pair of letters (T,S) defining the type of ‘Trend’ and ‘Seasonal’ components. For example, (A,M) is the method with an additive trend and multiplicative seasonality; (M,N) is the method with multiplicative trend and no seasonality; and so on.

Seasonal Component
Trend N A M
Component (None) (Additive) (Multiplicative)
N (None) (N,N) (N,A) (N,M)
A (Additive) (A,N) (A,A) (A,M)
Ad (Additive damped) (Ad,N) (Ad,A) (Ad,M)
M (Multiplicative) (M,N) (M,A) (M,M)
Md (Multiplicative damped) (Md,N) (Md,A) (Md,M)

Some of these methods we have already seen:

(N,N) = simple exponential smoothing
(A,N) = Holts linear method
(M,N) = Exponential trend method
(Ad,N) = additive damped trend method
(Md,N) = multiplicative damped trend method
(A,A) = additive Holt-Winters method
(A,M) = multiplicative Holt-Winters method
(Ad,M) = Holt-Winters damped method

This type of classification was proposed by Pegels (1969). It was later extended by Gardner (1985) to include methods with additive damped trend and by Taylor (2003) to include methods with multiplicative damped trend.

Table 7.8 gives the recursive formulae for applying all possible fifteen exponential smoothing methods. Each cell includes the forecast equation for generating $h$-step-ahead forecasts and the smoothing equations for applying the method. In Table 7.9 we present some strategies for selecting initial values for some of the most commonly applied exponential smoothing methods. We do not recommend that these strategies be used directly; rather, they are useful in providing starting values for the optimization process.

Table 7.8: Formulae for recursive calculations and point forecasts. In each case, $\ell_t$ denotes the series level at time $t$, $b_t$ denotes the slope at time $t$, $s_t$ denotes the seasonal component of the series at time $t$, and $m$ denotes the number of seasons in a year; $\alpha$, $\beta^*$, $\gamma$ and $\phi$ are smoothing parameters, $\phi_h = \phi+\phi^2+\dots+\phi^{h}$ and $h_m^{+} = \lfloor(h-1) \text{ mod } m\rfloor + 1$. (Click the table for a larger version.)

Method Initial values
(N,N) $\ell_0=y_1$
(A,N)  (Ad,N) $\ell_0=y_1$, $b_0=y_2-y_1$
(M,N)  (Md,N) $\ell_0=y_1$, $b_0=y_2/y_1$
(A,A)  (Ad,A) $\ell_0=\frac{1}{m}{\left(y_1+\dots+y_m\right)}$
$b_0=\frac{1}{m}\left[\frac{y_{m+1}-y_{1}}{m}+\dots+\frac{y_{m+m}-y_{m}}{m} \right]$
$s_0= y_m - \ell_0,~s_{-1} = y_{m-1} - \ell_0,~\dots,~s_{-m+1} = y_1 -\ell_0$
(A,M)  (Ad,M) $\ell_0=\frac{1}{m}{\left(y_1+\dots+y_m\right)}$
$b_0=\frac{1}{m}\left[\frac{y_{m+1}-y_{1}}{m}+\dots+\frac{y_{m+m}-y_{m}}{m} \right]$
$s_0 = y_m/\ell_0,~s_{-1} = y_{m-1}/\ell_0,~\dots,~s_{-m+1} = y_1 /\ell_0$