## 7.4 A taxonomy of exponential smoothing methods

Exponential smoothing methods are not restricted to those we have presented so far. By considering variations in the combinations of the trend and seasonal components, nine exponential smoothing methods are possible, listed in Table 7.5. 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; (A$$_d$$,N) is the method with damped trend and no seasonality; and so on.

Table 7.5: A two-way classification of exponential smoothing methods.
Trend Component
Seasonal Component
N A M
N (None) (N,N) (N,A) (N,M)
A$$_d$$ (Additive damped) (A$$_d$$,N) (A$$_d$$,A) (A$$_d$$,M)

Some of these methods we have already seen using other names:

Short hand Method
(N,N) Simple exponential smoothing
(A,N) Holt’s linear method
(A$$_d$$,N) Additive damped trend method
(A,M) Multiplicative Holt-Winters’ method
(A$$_d$$,M) Holt-Winters’ damped method

This type of classification was first proposed by , who also included a method with a multiplicative trend. It was later extended by to include methods with an additive damped trend and by to include methods with a multiplicative damped trend. We do not consider the multiplicative trend methods in this book as they tend to produce poor forecasts. See for a more thorough discussion of all exponential smoothing methods.

Table 7.6 gives the recursive formulas for applying the nine exponential smoothing methods in Table 7.5. Each cell includes the forecast equation for generating $$h$$-step-ahead forecasts, and the smoothing equations for applying the method. ### Bibliography

Gardner, E. S. (1985). Exponential smoothing: The state of the art. Journal of Forecasting, 4(1), 1–28. [DOI]
Hyndman, R. J., Koehler, A. B., Ord, J. K., & Snyder, R. D. (2008). Forecasting with exponential smoothing: The state space approach. Berlin: Springer-Verlag. http://www.exponentialsmoothing.net
Pegels, C. C. (1969). Exponential forecasting: Some new variations. Management Science, 15(5), 311–315. [DOI]
Taylor, J. W. (2003). Exponential smoothing with a damped multiplicative trend. International Journal of Forecasting, 19(4), 715–725. [DOI]