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## 4.3 ETS taxonomy

Building on the idea of time series components we can move to the ETS taxonomy. ETS stands for “Error-Trend-Seasonality” and defines how specifically the components interact with each other. Based on the type of error, trend and seasonality, (Pegels 1969) proposed a taxonomy, which was then developed further by (Hyndman et al. 2002) and refined by (Rob J. Hyndman et al. 2008). According to this taxonomy, error, trend and seasonality can be:

- Error: either “Additive” (A), or “Multiplicative” (M);
- Trend: either “None” (N), or “Additive” (A), or “Additive damped” (Ad), or “Multiplicative” (M), or “Multiplicative damped” (Md);
- Seasonality: either “None” (N), or “Additive” (A), or “Multiplicative” (M).

According to this taxonomy, the model (4.1) is denoted as ETS(A,A,A) while the model (4.2) is denoted as ETS(M,M,M), and (4.3) is ETS(M,A,M).

The main advantages of the ETS taxonomy are that the components have clear interpretations and that it is flexible, allowing to have 30 models with different types of error, trend and seasonality. The figure below shows examples of different time series with deterministic (they do not change over time) level, trend and seasonality, based on how they interact in the model. The first one shows the additive error case:Things to note from this plot:

- When the seasonality is multiplicative its amplitude increases with the level of the data while with additive seasonality the amplitude is constant. Compare, e.g., ETS(A,A,A) with ETS(A,A,M): the distance between the highest and the lowest points for the former in the first year is roughly the same as in the last year. In the case of ETS(A,A,M) the distance increases with the increase in the level.
- When the trend is multiplicative data with exponential growth / decay result. With ETS(A,M,N), for example, we say that there is roughly 5% growth in the data;
- The damped trend models slow down both additive and multiplicative trends;
- It is practically impossible to distinguish additive and multiplicative seasonality if the series does not trend because what distinguishes the two – see (1) – is not relevant (compare ETS(A,N,A) and ETS(A,N,M)).

They show roughly the same picture as the additive case, the main difference being that the variance of the error increases with the increase of the level of the data - this becomes clearer on ETS(M,A,N) and ETS(M,M,N) data. This property is called heteroscedasticity in statistics and (Rob J. Hyndman et al. 2008) argue that the main benefit of the multiplicative error models is being able to capture this feature.

In the next chapters we will discuss the most important members of the ETS taxonomy. Not all the models in this taxonomy are particularly sensible and some are typically ignored entirely. Although ADAM implements the entire taxonomy we will discuss potential issues and what to expect from them.

### References

Hyndman, Rob J., Anne B. Koehler, J. Keith Ord, and Ralph D. Snyder. 2008. *Forecasting with Exponential Smoothing*. Springer Berlin Heidelberg.

Hyndman, Rob J, Anne B Koehler, Ralph D Snyder, and Simone Grose. 2002. “A state space framework for automatic forecasting using exponential smoothing methods.” *International Journal of Forecasting* 18 (3): 439–54. https://doi.org/10.1016/S0169-2070(01)00110-8.

Pegels, C Carl. 1969. “Exponential Forecasting : Some New Variations.” *Management Science* 15 (5): 311–15. https://www.jstor.org/stable/2628137.