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## 4.1 Time series components

The main idea behind many forecasting techniques is that any time series can contain several unobservable components, such as:

**Level**of the series - the average value for specific period of time,**Growth**of the series - the average increase or decrease of the value over a period of time,**Seasonality**- a pattern that repeats itself with a fixed periodicity. This pattern need not literally be seasonal, like beer sales being higher in summer than they are in winter (season of year). Any pattern with a fixed periodicity works: the number of hospital visitors is higher on Mondays than on Saturdaya or Sundays because people tend to stay at home over the weekend (day of week seasonality), and sales are higher during daytime than they are at night (hour of the day seasonality).**Error**- unexplainable white noise.

Each textbook and paper will use slightly different names to refer to these components. For example, in classical decomposition (Warren M. Persons 1919) it is assumed that (1) and (2) jointly represent a “trend” component so a model will contain error, trend and seasonality. There are modifications of this, which also contain cyclical component(s). When it comes to ETS, the growth component (2) is called “trend”, so the model consists of the four components. We will use the ETS formulation in this textbook. According to this formulation the components can interact with each other in one of two ways: additively or multiplicatively. The pure additive model in this case can be summarised as: \[\begin{equation} y_t = l_{t-1} + b_{t-1} + s_{t-m} + \epsilon_t , \tag{4.1} \end{equation}\] where \(l_{t-1}\) is the level, \(b_{t-1}\) is the trend, \(s_{t-m}\) is the seasonal component with periodicity \(m\) (e.g. 12 for months of year data, implying that something is repeated every 12 months) - all these components are produced on the previous observations and are used on the current one. Finally, \(\epsilon_t\) is the error term, which follows some distribution and has zero mean. Similarly, the pure multiplicative model is: \[\begin{equation} y_t = l_{t-1} b_{t-1} s_{t-m} \varepsilon_t , \tag{4.2} \end{equation}\] where \(\varepsilon_t\) is the error term that has mean of one. The interpretation of the model (4.1) is that the different components add up to each other, so, for example, the sales in January typically increase by the amount \(s_{t-m}\), and that there is still some randomness that is not taken into account in the model. The pure additive models can be applied to data that can have positive, negative and zero values. In case of the model (4.2), the interpretation is similar, but the sales change by \((s_{t-m}-1)\)% from the baseline. These models only work on data with strictly positive values (data with purely negative values are also possible but rare in practice).

It is also possible to define mixed models in which, for example, the trend is additive but the other components are multiplicative: \[\begin{equation} y_t = (l_{t-1} + b_{t-1}) s_{t-m} \varepsilon_t \tag{4.3} \end{equation}\] These models work well in practice when the data has large values far from zero. In other cases, however, they might produce strange results (e.g. negative values on positive data) so the conventional decomposition techniques only consider the pure models.

### References

Warren M. Persons. 1919. “General Considerations and Assumptions.” *The Review of Economics and Statistics* 1 (1): 5–107. https://doi.org/10.2307/1928754.