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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:

  1. Level of the series - the average value for specific period of time,
  2. Growth of the series - the average increase or decrease of the value over a period of time,
  3. Seasonality - a pattern, which is observed from year to year (e.g. growth in sales of lager beer in Summer),
  4. Error - an unexplainable white noise.
Depending on a textbook or on a paper you are dealing with, you might have different names for these components. For example, in classical decomposition (Warren M. Persons 1919) it is assumed that (1) and (2) represent a specific component called "trend", so the typical model contains error, trend and seasonality. There are modifications of this, which also contain cyclical component. 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 notations in this textbook. According to it, the components can interact with each other differently: either via addition or multiplication. 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 for the 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 \((s_{t-m}-1) \text{%}\) from the baseline. These models only work with the data with positive values. Although they should also work on data with purely negative values as well, this is less often met in practice.

It is also possible to define mixed models, for example, when 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 high values, far from zero, but in the other cases they might produce contradicting results: e.g., generate negative values on positive data. So, the conventional decomposition techniques only consider the pure models.


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