If you have read the article Time series analysis for statistical forecasting, you already know that a time series is simply a sequence of values temporarily sorted. However, if such a sequence represents a real behaviour of something (for example the sales history of your business), it will be more than a simple sequence of values.
Therefore, a not trivial time series may be characterized by four different components:
- trend (T): represents the long term behaviour
- cycle (C): represents long term fluctuations due to external factors
- seasonality (S): represents repeating patterns in certain periods
- randomness (R): represents something which is not explicable
How can we merge those components in one time series?
One of the most used formulation is the additive formulation. It says that the time series is simply a sum of the four components. Hence, if Y is our time series, this formulation says that Y = T+C+S+R.
This is a suitable solution when the seasonal component does not vary its intensity with the level of the time series (as in figure above).
Otherwise, a multiplicative formulation could be the solution. As expected, this formulation says that the time series is a multiplication of the four components, so Y = T*C*S*R.
Methods for the estimation of the components
First of all, estimating the components of the time series is not trivial. There were several methods for extracting cycle-trend or seasonality components. For example, for extracting polynomial trend we can use a least squares approach.
If the only trend affects the time series, we can write
Y = a0 + a1 * t + a2 * t2 + … + an * tn.
The a values could be estimated using a least squares approach.
Suppose you have a monthly time series and you want to extract the seasonality. A possible way to do it, is to find a linear dependence between the current month and the previous ones. Hence, if you want to know the value for September 2017 the seasonal component might be
YSep 2017 = a0 + a1 * YSep 2016 + a2 YSep 2015.
In conclusion, altough the decomposition methods are not the state of art for statistical time series forecasting, they represent a milestone for the literature and cannot be ignored.