One of the fundamental purposes of the classical analysis of time series is to break down the series into its components, isolating them in order to study them better. Moreover, to be able to apply the stochastic approach to a time series, it is almost always necessary to eliminate the trend and the seasonality to have a steady process. As we have specified in the previous sections, the components of a time series are usually the following: trend, seasonality, cycle, and residual.

As already mentioned, they can be decomposed by an additive way:

Y(t) = τ(t) + S(t) + r(t)

They can also be decomposed by a multiplicative method:

Y(t) = τ(t) * S(t) * r(t)

In the following sections, we will look at how to derive these components using both these methods.