So far we have dealt with time series according to a classic approach to the topic. In this perspective, the classic models that try to simulate the phenomenon can be of two types:

• Composition models: The elementary components are known, and, by assuming a certain form of aggregation, the resulting series is obtained
• Decomposition models: From an observed series is hypothesized the existence of some elementary trends of which we want to establish the characteristics

The decomposition models are the most used in practice, and, for this reason, we will analyze them in detail.

The components of a time series can be aggregated according to different types of methods:

• Additive method: Y(t) = τ(t) + C(t) + S(t) + r(t)
• Multiplicative method: Y(t) = τ(t) * C(t) * S(t) * r(t)
• Mixed method: Y(t) = τ(t) * C(t) + S(t) * r(t)

In these formulas, the factors are defined as follows:

• Y(t) represents the time series
• τ(t) represents the trend component
• C(t) represents the cyclic component
• S(t) represents the seasonality component
• r(t) represents the residual component

The multiplicative model can be traced back to the additive model through a logarithmic transformation of the components of the series:

Y(t) = τ(t) * C(t) * S(t) * r(t)

This formula, by applying the logarithm function to all factors, becomes:

lnY(t) = lnτ(t) + lnC(t) + lnS(t) + lnr(t)