Trends, Seasons, and Cycles

Time series decomposition

Trends can be quite a nuisance for time series modeling, as they make our data nonstationary. We know that taking the first differences of a series can sometimes solve the problem of integration. However, there are cases in which it will be worth breaking up the series into separate components, each with different characteristics. Implementations usually distinguish between the trend, the season, and the cycle.

Let $y_t$ be a time series. We can define it in terms of the following equation: