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:

The above equation consists of the following three components:

• $g_t$: A trend, as we have already seen, is a persistent increase or decrease in the data for a long period of time. There can be changes in the trend, but generally speaking, a trend will make it impossible for the series to converge to a value over time. In some textbooks, this term is called the trend-cycle component. In some other texts, the authors will use the word cycle separately. While it is a bit confusing, remember that the trend (or trend-cycle) component reflects a path. On the contrary, a cycle or cyclical component will reflect a temporal perturbation of the system.

• $s_t$: The seasonal component of the series reflects regular changes in the data over fixed periods of time. Such periods can be defined in terms of minutes, hours, days, months, or any other unit of time. For instance, in the case of our temperatures time series, we expect a seasonal component: Summers are warmer than winters. Not all series with a trend will have a seasonal component (and vice versa).

• $c_t$: The cyclical component is what remains after adjusting for trend and seasonality in the data. We can also call it residual. Typically, a cycle will present irregular ups and downs. This is the covariance-stationary process that we hope we can model after getting rid of $g_t$ and $s_t$.

The chart below shows a decomposition like the one we just described. It plots the UK’s electricity demand in the first three weeks of 2023.

Note that, in reality, we only observe $y_t$. The components are an abstraction that we infer from the data. In other words, we assume that the data can be broken down into these elements. Additionally, in the formulation above, we are also assuming that ...