Seasonality and Random Determination

In this post we saw how the three components trend, seasonality and random of a time series. How to extract the trend was shown there, now we focus on how the seasonal component and the random component is determined.

Assume we have a detrended time series (we take here the AirPassengers time series and remove the trend). We assume a seasonality of a fixed period. In reality the assumption to have a fixed seasonality is too strict, as the period could shorten or change its structure over time. But under this assumption the determination of the seasonality is easy: To get the seasonal value of January, we take all values of January and build the average. This is the pattern we use for all periods.

The last step is to determine the random component $I$, we get it by simply removing the trend $T$ and seasonal component $S$ from the original time series $Y$, in an additive model this would be $I_t = Y_t - T_t - S_t$ and in a multiplicative model $I_t = Y_t / (T_t * S_t)$.
In our example, this is how the random components looks like:

What can we get out of it?
The random component shows the noise in the data, the values that do not fit the model. It helps to get a feeling how well the data is explained by the assumption to have a trend and a seasonality. The classical decomposition also could help to find outliers, which will show up with a high peek.

For completeness here again the whole picture holding all the steps discussed: