## 3.7 STL decomposition

STL is a versatile and robust method for decomposing time series. STL is an acronym for “Seasonal and Trend decomposition using Loess”, while Loess is a method for estimating nonlinear relationships. The STL method was developed by Cleveland, Cleveland, McRae, & Terpenning (1990).

STL has several advantages over the classical, SEATS and X11 decomposition methods:

- Unlike SEATS and X11, STL will handle any type of seasonality, not only monthly and quarterly data.
- The seasonal component is allowed to change over time, and the rate of change can be controlled by the user.
- The smoothness of the trend-cycle can also be controlled by the user.
- It can be robust to outliers (i.e., the user can specify a robust decomposition), so that occasional unusual observations will not affect the estimates of the trend-cycle and seasonal components. They will, however, affect the remainder component.

On the other hand, STL has some disadvantages. In particular, it does not handle trading day or calendar variation automatically, and it only provides facilities for additive decompositions.

It is possible to obtain a multiplicative decomposition by first taking logs of the data, then back-transforming the components. Decompositions between additive and multiplicative can be obtained using a Box-Cox transformation of the data with \(0<\lambda<1\). A value of \(\lambda=0\) corresponds to the multiplicative decomposition while \(\lambda=1\) is equivalent to an additive decomposition.

The best way to begin learning how to use STL is to see some examples and experiment with the settings. Figure 3.4 showed an example of an STL decomposition applied to the total US retail employment series. Figure 3.15 shows an alternative STL decomposition where the trend-cycle is more flexible, the seasonal pattern is fixed, and the robust option has been used.

```
%>%
us_retail_employment model(STL(Employed ~ trend(window=7) + season(window='periodic'),
robust = TRUE)) %>%
components() %>%
autoplot()
```

The two main parameters to be chosen when using STL are the trend-cycle window `trend(window = ?)`

and the seasonal window `season(window = ?)`

. These control how rapidly the trend-cycle and seasonal components can change. Smaller values allow for more rapid changes. Both trend and seasonal windows should be odd numbers; trend window is the number of consecutive observations to be used when estimating the trend-cycle; season window is the number of consecutive years to be used in estimating each value in the seasonal component. Setting the seasonal window to be infinite is equivalent to forcing the seasonal component to be periodic `season(window='periodic')`

(i.e., identical across years). This was the case in Figure 3.15.

By default, the `STL()`

function provides a convenient automated STL decomposition using a seasonal window of `season(window=13)`

, and the trend window chosen automatically from the seasonal period. The default setting for monthly data is `trend(window=21)`

. This usually gives a good balance between overfitting the seasonality and allowing it to slowly change over time. But, as with any automated procedure, the default settings will need adjusting for some time series. In this case the default trend window setting produces a trend-cycle component that is too rigid. As a result, signal from the 2008 global financial crisis has leaked into the remainder component, as can be seen in the bottom panel of 3.4. Selecting a shorter trend window as in Figure 3.15 improves this.

### Bibliography

Cleveland, R. B., Cleveland, W. S., McRae, J. E., & Terpenning, I. J. (1990). STL: A seasonal-trend decomposition procedure based on loess. *Journal of Official Statistics*, *6*(1), 3–33. http://bit.ly/stl1990