1. Exponential smoothing methods with trend and seasonality
Charles Holt also introduced a forecasting method that accounted for seasonality as well as trend. It has since become known as the Holt-Winters' method as Holt's student, Peter Winters, showed how to do the calculations efficiently.
There are actually two versions of the Holt-Winters' method: the additive version and the multiplicative version.
2. Holt-Winters' additive method
Here are the equations for the additive version. They are similar to the equations for Holt's trend method, but with an additional term for the seasonal component, and an additional equation showing how the seasonal component evolves over time.
There is one more smoothing parameter to estimate, gamma, and several more state parameters to estimate to account for the initial seasonal pattern.
but instead of adding or subtracting seasonality, we use multiplication and division. Also, the seasonal component averages to one rather than zero.
In this additive version, the seasonal component averages to zero.
3. Holt-Winters' multiplicative method
but instead of adding or subtracting seasonality, we use multiplication and division. Also, the seasonal component averages to one rather than zero.
Notice in the first equation that the trend is still linear, but the seasonality is multiplicative. It is probably more instructive to look at an example.
4. Example: Visitor Nights
Here is a plot of the number of nights spent by visitors in Australian accommodation such as motels, hotels and guesthouses. The data is quarterly and has a strong seasonal pattern as you would expect. It is nicer to holiday in Australia in summer.
The hw function produces forecasts using the Holt-Winters' method. The seasonal argument controls whether we want additive or multiplicative forecasts.
In this example, it doesn't make much difference which we use as the seasonal variation is much the same over the whole series. In cases where the seasonal variation increases with the level of the series, we would want to use the multiplicative method.
5. Taxonomy of exponential smoothing methods
We have looked at several exponential smoothing methods, beginning with simple exponential smoothing, then methods for trended data, and finally methods for trended and seasonal data.
The ses function handled methods with no trend or seasonality.
6. Taxonomy of exponential smoothing methods
The holt function handled methods with trend, while hw provides forecasts that account for trend and seasonality.
In this table, we can think of these exponential smoothing methods as a family of methods where the trend component is either missing, additive or damped, and the seasonal component is either missing, additive or multiplicative.
There are actually also some other methods with multiplicative trend, but they tend to produce poor forecasts and I wouldn't recommend you use them.
There are nine possible methods in this table, but only six of them are available using the functions we have introduced so far. I haven't given you any way of fitting time series with seasonality, but no trend. We will consider them later.
7. Let's practice!
For now, go ahead and use hw to forecast time series assuming there is both trend and seasonality.