1. Exponential smoothing methods with trend
Simple exponential smoothing works fine provided your data have no trend or seasonality.
2. Holt's linear trend
But we need to add some more features to it in order to handle these characteristics. First let me remind you of the equations we used for simple exponential smoothing. The first equation gives the forecasts as the last value of the estimated level. The second equation describes how the level changes over time as a function of the most recent observation, and the previous estimate of the level.
To deal with trend, we need to add a trend component to the equations.
The forecast equation is now a linear function of the forecast horizon. So it gives trended forecasts with slope equal to b_T.
The level is similar to what it was before, but we adjust it slightly to allow for the fact that the data are now trended.
And we add a third equation describing how the slope changes over time.
Because we allow the slope to change over time, this is often called a "local linear trend".
The beta* parameter controls how quickly the slope can change.
3. Holt's linear trend
A small beta* value means that the slope hardly changes, and so the data will have a trend that is close to linear throughout the series. A large beta* value means the slope changes rapidly, allowing for highly nonlinear trend.
We use beta* here rather than beta, because we will use beta later on.
There are now four parameters to estimate: the smoothing parameters alpha and beta*, and the state parameters l0 and b0. The Holt function will handle this for you.
4. Holt's method in R
This method is named after Charles Holt who developed it in the 1950s while working on forecasting for the US navy.
Here is an example applying the method to total air passenger traffic in Australia. With only 10 years of data, I am restricting my forecast to 5 years ahead. Like the ses function, holt will both estimate the parameters and compute the forecasts. The returned object contains information about the parameters, the forecasts and prediction intervals. Holt's method will produce forecasts where the trend continues at the same slope indefinitely into the future.
5. Damped trend method
A variation on this method is to allow the trend to dampen over time, so that it levels off to a constant value.
This is called the damped trend method, and is due to Ev Gardner from Texas, and Eddie McKenzie from Scotland. They proposed a variation of Holt's method, with one extra parameter phi to control the damping. The larger the value of phi, the less damping there is, with phi equals 1 being equivalent to Holt's method. Under this method, the short-run forecasts are trended, but the long-run forecasts are constant.
6. Example: air passengers
An example will help illustrate the idea.
Here we are forecasting 15 steps ahead using only 15 observations, which is not to be recommended, but it helps show the difference between the two methods.
To use the damped trend method, just add damped equals TRUE to the holt function.
As you can see, the damped trend method levels off, while the linear trend method continues at the same slope for all future periods.
The parameter $\phi$, which controls the damping, is estimated along with all the other parameters by the holt function.
So now we have two approaches to forecasting data with trend, one which uses local linear trends, and one which uses damped linear trends.
7. Let's practice!
It's time for another hands-on exercise, forecasting with the holt function.