Last time we saw how to do automated model selection using the ES function from the smooth package. Now I want to show how to produce combined forecasts from ETS.

Why bother?

There is a vast body of literature on forecast combinations (read this great review). The main idea is that you should not put all your eggs in one basket — the safer strategy is to combine forecasts from different models instead of selecting just one. Yes, it is more computationally expensive, but the trade-off is higher accuracy on average.

For ETS, a great solution was proposed by Stephan Kolassa in his 2011 paper: extract AIC values, calculate AIC weights (giving the highest weight to the best-performing model and lower ones to the rest), then combine the forecasts. The resulting forecasts tend to be more robust, because in practice it might be hard to tell the difference between, for example, ETS(M,A,M) and ETS(M,Md,M). So why choose one when you can have all? I implemented this mechanism in the smooth package for R years ago, and now it is also available in Python.

Here is how it works on an example using an M3 time series. I picked this specific one because it is seasonal, but the trend is not very well pronounced. The series is shown in the first image.

from smooth import ES
from fcompdata import M3

series = M3[1687]
y = series.y
freq = series.period

# Fit ETS models, combine forecasts
model = ES(model="CXC", lags=series.period, h=18, holdout=True)
model.fit(y)
model.predict(h=18)

The code above tells ES to fit all ETS models with additive and no trend (“X” in the middle), calculate AIC weights, produce forecasts from each one of them, and then combine them. The resulting point forecast is the weighted combination of the individual forecasts. If a prediction interval is required, the specific quantiles are combined directly (see the paper by Lichtendahl et al., 2013). This is inevitably slower than the default model selection mechanism, but is a safer approach. The point forecast and the prediction interval (grey lines) are shown in the attached image.

Note that the user can regulate the pool of combined models via the “model” parameter of the function. This wiki explains all the accepted options.

So why not go ahead and try it yourself, and see how it works for your data?

🔗 Install smooth: pip install smooth
📖 More on forecasts combination in ADAM.