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Chapter 16 Handling uncertainty in ADAM

So far, when we discussed forecasts from ADAM models, we have assumed that the smoothing parameters and initial values are known, even though we have acknowledged in the Chapter 11 that they are in fact estimated. This is the conventional assumption of ETS models from Hyndman et al. (2008) (and it also applies to ARIMA models). However, in reality the parameters are never known and are always estimated in sample. This means that with the change of sample size, the estimates of parameters will inevitably change as well. This uncertainty will impact the model fit, the point forecasts and prediction intervals. In order to overcome this issue Bergmeir et al. (2016) proposed bagging - the procedure that decomposes time series using STL (Cleveland et al., 1990), then recreates many time series by bootstrapping the remainder then fits best ETS models to each of the newly created time series and combines the forecasts from the models. This way (as was explained by Petropoulos et al., 2018a) the parameters of models will differ from one generated time series to another, thus the final forecasts will handle the uncertainty about the parameters. In addition, this approach also covers the model uncertainty element, which was discussed in Section 15.4. The main issue with the approach is that it is computationally expensive and assumes that STL decomposition is appropriate for time series and that the residuals from this decomposition are independent.

In this chapter we focus on discussion of uncertainty about the estimates of parameters of ADAM models, starting from dealing with confidence intervals for them and ending with propagating the parameters uncertainty to the states and fitted values of the model. We start with the discussion of covariance matrix of parameters, then move to the confidence intervals construction. Finally, we discuss a method that allows capturing this uncertainty and use it for fitted values and forecasts of the model.


• Bergmeir, C., Hyndman, R.J., Benítez, J.M., 2016. Bagging exponential smoothing methods using STL decomposition and Box-Cox transformation. International Journal of Forecasting. 32, 303–312.
• Cleveland, R.B., Cleveland, W.S., McRae, J.E., Terpenning, I., 1990. STL: A seasonal-trend decomposition procedure based on loess. Journal of Official Statistics. 6, 3–73.
• Hyndman, R.J., Koehler, A.B., Ord, J.K., Snyder, R.D., 2008. Forecasting with Exponential Smoothing. Springer Berlin Heidelberg.
• Petropoulos, F., Hyndman, R.J., Bergmeir, C., 2018a. Exploring the sources of uncertainty: Why does bagging for time series forecasting work? European Journal of Operational Research. 268, 545–554.