\( \newcommand{\mathbbm}[1]{\boldsymbol{\mathbf{#1}}} \)

Chapter 12 Multiple frequencies in ADAM

When we work with weekly, monthly, or quarterly data, we do not have more than one seasonal cycle. In this case, one and the same pattern can repeat itself only once a year. For example, we might see an increase in ski equipment sales over winter, so the seasonal component for December will typically be higher than the same component in August. However, we might see several seasonal patterns when moving to the data with higher granularity. For example, daily sales of the product will have a time of year seasonal pattern and a day of week one. If we move to hourly data, then the number of seasonal elements might increase to three: the hour of the day, the day of the week, and the time of year. Note that from the modelling point of view, these seasonal patterns should be called either “periodicities” or “frequencies” as the hour of the day cannot be considered a proper “season”. But it is customary to refer to them as “seasonality” in forecasting literature.

To correctly capture such a complicated structure in the data, we need to have a model that includes these multiple frequencies. In this chapter, we discuss how this can be done in the ADAM framework for both ETS and ARIMA. In addition, when we move to modelling high granularity data, there appear several fundamental issues related to how the calendar works and how human beings make their lives more complicated by introducing daylight saving-related time changes over the year. Finally, we will discuss a simpler modelling approach, relying on the explanatory variables (mentioned in Chapter 10).

Among the papers related to the topic, we should start with Taylor (2003a), who proposed an Exponential Smoothing model with double seasonality and applied it to energy data. Since then, the topic was developed by Gould et al. (2008), Taylor (2008), Taylor (2010), De Livera (2010), and De Livera et al. (2011). In this chapter, we will discuss some of the proposed models, how they relate to the ADAM framework and can be implemented.


• De Livera, A.M., 2010. Exponentially Weighted Methods for Multiple Seasonal Time Series. International Journal of Forecasting. 26, 655–657. https://doi.org/10.1016/j.ijforecast.2010.05.010
• De Livera, A.M., Hyndman, R.J., Snyder, R.D., 2011. Forecasting Time Series With Complex Seasonal Patterns Using Exponential Smoothing. Journal of the American Statistical Association. 106, 1513–1527. https://doi.org/10.1198/jasa.2011.tm09771
• Gould, P.G., Koehler, A.B., Ord, J.K., Snyder, R.D., Hyndman, R.J., Vahid-Araghi, F., 2008. Forecasting Time Series with Multiple Seasonal Patterns. European Journal of Operational Research. 191, 205–220. https://doi.org/10.1016/j.ejor.2007.08.024
• Taylor, J.W., 2010. Triple Seasonal Methods for Short-term Electricity Demand Forecasting. European Journal of Operational Research. 204, 139–152. https://doi.org/10.1016/j.ejor.2009.10.003
• Taylor, J.W., 2008. An Evaluation of Methods for Very Short-term Load Forecasting Using Minute-by-minute British Data. International Journal of Forecasting. 24, 645–658. https://doi.org/10.1016/j.ijforecast.2008.07.007
• Taylor, J.W., 2003a. Exponential Smoothing with a Damped Multiplicative Trend. International Journal of Forecasting. 19, 715–725. https://doi.org/10.1016/S0169-2070(03)00003-7