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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 will repeat itself only once a year. For example, we might see an increase in sales of ski equipment over Winter, thus the seasonal component for December will be typically higher than the same component in August. However, when we move to the data with higher granularity, we might see several seasonal patterns. For example, daily sales of product will not only have time of year seasonal pattern, but also the day of week one. If we move to hourly data, then the number of seasonal elements might increase to three: hour of day, day of week and 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 day cannot be considered as a proper “season.” But it is customary to refer to them as “seasonality” in forecasting literature.

In order to capture such complicated structure in the data correctly, we need to have a model that includes these multiple frequencies in it. In this chapter, we discuss how this can be done in 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 calendar works and how human beings make their lives more complicated by introducing daylight saving related time changes over the year. Finally, we move to the discussion of a simpler approach, relying on the explanatory variables (discussed in Chapter 10).

Among the papers related to the topic, we should start with James W. 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.
• 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.
• 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.
• Taylor, J.W., 2010. Triple seasonal methods for short-term electricity demand forecasting. European Journal of Operational Research. 204, 139–152.
• 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.
• Taylor, James W., 2003a. Exponential smoothing with a damped multiplicative trend. International Journal of Forecasting. 19, 715–725.