International Journal of Forecasting 26 (2010) 652–654 www.elsevier.com/locate/ijforecast Discussion Exponentially weighted methods for forecasting intraday time series with multiple seasonal cycles: Comments Haipeng Shen Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, United States First, Professor Taylor ought to be congratulated on another nice piece of work about exponential smooth- ing for intraday time series with multiple seasonal cy- cles. The methods discussed can be roughly grouped into two categories, depending on how one views the time series to be forecasted. Below I would like to briefly discuss the two perspectives, as well as some connection with functional data analysis and func- tional time series forecasting. Similar time series exist in multiple applications, including electricity demand and hospital emergency room patient arrivals, beside the call center volumes analyzed in the paper. A distinguishing feature of this type of data that needs to be incorporated by a suc- cessful forecaster is the multi-seasonality. For exam- ple, the call center arrival volume data contain both an intraday cycle and an intraweek cycle. A common way of analyzing such data is to view the data as a “long” univariate time series with double seasonality. This is the approach taken by the first four exponen- tially weighted methods of Taylor (2010), as well as the methods of Taylor (2008). Alternatively, one could view the basic cycle (i.e. each day) as the basic data unit, and split the univari- ate time series into daily segments. The sequence of E-mail address: haipeng@email.unc.edu. the resulting segments then forms a “fat” multivari- ate (or vector) time series. This formulation naturally separates the intraday cycle from the intraweek (or other longer) cycle, and allows one to model the in- traday dependence and the interday (time series) de- pendence separately. This viewpoint connects nicely with the relatively new area of functional data analy- sis, which treats functions or curves as the basic data units and aims to understand the characteristics of pop- ulations of curves. (Ramsay & Silverman, 2005, offer a comprehensive survey of the related methodologies and applications.) As a visual illustration, Figs. 1 and 2 offer this mul- tivariate (or functional) view of the data depicted in Figures 1 and 2 of Taylor (2010). In these figures, each curve plots the arrival volumes of the 48 half-hour in- tervals for one day, and different colors and line types are used to indicate the seven days of the week. Several insightful observations can already be drawn from this graphical view: (1) the intraday arrival patterns tend to be similar for the same weekday; (2) the patterns vary for different weekdays, and can be roughly grouped into four clusters: Monday, Tuesday–Friday, Saturday, and Sunday. Incidentally, these are the four clusters identified by Taylor (2010). (One can also treat Fri- days as a fifth cluster. A formal statistical test for the significance of the clusters can be carried out using the 0169-2070/$ - see front matter c  2010 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ijforecast.2010.05.011