Forecasting intraday time series with multiple seasonal cycles using parsimonious seasonal exponential smoothing
Introduction
This paper is concerned with univariate forecasting methods for time series recorded at an intraday frequency. It is common for such data to exhibit seasonal cycles within the week and also within the day. Examples of the type of series of interest are the two samples of half-hourly electricity load presented in Fig. 1. Both series show a repeating intraweek cycle of length m2=336 periods and also, at least for the weekdays, there is similarity between the intraday patterns of length m1=48 periods. The univariate methods that we consider aim to capture this feature of ‘double seasonality’. Although electricity load is often modelled in terms of weather variables [1], the use of univariate methods is common for lead times less than half a day, and for longer horizons in locations where weather data is not available. Other examples of intraday data exhibiting double seasonality are call centre arrivals, hospital admissions, transportation and Internet traffic [2], [3], [4]. In these applications, explanatory variables are often not available, and so univariate forecasting methods are required.
A variety of univariate methods have been proposed for forecasting intraday data. These include exponentially weighted regression [5]; seasonal ARMA [6]; artificial neural networks [7]; dynamic harmonic regression [2]; vector autoregression [8]; random effects models [9] and an approach based on singular value decomposition [10]. In many applications with intraday data, the forecasting method must be implemented within an automated system. A class of univariate methods that have been widely used in automated applications due to their robustness, intuitiveness and accuracy is exponential smoothing [11], [12]. Indeed, exponential smoothing methods have performed well in empirical studies with intraday data [13], [14]. This motivates us to develop, in this paper, the use of exponential smoothing for intraday data.
Two exponential smoothing methods have been presented in the literature for intraday data. Taylor's [15] adaptation of the Holt-Winters method involves the smoothing of an intraday cycle and intraweek cycle. We term this ‘HWT exponential smoothing’. Although empirical results for the method have been encouraging [13], [14], it has two features that are rather unappealing. Firstly, the method uses a common intraday cycle for all days of the week. Secondly, the method can be viewed as being of high dimension because the method's smoothing of an intraday cycle and an intraweek cycle requires the initialisation and updating of a large number of terms. Gould et al. [4] address these issues by introducing an exponential smoothing method that models the intraweek and intraday cycles through the use of a different intraday cycle for each distinct type of day of the week. For example, if the pattern of demand on Saturdays is different to the rest of the week, it is allocated its own intraday cycle. If all weekdays can be assumed to be the same, a common intraday cycle can be used for these five days. The result is a method with lower dimensionality than the HWT method. Due to its focus on the intraday cycle, the method has been termed ‘intraday cycle (IC) exponential smoothing’.
A limiting feature of IC exponential smoothing is that it allows only whole days to be treated as identical. We would argue that it often makes more sense to assume that just parts of days are identical. For example, it could be that during daylight hours the series differs on each day of the week, but that the pattern during night hours can be treated as identical on all days of the week. In this paper, we propose a new exponential smoothing formulation that has the flexibility to allow parts of different days of the week to be treated as identical. The new method is of lower dimension than the IC method. In view of this, we term the method ‘parsimonious seasonal exponential smoothing’. The HWT and IC methods can be viewed as special cases of this new method. A simple version of the method is briefly discussed by Hyndman et al. [11, Section 3.5.5], but our study is the first to consider it formally and to develop the method for modelling intraday data.
In this paper, we present two empirical studies. The first involves two series of electricity load; one for Great Britain and the other for France. Automated electricity demand prediction is required for the control and scheduling of power systems. The liberalisation of electricity markets has meant that load forecasting is also important for market participants to support energy transactions [16], [17]. We use the two load series to illustrate the implementation of methods in the earlier sections of the paper before using the series to compare the forecast accuracy of the various methods. The series in Fig. 1 are samples of the two electricity load series. The full series spanned the three-year period from Thursday 1 January 2004 to Sunday 31 December 2006, inclusive. We use the first two years of data for estimation of method parameters, and the remaining year for post-sample evaluation. As the estimation sample contains just two years of data, we make no attempt to model annual seasonality. Both of our empirical studies focus on lead times up to one day-ahead.
The second empirical study focuses on a half-hourly series of the number of calls arriving at a call centre of an Israeli bank. The efficient scheduling of call centre staff relies on accurate forecasts for the number of calls [18]. Although causal models are sometimes used for modelling call arrivals [19], [20] it is univariate methods that tend to be used for short-term forecasting of intraday call arrivals, which is our focus in this paper. The Israeli bank's call centre is open for a shorter duration at the weekends than on weekdays. This situation, where the number of periods on each day of the week is not the same, cannot be satisfactorily addressed with the HWT and IC methods. By contrast, it can be modelled in a straightforward way using parsimonious seasonal exponential smoothing. Our call centre forecasting study considers lead times up to one day-ahead.
In each of the three time series used in this paper, we identified a number of unusual days, termed ‘special’ days, on which the pattern of observations differed considerably from the regular seasonal pattern. These were either public holidays or days adjacent or close to public holidays. Online prediction systems are typically replaced for these days by forecasts prepared offline. As the methods considered in this paper are appropriate for automated online prediction, prior to fitting the methods, we smoothed out the special days. This smoothing used the average of the corresponding periods in the two adjacent weeks. In our post-sample evaluation of the forecasting methods, we did not include the special days occurring in the evaluation period.
In Section 2, we review the HWT and IC exponential smoothing methods. In Section 3, we introduce our new parsimonious seasonal exponential smoothing method. Section 4 uses the load data to compare post-sample forecast accuracy for the exponential smoothing methods, as well as ARMA modelling and an artificial neural network approach. In Section 5, we present our empirical study of intraday call centre arrivals. Section 6 summarises the paper, and provides concluding comments.
Section snippets
HWT exponential smoothing
Taylor's [15] HWT exponential smoothing extends the Holt-Winters method in order to accommodate the intraday and intraweek cycles in intraday data. It is presented in the following expressions:where lt is the smoothed level; dt is the seasonal index for the intraday seasonal cycle; wt is the seasonal index for the intraweek seasonal cycle that remains
Parsimonious seasonal exponential smoothing
A limiting feature of IC exponential smoothing is that it allows only whole days to be treated as identical. We would argue that it often makes more sense to allow parts of days to be treated as the same. For example, in the IC method of Section 2.2, a distinct intraday cycle was used for Fridays. However, for the British data, Fig. 2 shows that the average Friday pattern of load prior to about 11 am seems to be very similar to that on Tuesday, Wednesday and Thursday, and so it seems inefficient
Evaluation of forecasts for the electricity load time series
In this section, we use the two load series to compare the post-sample forecast accuracy of the exponential smoothing methods and several benchmark methods. Estimation was performed once using the first two years of the data, and the forecast origin was then rolled forward through the one-year post-sample evaluation period to produce a collection of forecasts for lead times up to a day ahead. Before evaluating forecast accuracy, we first describe the estimation of the exponential smoothing
The call centre times series
In this section, we consider the Israeli bank data introduced and analysed by Brown et al. [28], and also used by Taylor [14]. This data is taken from a relatively small call centre that is open from 7 am to midnight on weekdays (Sunday–Thursday in Israel), from 7 am to 2 pm on Fridays and from 8 pm to midnight on Saturdays. Our analysis focuses on the single time series of total arrivals minus Internet assistance calls for the period 1 August to 25 December 1999, inclusive, with the first 14 weeks
Summary and concluding comments
In comparison with the HWT method, the IC method has the appeal that it involves the initialisation and updating of fewer terms. This dimension reduction is essentially achieved by allowing the same intraday cycle to be used for days that can be assumed to be identical. However, an unappealing feature of the IC method is that it allows only whole days to be treated as identical. It often makes more sense to assume that just parts of days are identical. In this paper, we have introduced a new
Acknowledgements
We would like to thank Matthew Roberts of National Grid for supplying the British load data, and RTE for making the French load data available on their website. We acknowledge Avi Mandelbaum for making available the Israeli bank data. We are also grateful for the helpful comments of two referees.
References (30)
Doing something about the weather
Omega
(2008)- et al.
An unobserved component model for multi-rate forecasting of telephone call demand: the design of a forecasting support system
International Journal of Forecasting
(2002) - et al.
Forecasting time-series with multiple seasonal patterns
European Journal of Operational Research
(2008) - et al.
Forecasting the short-term demand for electricity—Do neural networks stand a better chance?
International Journal of Forecasting
(2000) Exponential smoothing: the state of the art—Part II
International Journal of Forecasting
(2006)- et al.
A comparison of univariate methods for forecasting electricity demand up to a day ahead
International Journal of Forecasting
(2006) - et al.
Efficient frontiers for electricity procurement by an LDC with multiple purchase options
Omega
(2006) On time series data and optimal parameters
Omega
(2004)- et al.
Financial crises and bank failures: a review of prediction methods
Omega
(2010) - et al.
Quantifying costs of forecast errors: a case study of the warehouse environment
Omega
(2009)
Exponentially weighted methods for forecasting intraday time series with multiple seasonal cycles
International Journal of Forecasting
Short-term hourly traffic Forecasts using Hong Kong Annual Traffic Census
Transportation
Short-term load forecasting using general exponential smoothing
IEEE Transactions on Power Apparatus and Systems
The time series approach to short-term load forecasting
IEEE Transactions on Power Systems
Bayesian modeling and forecasting of intraday electricity load
Journal of the American Statistical Association
Cited by (63)
Research areas and methods of interest in European intraday electricity market research—A systematic literature review
2024, Sustainable Energy, Grids and NetworksForecasting: theory and practice
2022, International Journal of ForecastingCitation Excerpt :The first methods capable of capturing multiple seasonality were evaluated by Taylor (2008) and included double seasonal exponential smoothing (Taylor, 2003b) and multiplicative double seasonal ARMA (SARMA). Since then several advanced time series methods have been developed and evaluated (De Livera et al., 2011; Taylor, 2010; Taylor & Snyder, 2012), including artificial neural networks (Li, Huang, & Gong, 2011; Millán-Ruiz & Hidalgo, 2013; Pacheco, Millán-Ruiz, & Vélez, 2009) and models for density forecasting (Taylor, 2012). Another family of models relies on the assumption of a time-inhomogeneous Poisson process adopting fixed (Brown et al., 2005a; Jongbloed & Koole, 2001; Shen & Huang, 2008a; Taylor, 2012) and mixed modelling (Aldor-Noiman, Feigin, & Mandelbaum, 2009; Avramidis et al., 2004; Ibrahim & L’Ecuyer, 2013) approaches to account for the overdispersed nature of the data, and in some cases, interday and intraday dependence.
Optimal selection of heterogeneous ensemble strategies of time series forecasting with multi-objective programming
2021, Expert Systems with ApplicationsForecasting IT security vulnerabilities – An empirical analysis
2020, Computers and SecurityLong term load projection in high resolution for all countries globally
2019, International Journal of Electrical Power and Energy Systems