Using composite moving averages to forecast sales
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#2002 Operational Research Society Ltd. All rights reserved. 0160-5682/02 $15.00 www.palgrave-journals.com/jors
Using composite moving averages to forecast sales DJ Robb1* and EA Silver2 1 Department of Management Science & Information Systems, The University of Auckland, Auckland, New Zealand; and 2 Faculty of Management, The University of Calgary, Calgary, Alberta, Canada
Combining moving averages has been suggested as a simple and practical means to improve sales forecasting. Here we present a natural extension whereby combinations of all possible moving averages up to a given number of periods are employed. We evaluate the method’s performance relative to other methods, such as simple moving averages and exponentially-weighted moving averages, on two industrial data sets. Particular attention is placed on methods for selecting the number of periods employed, and on handling noisy data. Journal of the Operational Research Society (2002) 53, 1281–1285. doi:10.1057=palgrave.jors.2601440 Keywords: forecasting; time series; combinations of moving averages
Introduction In recent papers, Johnston et al1,2 showed that the average of two moving averages can provide better forecasts than using either one of the single moving averages. In this note we propose an extension, namely a combination of all possible moving averages up to N periods, ie equal weights of 1=N to a one-period average, a two-period average, . . . , an N-period average. Denoting the demand for the most recent period ðtÞ as dt , then an N-period ‘‘Composite Moving Average’’ (CMA:N) prescribes the weight for period t j þ 1 as: wj ¼
N 1 1P N k¼j k
j ¼ 1; . . . ; N
and the CMA:N forecast for period t þ 1 ð ftþ1 Þ is thus given by: ftþ1 ¼
N P
wj dtjþ1 ¼
j¼1
N P N d 1P tjþ1 N j¼1 k¼j k
Table 1 provides the weights for each period for N 4 5. These weights, with the desirable property of decreasing with the age of the data, are readily calculated using a spreadsheet (eg, using EXCEL’s OFFSET( ) function) and=or by using the simple backward recursion formulae wN þ1 ¼ 0;
wNj ¼ wN jþ1 þ
1 ; N ðN jÞ
j ¼ 0; 1; 2; . . . ; N 1 *Correspondence: DJ Robb, Department of Management Science & Information Systems, The University of Auckland, Private Bag 92019, Auckland, New Zealand. E-mail: [email protected]
Two commonly used simple forecasting methods are simple exponentially-weighted moving averages (SEWMA) and simple moving averages (SMA). The one-period-ahead forecasts for these methods are ftþ1 ¼ P j1 a 1 ð1 aÞ d (where a is the smoothing coeffitjþ1P j¼1 cient) and ftþ1 ¼ 1=N Nj¼1 dtjþ1, respectively. One may compare CMA with SEWMA and SMA by equating the average age of the data, viz, ðN þ 3Þ=4 for CMA (see derivation in Appendix 1), 1=a for SEWMA, and ðN þ 1Þ=2 for SMA (for which the average age is, as expected, smaller for any N > 1). For example, consider N ¼ 5, for which the average age using CMA is 2 periods. A SEWMA with the same average age has a ¼ 4=ðN þ 3Þ ¼ 0:5, with resulting weights of 0.50, 0.25, 0.13, 0.06, 0.03, 0.02, 0.01
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