Forecasting seasonals and trends by exponentially weighted moving averages

doi:10.1016/j.ijforecast.2003.09.015

International Journal of Forecasting

Volume 20, Issue 1, January–March 2004, Pages 5-10

https://doi.org/10.1016/j.ijforecast.2003.09.015 Get rights and content

Abstract

The paper provides a systematic development of the forecasting expressions for exponential weighted moving averages. Methods for series with no trend, or additive or multiplicative trend are examined. Similarly, the methods cover non-seasonal, and seasonal series with additive or multiplicative error structures. The paper is a reprinted version of the 1957 report to the Office of Naval Research (ONR 52) and is being published here to provide greater accessibility.

Introduction

An exponentially weighted moving average is a means of smoothing random fluctuations that has the following desirable properties: (1) declining weight is put on older data, (2) it is extremely easy to compute, and (3) minimum data is required. A new value of the average is obtained merely by computing a weighted average of two variables, the value of the average from the last period and the current value of the variable. This paper utilizes these desirable properties both to smooth current random fluctuations and to revise continuously seasonal and trend adjustments. These may then be extrapolated into the future for forecasts. The flexibility of the method combined with its economy of computation and data requirements make it especially suitable for industrial situations in which a large number of forecasts are needed for sales of individual products.

The simplest application of an exponentially moving average would be to the following stochastic process. Consider the problem of making an expected value forecast of a random variable whose mean changes between successive drawings. The following rule might be proposed: take a weighted average of all past observations and use this as your forecast of the present mean of the distribution, i.e., $S ̄_{t} =B[S_{t} +AS_{t−1} +A^{2} S_{t−2} +A^{3} S_{t−3} +A^{4} S_{t−4} +…⌋⌋$ where B is a constant between 0 and 1, A is (1−B), the S's are observations of the variable and the t subscript indicates the time ordering of the observations. S̄_t is the estimate of the expected value of the distribution, ES_t. If the distribution mean is subject to large changes, A should be small so as to quickly attenuate the effect of old observations. However, if A is too small, S̄_t is subject to so much random variation that it would be a poor estimator of the mean. The following relation is convenient in minimizing computations: $S ̄_{t} =BS_{t} +(1−B) S ̄_{t−1}$ A weighted moving average with exponential weights is clearly a sensible mode of behavior in dealing with this simple forecasting problem. An exploration of the exact conditions under which this behavior is optimal will not be considered, but rather the question of whether this approach to forecasting seems to hold promise in coping with trends and seasonals in forecasting.

Section snippets

Forecasting ratio seasonals

Let S_t=sales in period t and let S̄_t=smoothed and seasonally adjusted sales rate in period t. This is an estimate of ES̄_t. P_t=periodic (seasonal) adjustment ratio for the tth period. This is an estimate of ES̄_t/ES_t.

The sales rate is obtained by combining the current seasonally adjusted sales with the sales rate from the previous period $S ̄_{t} =AP_{t} S_{t} +(1−A) S ̄_{t−1}$ where the constant, A, determines how fast the exponential weights decline over the past consecutive periods. $0≤A≤1.$

The current seasonal

Forecasting a ratio trend

In order to explore the application of the exponentially weighted moving average to forecasting a trend, we will first consider the simplest case in which there is no seasonal fluctuation. The sales rate is obtained by combining the current sales with the sales rate from the previous period corrected for trend. $S ̄_{t} =AS_{t} +(1−A)R_{t} S ̄_{t−1},$ where R_t is the trend adjustment ratio for the tth period. This is an estimate of ES̄_t/ES_t−1. Note the implicit assumption that the trend has a constant percentage

Forecasting a ratio trend and seasonals

The sales rate is obtained by combining the current sales adjusted for seasonal with the sales rate of the previous period adjusted for trend. $S ̄_{t} =AS_{t} P_{t} +(1−A)R_{t} S ̄_{t−1} .$ Each seasonal ratio is revised every N periods as before in Eq. (2). The trend ratio is revised each period as before in Eq. (8). Now substituting , in Eq. (12) we obtain: $S ̄_{t} = A(1−B) 1−AB−(1−A)C S_{t} P_{t−N} + (1−A)(1−C) 1−AB−(1−A)C S ̄_{t−1} R_{t−1} .$ Substituting Eq. (13) in Eq. (2) yields: $P_{t} = [1−B][1−(1−A)C] 1−AB−(1−A)C P_{t−N} + B(1−A)(1−C) 1−AB−(1−A)C S ̄_{t−1} S_{t} R$

Forecasting a linear trend and additive seasonals

We consider the case of both trend and seasonal, but as before either of these can be worked out separately. The sales rate is obtained from the following relation: $S ̄_{t} =a(S_{t} +P_{t})+(1−a)(S ̄_{t−1} +r_{t}).$ where P_t=periodic (seasonal) adjustment increment for the tth period. This is an estimate of ES̄_t−ES_t. r_t=trend adjustment increment for the tth period. This is an estimate of $E S ̄_{t} −E S ̄_{t−1} .$ Note that the lower case letters are used to denote the additive case while upper case letters denote the

Forecasting a linear trend and ratio seasonals

The sales rate is estimated by the following relation: $S ̄_{t} =AS_{t} P_{t} +(1−A)(S ̄_{t−1} +r_{t})$ The seasonal rate is estimated by Eq. (2), and the trend increment is estimated by Eq. (19). Substituting these in Eq. (24) yields the formula for calculating S̄_t: $S ̄_{t} = A(1−B) 1−AB−(1−A)c S_{t} P_{t−N} + (1−A)(1−c) 1−AB−(1−A)c [S ̄_{t−1} +r_{t−1}].$ Values of P_t and r_t may then be calculated numerically by substitutions in , , respectively. Forecasts for sales T periods in the future would be obtained by the formula: $ES_{t+T} = S ̄_{t} +r_{t} T P_{t+T−N}$

Comments on the theory

The foregoing derivations illustrate the flexibility of exponentially weighted moving averages in forecasting. They make possible a simple integrated approach to the estimation of both trends and seasonals.

The underlying stochastic theory has not been explored here. A great deal of work has been done on optimal filter design for stationary time series using the criteria of minimizing the sum of the square of the errors.²

Conclusion

This exploratory analysis indicates the great flexibility of exponentially weighted moving averages in dealing with forecasts of seasonals and trends. Further study seems fully justified both on empirical and theoretical levels.

Biography: Charles C. HOLT is Professor of Management Emeritus at the Graduate School of Business, University of Texas at Austin. His current research is on quantitative decision methods, decision support systems, and financial forecasting. Previously he has done research and teaching at M.I.T., Carnegie Mellon University, the London School of Economics, the University of Wisconsin, and the Urban Institute. He has been active in computer applications since 1947, and has done research on

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