Foregasting with Abaptive Filtering

— During the past decade Régression Analysis has gained wide acceptance as a method for preparing medium and long range forecasts for time series. However/for a shortterm forecasting situation or when the number of observations is small, régression analysis is costly and of ten impractical. Exponential smoothing is the forecasting method most of ten used in these latter situations, but it has some major shortcomings ioo. Rather than trying to distinguish some underlying pattern from the noise (randomness) includedin observed data, exponential smoothing simpfy « smooths » the extreme values in preparing a forecast, which in many cases is not completely suitable. Thus there are a number of medium range forecasting situations and cases for which not much data is available where neither régression analysis nor exponential smoothing methods are appropriate. This paper briefly examines the gênerai class of forecasting methods that are based on a weighting of past observations and then présents the theoretical and practical aspects of adaptive filtering, a method for determining an appropriate set ofweights. Adaptive Filtering* a technique prevlously developed in télécommunications engineering, is attractive in many forecasting situations involving time series because it does discriminate between noise and an underlying pattern, it is conceptually appealing and easy to apply, it can be used with a relatively small amount of data, and the accuracy and reliability of its forecasts compare very favorably with other techniques. Some Existing Techniques for Forecasting There are numerous situations which arise in the opération of a business that require the development of a forecast for a time series. One of the most common of these involves the area of production scheduling and inventory control. In order to control out-of-stock costs and keep inventory costs within reason, firms must forecast demand for individual products and groups of products and then use those forecasts in making production décisions. Similarly, in the areas of finance, budgeting and marketing, forecasts must be prepared for working capital, cash flow, prices and other time series. While most of these situations involve short or medium term forecasts, firms also are faced with requirements for longer term projections in areas such as capacity utilization, capital requirements, and market growth. (1) Harvard Business School, Boston, Massachusetts. (2) INSEAD, Fontainebleau, France. Revue Française d* Automatique, Informatique et Recherche Opérationnelle n° mars 1973, V-l. 32 S. WHEELWRIGHT ET S. MAKRIDAKIS In order to meet these forecasting requirements, a number of methods have been developed for managers. These have been adopted to varying degrees, based largely on the manager's évaluation of their accuracy, their cost, and his ability to understand what they actually do (*). The majority of these methods are based on the idea that past observations contain information about some underlying pattern of the time series. The purpose of the forecasting method is then to distinguish that pattern from any noise (randomness) that also may be contained in past observations and then to use that pattern to predict future values in the series. A gênerai class of widely used forecasting methods that attempts to deal with both causes of fluctuations in a time series is that of smoothing. Spécifie techniques of this type assume that the extreme values in a series represent the randomness and thus by « smoothing » these extrêmes, the basic pattern can be identified. The two methods of this type that are used most often are moving averages and exponential smoothing. The technique of moving averages consists of taking the n most recent observations of a time series, finding the average of those values, and using that average as a forecast for the next value in the series. That is (), Jf+l = [ * * + **-l + . . +*,-(„-!)] where st +1 = the moving average forecast for period t + 1 based on the previous n observations n = the number of observations included in the average x% = the observed value in period i (i = 1, 2,... t). This approach to short term forecasting is referred to as moving averages because n is held constant and for each new forecast, t is incremented by 1 and the average is recomputed by dropping the oldest observation and picking up a new observation. The value of n détermines how much ofthe fluctuations in observed values is carried into the smoothed value, st+i : a larger value of n giving a more smoothed forecast than a smaller value of n. A major drawback of moving averages is that it assigns equal weight to each of the past n observations and no weight to observations before that. It can often be argued that the most recent observations in a series contain more information than the older values. Following this line of reasoning, many managers have adopted the technique of exponential smoothing which gives decreasing importance (smaller weights) to older observations. (1) As has become evident during the past few years, the ease with which a manager can understand a forecasting method is a major factor in determining its use in practice. (2) The notation used throughout this paper is that lower case letters represent scalar quantities and upper case letters represent vectors. The only exception to this is that q> is used to represent a single cross corrélation, $(x, d) is used to represent a vector, and [o(x, x)] is used to represent a matrix of these coefficients. Finally, where the range of values for a summation index is not given, it is from t — n + 1 to t. Revue Française d'Automatique* Informatique et Recherche Opérationnelle FORECASTING WITH ADAFITVE FILTERING 33 Exponential smoothing can be described mathematieally as st+t = oexf + (l —'a)st where st+ x = the exponentially smoothed value to be used as a forecast for period t + 1 a =s the smoothing constant (0 ^ a ^ 1) Xi = the observed value in period i (i — 1,2,... f). This gênerai équation can be expanded by replacing st with its computed value. Carrying out this expansion gives st+t = axt + a(l — <x)xt-x + a(l — a) x*__2 + (* — «)**-3 + ... From this expanded form it can be seen that since a is between ö and 1, decreasing weights are being given to older observations and the size of oc détermines the relative value of these weights. A larger a (close to 1) gives most of the weight to very recent observations whereas a small a (close to 0) does not give much weight to any single observation, thus giving a much more smoothed value for st+1. Exponential smoothing has been widely used by managers because it is easy to understand, inexpensive to apply and intuitively appealing because the manager has some control over the weights through assignment of a value for ou Ho wever» a major drawback of this method is that there is no easy way to détermine the most appropriate value of a. Some work on this problem has been done under the title of adaptive smoothing, aimed at examining alternative rules that might be used to détermine when and by how much the value of a should be varied [1]. Another author, Brown, has also looked at this problem and has developed rules that can be used to trade off the cost of variance in the forecast with the cost of response time to changes in the underlying pattern [2). To further improve on this smoothing technique, higher forms of exponential smoothing have been developed. These higher forms can handle time series models other than the constant model assumed in simple exponential smoothing. (For example, double exponential smoothing assumes a trend model.) However, even with these additions, exponential smoothing is still not completelya dequate in many forecasting situations because it does smooth the observed values rather than explicitly looking for the underlying pattern. An approach to forecasting that is based on a weigthing of past observations but avoids some of the weaknesses of exponential smoothing is polynomial fitting. (Although this method has only been widely used in the area of satellite tracking, it will be discussed briefly here because it illustrâtes the relationship between smoothing techniques and adaptive filtering.) The method of polynomial fitting consists of taking the n + 1 most recent observations and fitting