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Sensitivity analysis of the response characteristics of pattern search techniques applied to exponentially smoothed forecasting models Bitz, Brent William John

Abstract

The purpose of this study was to undertake a sensitivity analysis of selected input parameters of the pattern search-exponential smoothing forecasting system. The inputs subjected to the analysis were: 1) maximum number of pattern moves, 2) minimum step size, 3) pattern search step size, 4) step size reduction factor, 5) exponential smoothing constants (A, B and C). As the values of these input parameters were changed during the course of the analysis the resultant changes in certain criterion variables of the system were noted. These variables were: 1) forecast error standard deviation, 2) number of iterations (or pattern moves), 3) exponential smoothing constants (A, B and C). The three separate time series that were used in this study were furnished by the Frazer Valley Milk Producer's Association. The data series are composed of unit sales of fluid milk segregated according to container size, butterfat content and channel of distribution. Each of the time series analysed represents a different type of trend factor. One each for rising, falling and stable trend factors. The three time series were subjected to identical analytical procedures. The results were then compared across the three time series in order to determine if the response patterns of the pattern search system were sensitive to changes in the series trend. As measured by the response patterns of the criterion variables, the accuracy of the system is not influenced significantly by changes in the input parameters. Throughout the sensitivity analysis there developed a consistent pattern of minimal change in the forecast error standard deviation and the exponential smoothing constants. The search process was able to consistently reach very similar forecast error standard deviation values and exponential smoothing constant values, given the range of input values tested. The only dependent variable that experienced any marked change was the number of iterations. There does appear to be certain input values that minimize the number of iterations that the pattern search system needs, to arrive at solution values. Neither the maximum number of pattern moves nor the minimum step size exerted much of an effect on the size of the forecast error standard deviation or the "optimum values" for the exponential smoothing constants. However, changes in the minimum step size do affect the number of iterations the pattern search system makes before reaching a minimum forecast error standard deviation. If the minimum step size is decreased the number of iterations is increased. The opposite is also true, and if the minimum step size is increased, the number of iterations is decreased. Changes in the maximum number of pattern moves have no effect on the number of iterations. The pattern search system also appears to be unresponsive to changes in the pattern search step size. Neither the forecast error standard deviation nor the expotential smoothing, constant values can be improved through the use of different pattern search step sizes. The number of iterations is somewhat more responsive. Both large and small pattern search step size yield larger numbers of iterations than do middle values i.e. .10 - .20. Like the other inputs, the step size reduction factor, also does not elicit change in the results of the search process. Movements in the forecast error standard deviation and the exponential smoothing constants are small enough to be considered insignificant. Step size reduction factor values from .100 to .500 minimize the number of iterations, although within this interval there is little change. Larger values of the step size reduction factor tend to increase the number of iterations. There is little responsiveness in the pattern search system to changes in the initial values for the exponential smoothing constants. Between the three time series used, there is little consistency with regards to the effects of changes in the initial constant values on the number of iterations. The rising series benefits most from small values i.e. .250. The falling series benefited most with a middle value i.e. .500. The stable series reacted opposite to the falling one and benefited most with values at the extremes i.e. .250 and .750. One important finding is that most of the responsiveness of the pattern search system takes place before the first step size reduction. The bulk of all improvement in the forecast error standard deviation and the majority of all change in the exponential smoothing constants occurs in this first set of pattern moves. This is an important result as it explains the insensitivity of the search system to changes in the maximum number of pattern moves, the minimum step size and the step size reduction factor.

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