Exponential Smoothing Models

Exponential Smoothing Models

Alternative Moving Average Forecasting Models

To analyze the accuracy of these models, we can define error metrics, which compare quantitatively the forecast with the actual observations. Three metrics that are commonly used are the mean absolute deviation, mean square error, and mean absolute percentage error. The mean absolute deviation (MAD) is the absolute difference between the actual value and the forecast, averaged over a range of forecasted values:

where At is the actual value of the time series at time t, Ft is the forecast value for time t, and n is the number of forecast values (not the number of data points since we do not have a forecast value associated with the first k data points). MAD provides a robust measure of error and is less affected by extreme observations.

Mean square error (MSE) is probably the most commonly used error metric. It penalizes larger errors because squaring larger numbers has a greater impact than squaring smaller numbers. The formula for MSE is:

Again, n represents the number of forecast values used in computing the average. Sometimes the square root of MSE, called the root mean square error (RMSE), is used.

Table 7.2 Error Metrics for Moving Average Models of Burglary Data

k = 2 k = 3 3-Period Weighted

MAD 13.63 14.86 13.70

MSE 254.38 299.84 256.31

MAPE 23.63% 26.53% 24.46%

A third commonly used metric is mean absolute percentage error (MAPE). MAPE is the average of absolute errors divided by actual observation values.

The values of MAD and MSE depend on the measurement scale of the time-series data. For example, forecasting profit in the range of millions of dollars would result in very large MAD and MSE values, even for very accurate forecasting models. On the other hand, market share is measured in proporti The values of MAD and MSE depend on the measurement scale of the time-series data. For example, forecasting profit in the range of millions of dollars would result in very large MAD and MSE values, even for very accurate forecasting models. On the other hand, market share is measured in proportions; therefore, even bad forecasting models will have small values of MAD and MSE. Thus, these measures have no meaning except in comparison with other models used to forecast the same data. Generally, MAD is less affected by extreme observations and is preferable to MSE if such extreme observations are considered rare events with no special meaning. MAPE is different in that the measurement scale is eliminated by dividing the absolute error by the time-series data value. This allows a better relative comparison ons; therefore, even bad forecasting models will have small values of MAD and MSE. Thus, these . Although these comments provide some guidelines, there is no universal agreement on which measure is best.

These measures can be used to compare the moving average forecasts in Figure 7.8. The results, shown in Table 7.2, verify that the two-period moving average model provides the best forecast among these alternatives.

Exponential Smoothing Models

A versatile, yet highly effective approach for short-range forecasting is simple exponential smoothing. The basic simple exponential smoothing model is: where Ft + 1 is the forecast for time period t + 1, Ft is the forecast for period t, At is the observed value in period t, and α is a constant between 0 and 1, called the smoothing constant. To begin, the forecast for period 2 is set equal to the actual observation for period 1.

Using the two forms of the forecast equation just given, we can interpret the simple exponential smoothing model in two ways. In the first model, the forecast for the next period, Ft + 1, is a weighted average of the forecast made for period t, Ft, and the actual observation in period t, At. The second form of the model, obtained by simply rearranging terms, states that the forecast for the next period, Ft + 1, equals the forecast for the last period, plus a fraction α of the forecast error made in period t, At − Ft. Thus, to make a forecast once we have selected the smoothing constant, we need only know the previous forecast and the actual value. By repeated substitution for Ft in the equation, it is easy to demonstrate that Ft + 1 is a decreasingly weighted average of all past time-series data. Thus, the forecast actually reflects all the data, provided that is strictly between 0 and 1.

For the burglary data, the forecast for month 43 is 88, the actual observation for month 42. Suppose we choose α = 0.7; then the forecast for month 44 would be:

The actual observation for month 44 is 60; thus, the forecast for month 45 would be:

Since the simple exponential smoothing model requires only the previous forecast and the current time-series value, it is very easy to calculate; thus, it is highly suitable for environments such as inventory systems where many forecasts must be made. The smoothing constant is usually chosen by experimentation in the same manner as choosing the number of periods to use in the moving average model. Different values of α affect how quickly the model responds to changes in the time series. For instance, a value of α = 1 would simply repeat last period’s forecast, while α = 1 would forecast last period’s actual demand. The closer α is to 1, the quicker the model responds to changes in the time series because it puts more weight on the actual current observation than on the forecast. Likewise, the closer is to 0, the more weight is put on the prior forecast, so the model would respond to changes more slowly.

An Excel spreadsheet for evaluating exponential smoothing models for the burglary data using values of between 0.1 and 0.9 is shown in Figure 7.9. A smoothing constant of α = 0.6 provides the lowest error for all three metrics. Excel has a Data Analysis tool for exponential smoothing (see Excel Note: Forecasting with Exponential Smoothing).

EXCEL NOTE Forecasting with Exponential Smoothing

From the Analysis group, select Data Analysis then Exponential Smoothing. In the dialog (Figure 7.10), as in the Moving Average dialog, you must enter the Input Range of the time-series data, the Damping Factor (1 − α)—not the smoothing constant as we have defined it (!)—and the first cell of the Output Range, which should be adjacent to the first data point. You also have options for labels, to chart output, and to obtain standard errors. As opposed to the Moving Average tool, the chart generated by this tool does correctly align the forecasts with the actual data, as shown in Figure 7.11. You can see that the exponential smoothing model follows the pattern of the data quite closely, although it tends to lag with an increasing trend in the data.

Figure 7.10 Exponential Smoothing Tool Dialog

Figure 7.11 Exponential Smoothing Forecasts for α = 0.6

Forecasting Models for Time Series with Trend and Seasonality

When time series exhibit trend and/or seasonality, different techniques provide better forecasts than the basic moving average and exponential smoothing models we have described. The computational theory behind these models are presented in the appendix to this chapter as they are quite a bit more complicated than the simple moving average and exponential smoothing models. However, a basic understanding of these techniques is useful in order to apply CB Predictor software for forecasting, which we introduce in the next section.

Models for Linear Trends

For time series with a linear trend but no significant seasonal components, double moving average and double exponential smoothing models are more appropriate. Both methods are based on the linear trend equation:

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