The Art of Mathematics in Business. Dr Jae K Shim
23.1 and Figure 23.1 These data show sales of DVD (in thousands of units) for a particular maker over the past four years.
We begin our analysis by showing how to identify the seasonal component of the time series. Looking at Figure 23.1, we can easily see a seasonal pattern of DVD sales. Specifically, we observe that sales are lower in the second quarter of each year, followed by higher sales in quarters 3 and 4. The computational procedure used to eliminate the seasonal component is explained below, step by step.
Step 1. We use a moving average to measure the combined trend of cyclical (TC) components of the time series. This way we eliminate the seasonal and random components, S and R.
More specifically, Step 1 involves the following sequences of steps:
Table 23.1: Quarterly Sales Data for DVDs over the Past 4 Years
| Year | Quarter | Sales |
| 1 | 1 | 5.8 |
| 2 | 5.1 | |
| 3 | 7.0 | |
| 4 | 7.5 | |
| 2 | 1 | 6.8 |
| 2 | 6.2 | |
| 3 | 7.8 | |
| 4 | 8.4 | |
| 3 | 1 | 7.0 |
| 1 | 6.6 | |
| 3 | 8.5 | |
| 4 | 8.8 | |
| 4 | 1 | 7.3 |
| 2 | 6.9 | |
| 3 | 9.0 | |
| 4 | 9.4 |
Figure 23.1: Quarterly DVD sales time series
a)Calculate the 4-quarter moving average for the time series, which we discussed in the above. However, the moving average values computed do not correspond directly to the original quarters of the time series.
b)We resolve this difficulty by using the midpoints between successive moving-average values. For example, since 6.35 corresponds to the first half of quarter 3 and 6.6 corresponds to the last half of quarter 3, we use (6.35 + 6.6)/2 = 6.475 as the moving average value of quarter 3. Similarly, we associate (6.6+6.875)/2 = 6.7375 with quarter 4. A complete summary of the moving-average calculation is shown in Table 23.2.
Table 23.2: Moving Average Calculations for the DVD Sales Time Series
c)Next, we calculate the ratio of the actual value to the moving average value for each quarter in the time series having a 4-quarter moving average entry. This ratio in effect represents the seasonal-random component, SR=Y/TC. The ratios calculated this way appear in Table 23.3.
Table 23.3: Seasonal Random Factors for the Series
d)Arrange the ratios by quarter and then calculate the average ratio by quarter in order to eliminate the random influence.
For example, for quarter 1
(0.975 + 0.929 + 0.920)/3 = 0.941
e)The final step, shown below, adjusts the average ratio slightly (for example, for quarter 1, 0.941 becomes 0.940), which will be the seasonal index, as shown in Table 23.4.
Table 23.4: Seasonal Component Calculations
Step 2: After obtaining the seasonal index, we must first remove the effect of season from the original time series. This process is referred to as deseasonalizing the time series. For this, we must divide the original series by the seasonal index for that quarter. This is shown in Table 23.5 and graphed in Fig. 23.2.
Step 3: Looking at the graph in Figure 23.2, we see the time series seem to have an upward linear trend. To identify this trend, we develop the least squares trend equation. This procedure is also shown in Table 23.5.
Figure 23.2: Quarterly DVD sales time series − original versus deseasonalized
Table 23.5: Deseasonalized Data
which means y = 6.1147 + 0.1469 t for the forecast periods:
| t= | 17 |
| 18 | |
| 19 | |
| 20 |
Table 23.6: Quarter-To-Quarter Sales Forecasts for Year 5
Note: (a) y = 6.1147 + 0.1469 t = 6.1147 + 0.1469 (17) = 8.6128
Step 4: Develop the forecast using the trend equation and adjust these forecasts to account for the effect of season. The quarterly forecast, as shown in Table 6.7, can be obtained by multiplying the forecast based on trend times the seasonal factor.
How is it used and applied?
The classical decomposition model is time-series model used for forecasting. This means that the method can used only to fit the time-series data, whether it is monthly, quarterly, or annual. The types of time-series data the company deals with include earnings, cash flows, market share, and costs. As long as the time series displays the patterns of seasonality and cyclicality, the model constructed should be very effective in projecting the future variable.
24. Measuring Accuracy of Forecasts
Introduction
The performance of a forecast should be checked against its own record or against that of other forecasts. There are various statistical measures that can be used to measure performance of the model.
How is it computed?
The performance is measured in terms of forecasting error, where error is defined as the difference between a predicted value and the actual result. Error (e) = Actual (A) - Forecast (F)
There are three common measures for summarizing historical errors.
1.Mean absolute deviation (MAD) is the mean or average of the sum of the errors for a given data set taken without regard to sign. The formula for calculating MAD is:
2.Mean squared error (MSE) is a measure of accuracy computed by squaring the individual errors for each item and then finding the average value of the sum of those squares. MSE gives greater weight to large errors than to small errors, since the errors are squared before being summed. The formula for calculating MSE is:
3.Mean absolute percentage error (MAPE): Sometimes it is more useful to compute the forecasting errors in percentages rather than in amounts. The MAPE is calculated by finding the