1. Naive Extrapolation1

2. In this part of the course, we want to begin to explicitly model changes that depend not only on changes in a sample or sampling scheme, but rather, changes as explained by some other variable. One of the simplest “other variables” is time. Naive Extrapolation2

3. 3 After each five data, average them. This is a common quality control approach.

4.  What we may want to do instead is to construct some variable that doesn’t require waiting on some fixed sample, but rather responds immediately to each item sampled.  In this case, construct the Moving Average of the last 5 samples, MA5. Naive Extrapolation 4

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6.  The number of time periods over which the average occurs is termed the SPAN.  The general formula for a moving average at time t with span S is: Naive Extrapolation 6

7.  The moving average may also be used as a forecast for any number of periods ahead.  If k represents the period of the forecast then: Naive Extrapolation 7

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9.  The moving average is a naïve forecasting method meaning that it offers no explanation for changes in values over time other than past variation. The future will be the same as the past, on average.  A particular vulnerability of using a moving average for forecasts is when predictable trends occur over time, i.e. inflation. In this situation, the forecasts using a moving average will always LAG behind actual changes. Naive Extrapolation 9

10.  When using any process for forecasting, it is possible to estimate the forecast error rate. Three common measures of forecast error are: MAE, MAPE, and RMSE  MAE is the Mean Absolute Error  MAPE is the Mean Absolute Percentage Error  RMSE is the Root Mean Square Error Naive Extrapolation 10

11. To calculate the MAE: ◦ Construct the forecast values, F i ◦ Calculate the differences between the actual values and the forecasted values, the Errors, E i = X i – F i ◦ Calculate the absolute values of the errors, the Absolute Errors, AE i = |E i | ◦ Calculate the Mean of the Absolute Errors, MAE Naive Extrapolation 11

12. To calculate the MAPE: ◦ Construct the forecast values, F i ◦ Calculate the differences between the actual values and the forecasted values, the Errors, E i = X i – F i ◦ Calculate the errors as a percentage of the actual values, PE i = AE i / X i ◦ Calculate the absolute values of the percentage errors, ◦ APE i = |PE i | ◦ Calculate the Mean of the APEs, MAPE Naive Extrapolation 12

13. To calculate the RMSE: ◦ Construct the forecast values, F i ◦ Calculate the squared errors, the square of the difference between the actual values and the forecasted values, (X i – F i ) 2 = E i 2 ◦ Calculate the mean of the squared errors, MSE ◦ Take the square root, RMSE ◦ If we express the RMSE as a percentage of the mean of the X-data, this is termed the coefficient of variation. Naive Extrapolation 13

14. Finally, we note that the choice of span is a modeling parameter that may be changed. In particular, in using a moving average it may be necessary to construct these calculations for different spans. To determine the best span, compare the MAE or the MAPE or the RMSE. Naive Extrapolation 14

15.  Moving Average Formula (varies with span)  Forecast Formula (varies with forecast period)  Error Formulas ◦ MAE ◦ MAPE ◦ RMSE  Optimization of Span Naive Extrapolation15

16.  For the following data, construct a moving average with a span of 2. Forecasting one period ahead, calculate the MAE, MAPE, and RMSE. 2002128 2003144 2004158 2005179 2006195 Naive Extrapolation16