Naive Extrapolation1. 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.

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Presentation transcript:

Naive Extrapolation1

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

 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

5

 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

 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

8

 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

 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

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

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

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

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

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

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