Commonly used Mean Absolute Percentage Errors (MAPE), and the authors’ revised Mean Absolute Percentage Errors (RMAPE) are applied to measure the forecasting accuracy from different Moving Average Methods for independent time series.

Data/Method

20,000 random data are simulated from each of Normal distribution, T-distribution with 3 degrees of freedom, Uniform distribution in interval (-½, ½), and Chi-squared distribution with 2 degrees of freedom. To have their mean be 1 and the standard deviation be the coefficients of variation (c.v.), we make the following transformations.

(1) For normal distributions, we will let N=1+N(0, c.v.²), where N(0, σ²) represents a normal probability distribution with a mean of 0 and a standard deviation of σ.

(2) For T-distributions, we will let T=1+ , where T(3) is the T-distribution of 3 degrees of freedom with a mean of 0 and a standard deviation of .

(3) For Uniform distributions, we will make U=1+ U(-½, ½), where U(-½, ½) is the Uniform probability distribution in interval (-½, ½) with a mean of 0 and a standard deviation of .

(4) For Chi-squared distributions, we will have have K=1+c.v.[ ], where is the Chi-squared probability distribution

of 2 degrees of freedom with a mean of 2 and a standard deviation of 2 also.

Data is then grouped into 1,000 groups with 20 observations each. The first nine (9) observations in each group are treated as historical observations, and the tenth (10^th) to twentieth (20^th) observations are treated as the future 11 observations. Moving average methods with moving period of 1, 3, 5, 7, and 9 are applied to historical observations and their forecasts compared with the first future observation (the 10^th observation). Absolute Percentage Deviation , is calculated for the first future

observation. Now, include the 11^th observation into the new historical group. Moving average methods with moving period of 1, 3, 5, 7, and 9 are applied to the most current 9 historical observations counted back from the 10^th observation (i.e., the first observation in the old historical data group is eliminated), and their forecasts are compared with the “new” first future observation (the 11^th observation). Absolute Percentage Deviation, , is calculated. Continue the above process until we find out the 11^th Absolute Percentage Deviation,, , .

Revised Mean Absolute Percentage Errors (RMAPE), can then be calculated for k from 1 to 11, accordingly.

Conclusion

Simulation results show that both MAPE and RMAPE can only provide sensitive forecasting accuracy measurements on Moving Average Methods when coefficients of variation (c.v.) is smaller than 0.4 or is much greater than 4.0 for those independent time series. For independent time series with moderate c.v.’s, the complexity from the ratios of MAPE and RMAPE will mislead researchers on distinguishing the forecasting accuracies from different Moving Average Methods. The complexity from the ratios will be released only when the c.v. is very small, or when the c.v. is very large. Therefore, when data are from independent time series, the Mean Absolute Deviation (MAD) reveals valid the forecasting accuracies from various Moving Average Methods, but not from MAPE or RMAPE.