X is the eventual outcome time series sample data (a one-dimensional array of cells e.g. row or column).

F is the forecast time series data (a one-dimensional array of cells (e.g. row or column).

1. The time series is homogeneous or equally spaced.
2. The time series X and F must be of identical size
3. The time series X or F may include observations with missing values (e.g. #N/A or blank).
4. Observations with missing values in Y or F are excluded from the MAAPE calculation.
5. The arctangent absolute percentage error (AAPE) for a given observation is defined as follows:
   
   $$ {\displaystyle {\mathrm{AAPE_t}=\mathrm{arctan}(\left | \frac{y_t-f_t}{y_t}\right |)}} $$
   Where:
   
   • $\{y_t\}$ is the actual outcome value at period t.
   • $\{f_t\}$ is the forecast value at period t.
6. Unlike the regular absolute percentage error (APE), the arctangent absolute error approaches to $\frac{\pi}{2}$ when division by zero occurs.
7. The AAPE is undefined when $y_t=f_t=0$, which can be found often in an intermittent demand time series.
8. The mean arctangent absolute percentage error (MAAPE) is defined as follows:
   
   $${\displaystyle {\mathrm{MAAPE}= \frac{1}{N}\sum_{t=1}^N \mathrm{AAPE_t}=\frac{1}{N}\sum_{t=1}^N{\mathrm{arctan}(\left |\frac{y_t-f_t}{y_t} \right |)}}}$$
9. Although MAAPE is finite when response variable (i.e. $y_t$) equals zero, it has a nice trigonometric representation. However, because MAAPE’s value is expressed in radians, this makes MAAPE less intuitive.
10. Please note that MAAPE does not have a symmetric version, since division by zero is no longer a concern.
11. The MAAPE is also scale-free because its values are expressed in radians.
12. The MAAPE function is available starting with version 1.65 HAMMOCK.

Example 1:

Files Examples

Related Links

• Wikipedia - Mean absolute percentage error (MAPE)
• Errors on percentage errors

References

• R.J. Hyndman, A.B. Koehler, "Another look at measures of forecast accuracy", International Journal of Forecasting, 22 (2006), pp. 679-688
• James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896
• Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740
• D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906