MAAPE - Mean Arctangent Absolute Percentage Error

Calculates the mean arctangent percentage error (MAAPE) between the forecast and the eventual outcomes.



is the eventual outcome time series sample data (a one-dimensional array of cells e.g., row or column).
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 the 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 has no 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.

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