GMRAE – Geometric Mean Relative Absolute Error

Calculates the geometric mean relative absolute error (GMRAE) between the forecast and the eventual outcomes.

Syntax

GMRAE(X, F, M)

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., rows or columns)).

M

is the seasonal period in X. For non-seasonal time series, set M = 1 (default), or leave it blank.

Remarks

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 GMRAE calculation.
5. The relative absolute error for a given observation is defined as follows: $$r_t=\left | \frac{y_t - f_t }{ y_t - f_t^*} \right |$$ Where:
   • $\{y_t\}$ is the actual outcome value at time $t$.
   • $\{f_t\}$ is the forecast value at time $t$.
   • $\{f_i^*\}$ is the forecast value of the benchmark model at time $t$.
6. NumXL uses the naïve forecasting model as a benchmark. The forecast value of the benchmark is defined as follows: $${\displaystyle f_t^*={\left\{\begin{matrix} y_{t-1} \\ y_{t-M} \end{matrix}\right. \begin{matrix} \mathrm{Non-Seasonal}\\ \mathrm{Seasonal} \end{matrix}}}$$
7. The geometric mean relative absolute error is given by the following formula: $${\displaystyle {\mathrm{GMRAE}=\sqrt[m]{\prod_{t=1}^{m}\left | \frac{y_t-f_t}{y_t-f_t^*}\right |}}}$$
8. The GMRAE is sensitive to extreme values (i.e., outliers) and low values.
9. Outliers influence the GMRAE to a much lesser extent than the MRAE.
10. The MRAE function is available starting with version 1.65 HAMMOCK.

Files Examples

Related Links

• Wikipedia - Mean absolute scaled error.

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.