# 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: $$f_t^*={\left\{\begin{matrix} y_{t-1} \\ y_{t-M} \end{matrix}\right. \begin{matrix} \mathrm{Non-Seasonal}\\ \mathrm{Seasonal} \end{matrix}}$$
7. The geometric mean relative absolute error is given by the following formula: $${\mathrm{GMRAE}=\sqrt[m]{\prod_{t=1}^{m}\left | \frac{y_t-f_t}{y_t-f_t^*}\right |}}$$
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.