# 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.

## Examples

Example 1:

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A B C
Date Data Forecast
2008-01-01 -2.9 -2.95
2008-02-01 -2.83 -2.7
2008-03-01 -0.95 -1.00
2008-04-01 -0.88 -0.68
2008-05-01 1.21 1.50
2008-06-01 -1.67 -1.00
2008-07-01 0.83 0.90
2008-08-01 -0.27 -0.37
2008-09-01 1.36 1.26
2008-10-01 -0.34 -0.54
2008-11-01 0.48 0.58
2008-12-01 -2.83 -2.13
2009-01-01 -0.95 -0.75
2009-02-01 -0.88 -0.89
2009-03-01 1.21 1.25
2009-04-01 -1.67 -1.65
2009-05-01 -2.99 -3.20
2009-06-01 1.24 1.29
2009-07-01 0.64 0.60

Formula Description (Result)
=GMRAE($B$3:$B$21,$C$3:$C$21,1) GMRAE (0.0967)