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

- The time series is homogeneous or equally spaced.
- The time series X and F must be of identical size.
- The time series X or F may include observations with missing values (e.g., #N/A or blank).
- Observations with missing values in Y or F are excluded from the GMRAE calculation.
- 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 $t$.

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

## Files Examples

## Related Links

## 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.

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