# MRAE – Mean Relative Absolute Error

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

## Syntax

MRAE(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. row or column).
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 MRAE 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 mean relative absolute error is given by the following formula:
$$\mathrm{MRAE}=\frac{1}{N}\sum_{i=1}^N\left |\frac{y_t-f_t}{y_t - f_t^*} \right |$$
8. The MRAE (mean relative absolute error) is sensitive to extreme values (i.e. outliers), and to low values.
9. Division by zero may occur if the predictive value obtained by the reference (benchmark) model is equal to the actual value. In this case, the MRAE function returns #VALUE!
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)
=MRAE($B$3:$B$21,$C$3:$C$21,1) MRAE (0.348)