Calculates the mean absolute scaled error (MASE) between the forecast and the eventual outcomes.

## Syntax

**MASE**(**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 (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 MASE calculation.
- The absolute scaled error is defined as follows: $$q_t = \left\{\begin{array}{l} \frac{\left | e_t \right |}{\frac{1}{n-1}\times\sum_{t=2}^n \left | y_i - y_{i-1}\right |} \\ \frac{\left |e_t \right |}{\frac{1}{n-M}\times\sum_{t=M+1}^n \left | y_i - y_{i-M}\right |} \end{array} \right. \begin{array}{l} \{y_i\}\,\mathrm{is \, non-seasonal} \\ \{y_i\}\,\mathrm{is \, seasonal} \end{array}$$ Where:
- $\{y_t\}$ is the actual observations time series.
- $\{e_t\}$ is the forecast error for a given period.

- The absolute scaled error is generally the same as the absolute error divided (i.e., scaled) by the Mean Absolute error (MAE) of the naïve benchmark model. $$q_t=\left\{\begin{array}{l} \frac{\left |e_t \right |}{\mathrm{MAE^*}} \\ \frac{\left |e_t \right |}{\mathrm{MAE_M^*}} \end{array} \right. \begin{array}{l} \{y_i\}\,\mathrm{is \, non-seasonal} \\ \{y_i\}\,\mathrm{is \, seasonal} \end{array}$$ Where:
- $\mathrm{MAE^*}$ is the mean absolute error of the naïve (aka naive 1) benchmark forecast model calculated in-sample.
- $\mathrm{MAE_M^*}$ is the in-sample mean absolute error of the seasonal naïve (aka naive 2) benchmark forecasting model calculated in-sample.

- The mean absolute scaled error (MASE) is calculated according to: $${\displaystyle {\mathrm{MASE}=\frac{1}{N}\sum_{i=1}^N q_i=\frac{\mathrm{MAE}}{\mathrm{MAE^*}}}}$$
- The MASE measure is symmetrical and resistant to outliers.
- Division by zero occurs only in one trivial case where all the values in the input time series are equal (i.e., constant).
- $\mathrm{MASE} \gt 1$ implies that the actual forecast does worse than a naïve benchmark forecasting method calculated in-sample.
- $\mathrm{MASE} \lt 1\quad$ implies that actual forecast performance better than a naïve method.
- We can use the MASE values for comparing different forecasting methods. The lower the MASE value, the lower the relative absolute forecast error, and the better the method.
- The MASE 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.

## Comments

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