MASE - Mean Absolute Scaled Error

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

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 MASE calculation.
5. 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.
6. 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.
7. The mean absolute scaled error (MASE) is calculated according to: $${\displaystyle {\mathrm{MASE}=\frac{1}{N}\sum_{i=1}^N q_i=\frac{\mathrm{MAE}}{\mathrm{MAE^*}}}}$$
8. The MASE measure is symmetrical and resistant to outliers.
9. Division by zero occurs only in one trivial case where all the values in the input time series are equal (i.e., constant).
10. $\mathrm{MASE} \gt 1$ implies that the actual forecast does worse than a naïve benchmark forecasting method calculated in-sample.
11. $\mathrm{MASE} \lt 1\quad$ implies that actual forecast performance better than a naïve method.
12. 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.
13. The MASE function is available starting with version 1.65 HAMMOCK.

Files Examples

Related Links

• Wikipedia - Mean absolute scaled error.

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.