Mean absolute scaled error

In statistics, the mean absolute scaled error (MASE) is a measure of the accuracy of forecasts . It was proposed in 2005 by statistician Rob J. Hyndman and Professor of Decision Sciences Anne B. Koehler, who described it as a "generally applicable measurement of forecast accuracy without the problems seen in the other measurements." The mean absolute scaled error has favorable properties when compared to other methods for calculating forecast errors, such as root-mean-square-deviation, and is therefore recommended for determining comparative accuracy of forecasts.

Non seasonal time series

For a non-seasonal time series, the mean absolute scaled error is estimated by

$\mathrm {MASE} ={\frac {1}{T}}\sum _{t=1}^{T}\left({\frac {\left|e_{t}\right|}{{\frac {1}{T-1}}\sum _{t=2}^{T}\left|Y_{t}-Y_{t-1}\right|}}\right)={\frac {\sum _{t=1}^{T}\left|e_{t}\right|}{{\frac {T}{T-1}}\sum _{t=2}^{T}\left|Y_{t}-Y_{t-1}\right|}}$

where the numerator e_t is the forecast error for a given period, defined as the actual value (Y_t) minus the forecast value (F_t) for that period: e_t = Y_t − F_t, and the denominator is the mean absolute error of the one-step "naive forecast method" on the training set, which uses the actual value from the prior period as the forecast: F_t = Y_t−1

Seasonal time series

For a seasonal time series, the mean absolute scaled error is estimated in a manner similar to the method for non-seasonal time series:

$\mathrm {MASE} ={\frac {1}{T}}\sum _{t=1}^{T}\left({\frac {\left|e_{t}\right|}{{\frac {1}{T-m}}\sum _{t=m+1}^{T}\left|Y_{t}-Y_{t-m}\right|}}\right)={\frac {\sum _{t=1}^{T}\left|e_{t}\right|}{{\frac {T}{T-m}}\sum _{t=m+1}^{T}\left|Y_{t}-Y_{t-m}\right|}}$

The main difference with the method for non-seasonal time series, is that the denominator is the mean absolute error of the one-step "seasonal naive forecast method" on the training set, which uses the actual value from the prior season as the forecast: F_t = Y_t−m, where m is the seasonal period.

This scale-free error metric "can be used to compare forecast methods on a single series and also to compare forecast accuracy between series. This metric is well suited to intermittent-demand series because it never gives infinite or undefined values except in the irrelevant case where all historical data are equal.

When comparing forecasting methods, the method with the lowest MASE is the preferred method.

See also

• Mean squared error
• Mean absolute error
• Mean absolute percentage error
• Root-mean-square deviation
• Test Set