Forecast Accuracy Metrics: MAPE, sMAPE and MASE

Forecast accuracy metrics turn prediction errors into summaries, but each emphasizes different properties. Mean absolute error averages absolute errors in the target's units. MAPE averages absolute error divided by the absolute actual value, often multiplied by 100. It is easy to read as a percentage, but it is undefined when actual values are zero and can be dominated by small actuals. It also treats over- and under-forecast errors asymmetrically in some settings because actuals appear only in the denominator.

sMAPE attempts a symmetric percentage by scaling absolute error relative to the magnitudes of actual and forecast. Several formulas exist, some use a factor of two and others not, and zero-over-zero cases require conventions. Despite its name, sMAPE can behave unexpectedly around zero and does not guarantee fair comparisons across all scales. Always state the exact implementation.

Mean absolute scaled error (MASE) divides a model's test-set MAE by the in-sample MAE of a naive forecast, typically the one-step persistence forecast for nonseasonal data. For seasonal data, a seasonal-naive denominator may be more appropriate. A MASE below one means the model's average absolute error on the evaluation cases is lower than the chosen naive in-sample scale. It does not mean the model always beats the baseline on every observation, and the denominator can be zero for perfectly constant training data.

Use an evaluation period that reflects the intended forecast horizon, preserve time order and compare models on identical cases. Report metric definitions, units, treatment of zeros, aggregation weights and baseline. If many series are combined, averaging percentage errors may give tiny-volume series disproportionate influence. Pair a headline metric with error distributions and business costs, such as stockouts versus overstock. No single metric captures calibration of prediction intervals, bias, tail risk and operational impact at once.