What are MAPE and SMAPE metrics? Where might they be ineffective?

MAPE (Mean Absolute Percentage Error) is the average absolute percentage error. It is calculated as the mean of the absolute forecast errors expressed as a percentage of the actual values:

[ \text{MAPE} = \frac{100%}{n} \sum_{t=1}^n \left| \frac{A_t - F_t}{A_t} \right| ]

where (A_t) is the actual value, (F_t) is the forecasted value.

SMAPE (Symmetric Mean Absolute Percentage Error) is a symmetric version of MAPE that considers both actual and forecasted values in the denominator to avoid issues with zero values:

[ \text{SMAPE} = \frac{100%}{n} \sum_{t=1}^n \frac{|F_t - A_t|}{(|A_t| + |F_t|)/2} ]

Where they can be ineffective:

• When actual values are zero or very close to zero, MAPE becomes unstable or infinite because division by zero is impossible.
• MAPE and SMAPE can give distorted results with highly asymmetric errors.
• SMAPE tries to mitigate the zero problem but can still be unstable with very small values.
• Both metrics are sensitive to the scale of data and are not suitable if absolute error is important rather than relative error.

In such cases, it is better to use other metrics, such as RMSE or MAE.