The value of this calculation is summed for every fitted point t and divided again by the number of fitted points n. The earliest reference to a similar formula appears to be Armstrong (1985, p. 348), where it is called "adjusted MAPE" and is defined without the absolute values in the denominator.
Therefore, the currently accepted version of SMAPE assumes the absolute values in the denominator.
In contrast to the mean absolute percentage error, SMAPE has both a lower and an upper bound.
One supposed problem with SMAPE is that it is not symmetric since over- and under-forecasts are not treated equally.
The following example illustrates this by applying the second SMAPE formula: However, one should only expect this type of symmetry for measures which are entirely difference-based and not relative (such as mean squared error and mean absolute deviation).
There is a third version of SMAPE, which allows measuring the direction of the bias in the data by generating a positive and a negative error on line item level.
Furthermore, it is better protected against outliers and the bias effect mentioned in the previous paragraph than the two other formulas.
Provided the data are strictly positive, a better measure of relative accuracy can be obtained based on the log of the accuracy ratio: log(Ft / At) This measure is easier to analyze statistically and has valuable symmetry and unbiasedness properties.
When used in constructing forecasting models, the resulting prediction corresponds to the geometric mean (Tofallis, 2015).