# Mean absolute percentage error

Mean absolute percentage error (MAPE) is measure of accuracy in a fitted time series value in statistics, specifically trending. It usually expresses accuracy as a percentage, and is defined by the formula:

$\mbox{M} = \frac{1}{n}\sum_{t=1}^n \left| \frac{A_t-F_t}{A_t}\right|$

where At is the actual value and Ft is the forecast value.

The difference between At and Ft is divided by the actual value At again. The absolute value of this calculation is summed for every fitted or forecast point in time and divided again by the number of fitted points n. This makes it a percentage error so one can compare the error of fitted time series that differ in level.

Although the concept of MAPE sounds very simple and convincing, it has two major drawbacks in practical application:

• If there are zero values (which sometimes happens for example in demand series) there will be a division by zero
• When having a perfect fit, MAPE is zero. But in regard to its upper level the MAPE has no restriction. When calculating the average MAPE for a number of time series there might be a problem: a few number of series that have a very high MAPE might distort a comparison between the average MAPE of time series fitted with one method compared to the average MAPE when using another method. In order to avoid this problem other measures have been defined, for example the sMAPE (symmetrical MAPE), weighted absolute percentage error (WAPE), real aggregated percentage error (RAPE),or a relative measure of accuracy (ROMA).

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