Mean absolute error

In statistics, the mean absolute error (MAE) is a quantity used to measure how close forecasts or predictions are to the eventual outcomes. The mean absolute error is given by

${\mathrm {MAE}}={\frac {1}{n}}\sum _{{i=1}}^{n}\left|f_{i}-y_{i}\right|={\frac {1}{n}}\sum _{{i=1}}^{n}\left|e_{i}\right|.$

As the name suggests, the mean absolute error is an average of the absolute errors $|e_{i}|=|f_{i}-y_{i}|$, where $f_{i}$ is the prediction and $y_{i}$ the true value. Note that alternative formulations may include relative frequencies as weight factors.

The mean absolute error used the same scale as the data being measured. This is known as a scale-dependent accuracy measure and therefore cannot be used to make comparisons between series using different scales.^[1]

The mean absolute error is a common measure of forecast error in time series analysis,^[2] where the terms "mean absolute deviation" is sometimes used in confusion with the more standard definition of mean absolute deviation. The same confusion exists more generally.

Related measures

The mean absolute error is one of a number of ways of comparing forecasts with their eventual outcomes. Well-established alternatives are the mean absolute scaled error (MASE) and the mean squared error. These all summarize performance in ways that disregard the direction of over- or under- prediction; a measure that does place emphasis on this is the mean signed difference.

Where a prediction model is to be fitted using a selected performance measure, in the sense that the least squares approach is related to the mean squared error, the equivalent for mean absolute error is least absolute deviations.

See also

• Least absolute deviations
• Mean absolute percentage error
• Mean percentage error
• Symmetric mean absolute percentage error

References

1. "2.5 Evaluating forecast accuracy | OTexts". www.otexts.org. Retrieved 2016-05-18.
2. Hyndman, R. and Koehler A. (2005). "Another look at measures of forecast accuracy"