Mean absolute error
In
Quantity disagreement and allocation disagreement
In remote sensing the MAE is sometimes expressed as the sum of two components: quantity disagreement and allocation disagreement. Quantity disagreement is the absolute value of the mean error:[4]
It is also possible to identify the types of difference by looking at an plot. Quantity difference exists when the average of the X values does not equal the average of the Y values. Allocation difference exists if and only if points reside on both sides of the identity line.[4][5]
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
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.
MAE is not identical to root-mean square error (RMSE), although some researchers report and interpret it that way. The MAE is conceptually simpler and also easier to interpret than RMSE: it is simply the average absolute vertical or horizontal distance between each point in a scatter plot and the Y=X line. In other words, MAE is the average absolute difference between X and Y. Furthermore, each error contributes to MAE in proportion to the absolute value of the error. This is in contrast to RMSE which involves squaring the differences, so that a few large differences will increase the RMSE to a greater degree than the MAE.[4]
Optimality property
The mean absolute error of a real variable c with respect to the random variable X is
More generally, a median is defined as a minimum of
Proof of optimality
Statement: The classifier minimising is .
Proof:
The Loss functions for classification is
See also
- Least absolute deviations
- Mean absolute percentage error
- Mean percentage error
- Symmetric mean absolute percentage error
References
- doi:10.3354/cr030079.
- ^ "2.5 Evaluating forecast accuracy | OTexts". www.otexts.org. Retrieved 2016-05-18.
- ^ Hyndman, R. and Koehler A. (2005). "Another look at measures of forecast accuracy" [1]
- ^ S2CID 21427573.
- S2CID 15407960.
- ISBN 978-0-521-13250-3.
- MR 0356303.