Mean absolute error
In statistics, mean absolute error (MAE) is a measure of errors between paired observations expressing the same phenomenon. Examples of Y versus X include comparisons of predicted versus observed, subsequent time versus initial time, and one technique of measurement versus an alternative technique of measurement. MAE is calculated as:[1]
Quantity disagreement and allocation disagreement[]
It is possible to express MAE as the sum of two components: Quantity Disagreement and Allocation Disagreement. Quantity Disagreement is the absolute value of the Mean Error given by:[4]
Allocation Disagreement is MAE minus Quantity Disagreement.
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 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.
MAE is not identical to root-mean square error (RMSE), although some researchers report and interpret it that way. 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] See the example above for an illustration of these differences.
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
This optimization-based definition of the median is useful in statistical data-analysis, for example, in k-medians clustering.
Proof of optimality[]
Statement: The classifier minimising is .
Proof:
The Loss functions for classification is
Differentiating with respect to a gives
This means
Hence
See also[]
- Least absolute deviations
- Mean absolute percentage error
- Mean percentage error
- Symmetric mean absolute percentage error
References[]
- ^ Willmott, Cort J.; Matsuura, Kenji (December 19, 2005). "Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance". Climate Research. 30: 79–82. 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]
- ^ a b c Pontius Jr., Robert Gilmore; Thontteh, Olufunmilayo; Chen, Hao (2008). "Components of information for multiple resolution comparison between maps that share a real variable". Environmental and Ecological Statistics. 15 (2): 111–142. doi:10.1007/s10651-007-0043-y.
- ^ Willmott, C. J.; Matsuura, K. (January 2006). "On the use of dimensioned measures of error to evaluate the performance of spatial interpolators". International Journal of Geographical Information Science. 20: 89–102. doi:10.1080/13658810500286976.
- ^ Stroock, Daniel (2011). Probability Theory. Cambridge University Press. pp. 43. ISBN 978-0-521-13250-3.
- ^ Nicolas, André (2012-02-25). "The Median Minimizes the Sum of Absolute Deviations (The $ {L}_{1} $ Norm)". StackExchange.
- Point estimation performance
- Statistical deviation and dispersion
- Time series
- Errors and residuals