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mean absolute error

nounAlso: MAE, L1 loss, absolute error loss, least absolute deviations, mean absolute percentage error, MAPE, mean absolute scaled error, MASE

In plain wordsA way of scoring numerical predictions: average how far off each prediction is, ignoring direction, so the score reads as 'typically off by this much'. Lower is better.

  1. The average of the absolute differences between predicted and true values. Expressed in the target's own units, it reads naturally as the typical size of an error. Because errors count linearly rather than squared, MAE is more robust to outliers than mean squared errormean squared error, n.: A way of scoring numerical predictions: square each prediction's error, then average them, so big misses count much more than small ones. Lower is better., and minimizing it yields predictions of the conditional median rather than the mean; as a training loss (L1 loss) its gradient has constant size and is undefined at zero, so smooth compromises such as the Huber loss are common. In time series forecastingtime series forecasting, n.: Predicting future values of something measured over time, such as sales, electricity demand, prices or rainfall, from its past values and related information., scale-free variants allow comparison across series: the mean absolute percentage error (MAPE), which is undefined when actual values are zero and treats over- and under-forecasts asymmetrically, and the mean absolute scaled error (MASE), which divides by the in-sample error of a naive forecast. The pinball loss used in quantile regressionquantile regression, n.: Predicting not just a typical value but a range — say, the delivery time that 90% of orders beat — by fitting models for chosen percentiles of the outcome instead of its average. generalizes absolute error to quantiles other than the median.

    MAE is 3.1 minutes, so a typical delivery-time estimate is off by about three minutes either way.

MAE=1n∑i=1n∣yi−y^i∣\mathrm{MAE} = \frac{1}{n} \sum_{i=1}^{n} \left| y_i - \hat{y}_i \right|

Origin

Fitting by least absolute deviations was proposed by Roger Joseph Boscovich in 1757 and developed by Pierre-Simon Laplace, half a century before least squares. Squared error displaced it because it was easier to compute, until linear programming and robust statistics revived absolute-error methods in the mid-20th century. Rob Hyndman and Anne Koehler proposed MASE in 2006.

Related

See also
  • mean squared errormean squared error, n.: A way of scoring numerical predictions: square each prediction's error, then average them, so big misses count much more than small ones. Lower is better.
  • quantile regressionquantile regression, n.: Predicting not just a typical value but a range — say, the delivery time that 90% of orders beat — by fitting models for chosen percentiles of the outcome instead of its average.
  • time series forecastingtime series forecasting, n.: Predicting future values of something measured over time, such as sales, electricity demand, prices or rainfall, from its past values and related information.
  • coefficient of determinationcoefficient of determination, n.: A score for numerical predictions showing how much of the variation in the true values a model accounts for, compared with always guessing the average: 1 is perfect, 0 is no better than the average.
  • loss functionloss function, n.: A score that measures how wrong a model's predictions are. Training means adjusting the model, step by step, to push this number down.
  • regressionregression, n.: Predicting a number rather than a category — a house's price, tomorrow's temperature, a patient's blood pressure — by learning from past examples where the answer is known.
Contrast with
  • mean squared errormean squared error, n.: A way of scoring numerical predictions: square each prediction's error, then average them, so big misses count much more than small ones. Lower is better.

Sources

  1. Hyndman, R. J. & Koehler, A. B. (2006). Another Look at Measures of Forecast Accuracy. International Journal of Forecasting 22(4).
  2. Willmott, C. J. & Matsuura, K. (2005). Advantages of the Mean Absolute Error (MAE) over the Root Mean Square Error (RMSE) in Assessing Average Model Performance. Climate Research 30.

Mentioned in

regressionregression, n.: Predicting a number rather than a category — a house's price, tomorrow's temperature, a patient's blood pressure — by learning from past examples where the answer is known., mean squared errormean squared error, n.: A way of scoring numerical predictions: square each prediction's error, then average them, so big misses count much more than small ones. Lower is better., time series forecastingtime series forecasting, n.: Predicting future values of something measured over time, such as sales, electricity demand, prices or rainfall, from its past values and related information., Huber lossHuber loss, n.: A way of measuring prediction errors that treats small errors like squared distance but large ones more gently, so a few wild data points cannot drag the whole model off course.