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Mean absolute percentage error

The mean absolute percentage error (MAPE), also known as mean absolute percentage deviation (MAPD), is a measure of prediction accuracy of a forecasting method in statistics. It usually expresses the accuracy as a ratio defined by the formula: where At is the actual value and Ft is the forecast value. Their difference is divided by the actual value At. The absolute value of this ratio is summed for every forecasted point in time and divided by the number of fitted points n. Mean absolute percentage error is commonly used as a loss function for regression problems and in model evaluation, because of its very intuitive interpretation in terms of relative error. Consider a standard regression setting in which the data are fully described by a random pair with values in , and n i.i.d. copies of . Regression models aim at finding a good model for the pair, that is a measurable function g from to such that is close to Y. In the classical regression setting, the closeness of to Y is measured via the L2 risk, also called the mean squared error (MSE). In the MAPE regression context, the closeness of to Y is measured via the MAPE, and the aim of MAPE regressions is to find a model such that: where is the class of models considered (e.g. linear models). In practice In practice can be estimated by the empirical risk minimization strategy, leading to From a practical point of view, the use of the MAPE as a quality function for regression model is equivalent to doing weighted mean absolute error (MAE) regression, also known as quantile regression. This property is trivial since As a consequence, the use of the MAPE is very easy in practice, for example using existing libraries for quantile regression allowing weights. The use of the MAPE as a loss function for regression analysis is feasible both on a practical point of view and on a theoretical one, since the existence of an optimal model and the consistency of the empirical risk minimization can be proved. WMAPE (sometimes spelled wMAPE) stands for weighted mean absolute percentage error.

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