Time Series Forecast Error

We use the following terminology: if y₁, …, y_n represents a time series, then ŷ_i represents the i^th forecasted value, where i ≤ n. For i ≤ n, the i^th error e_i is then

Our goal is to find a forecast that minimize the errors. A number of measures are commonly used to determine the accuracy of a forecast, including the mean absolute error (MAE), mean squared error (MSE) and root mean squared error (RMSE).

Note that MAE is also commonly called mean absolute deviation (MAD). This version of MAD should not be confused with the median absolute deviation (MAD) described in Measures of Variability.

Some other measurements are mean absolute percentage error (MAPE), mean absolute scaled error (MASE) and symmetric mean absolute percentage error (SMAPE).

For data with seasonality (see Holt-Winter Forecasting) where the periodicity is c, the formula for MASE becomes