Statistical errors
Mean Absolute Error (MAE)
$$ \text{MAE}=\frac{1}{n}\sum_{i=1}^n|y_i-\hat{y}_i| $$
Mean Squared Error (MSE)
$$ \text{MSE}=\frac{1}{n}\sum_{i=1}^n\left(y_i-\hat{y}_i\right)^2 $$
Residual Sum of Squares (RSS)
$$ \text{RSS}=\sum_{i=1}^n\left(y_i-\hat{y}_i\right)^2 $$
Root Mean Square [Error/Deviation] (RMSE, RMS)
$$ \text{RMS}=\sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2} $$
$$ \text{RMSD}=\sqrt{\text{MSE}}=\sqrt{\frac{1}{n}\sum_{i=1}^n\left(y_i-\hat{y}_i\right)^2} $$
Mean Absolute Percentage Error (MAPE)
$$ \text{MAPE}=\frac{100%}{n}\sum_{i=1}^n\left|\frac{y_i-\hat{y}_i}{y_i}\right| $$
| MAPE | Interpretation |
|---|---|
| <10% | Highly accurate forecasting |
| 10-20% | Good forecasting |
| 20-50% | Reasonable forecasting |
| >50% | Inaccurate forecasting |
(Lewis, 1982, p.40)
Mean Square Percentage Error (MSPE)
$$ \text{MSPE}=\frac{1}{n}\sum_{i=1}^n\left(\frac{\hat{y_i}-y_i}{y_i}\right)^2 $$
Root Mean Square Percentage Error (RMSPE)
Swanson et al., Fomby, Shcherbakov et al.
$$ \text{RMSPE}=\sqrt{\text{MSPE}}=\sqrt{\frac{1}{n}\sum_{i=1}^n\left(\frac{\hat{y_i}-y_i}{y_i}\right)^2} $$
Mean Absolute Scaled Error (MASE)
$$ \text{MASE}=\frac{\text{MAE}}{Q} $$
with $Q$ as scaling constant.
Median Absolute Deviation (MAD)
$$ \text{MAD}=\text{median}\left(|y_i-\text{median}(y)|\right) $$
Symmetric Mean Absolute Percentage Error (sMAPE)
$$ \text{sMAPE}=\frac{100}{n}\sum_{i=1}^{n}\frac{|\hat{y}_i-y_i|}{(|y_i|+|\hat{y}_i|)/2} $$