Statistical errors

Mean Absolute Error (MAE)

$MAE = \frac{1}{n} \sum_{i = 1}^{n} | y_{i} - {\hat{y}}_{i} |$

Mean Squared Error (MSE)

$MSE = \frac{1}{n} \sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}$

Residual Sum of Squares (RSS)

$RSS = \sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}$

Root Mean Square [Error/Deviation] (RMSE, RMS)

$RMS = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2}}$

$RMSD = \sqrt{MSE} = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}}$

Mean Absolute Percentage Error (MAPE)

$MAPE = \frac{100 %}{n} \sum_{i = 1}^{n} | \frac{y_{i} - {\hat{y}}_{i}}{y_{i}} |$

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)

$MSPE = \frac{1}{n} \sum_{i = 1}^{n} {(\frac{\hat{y_{i}} - y_{i}}{y_{i}})}^{2}$

Root Mean Square Percentage Error (RMSPE)

Swanson et al., Fomby, Shcherbakov et al.

$RMSPE = \sqrt{MSPE} = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(\frac{\hat{y_{i}} - y_{i}}{y_{i}})}^{2}}$

Mean Absolute Scaled Error (MASE)

$MASE = \frac{MAE}{Q}$

with $Q$ as scaling constant.

Median Absolute Deviation (MAD)

$MAD = median (| y_{i} - median (y) |)$

Symmetric Mean Absolute Percentage Error (sMAPE) $sMAPE = \frac{100}{n} \sum_{i = 1}^{n} \frac{| {\hat{y}}_{i} - y_{i} |}{(| y_{i} | + | {\hat{y}}_{i} |) / 2}$

Resources

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