Table 3.

List of main Error Measures. Arithmetic mean and absolute errors are used to calculate these measures in which positive and negative deviations do not cancel each other out and measures do not provide any information about the direction of errors

Measure name	Formula	Description	Scaled	Outlier Protection	Other forms	Penalize extreme deviation	Other Specification
Mean Absolute Error (MAE)	$\mathit{\text{MAE}}=\frac{1}{T}\sum _{t=1}^T\left|e_t\right|$	Demonstrates the magnitude of overall error	No	Not Good	GMAE	No	-
Root Mean Squared Error (RMSE)	$\mathit{\text{RMSE}}=\sqrt{\frac{\sum _{t=1}^T{e}_t^2}{T}}$	Root square of average squared error	No	Not Good	MSE	Yes	-
Mean Absolute Percentage Error (MAPE)	$\mathit{\text{MAPE}}=\frac{1}{T}\sum _{t=1}^T\left|\frac{e_t}{y_t}\right|$	Measures the average of absolute percentage error	Yes	Not Good	MdAPE a, RMSPE b	No	-
symmetric Mean Absolute Percentage Error (sMAPE)	$\mathit{\text{sMAPE}}=\frac{2}{T}\sum _{t=1}^T\left|\frac{e_t}{y_t+x_t}\right|$	Scale the error by dividing it by the average of y t and x t	Yes	Good	MdsAPE	No	Less possibility of division by zero rather than MAPE.
Mean Absolute Relative Error (MARE)	$\mathit{\text{MARE}}=\frac{1}{T}\sum _{t=1}^T\left|\frac{e_t}{e_{\mathit{\text{RWt}}}}\right|$	Measures the average ratio of absolute error to Random walk error	Yes	Fair	MdRAE, GMRAE	No	-
Relative Measures: e.g. RelMAE (RMAE)	$\mathit{\text{RMAE}}=\frac{\mathit{\text{MAE}}}{\mathit{\text{MA}}{E}_{\mathit{\text{RW}}}}=\frac{\sum _{t=1}^T\left|e_t\right|}{\sum _{t=1}^T|e_{\mathit{\text{RWt}}|}}$	Ratio of accumulation of errors to cumulative error of Random Walk method	Yes	Not Good	RelRMSE, LMR [43], RGRMSE [44]	No	-
Mean Absolute Scaled Error (MASE)	$\mathit{\text{MASE}}=\frac{1}{T}\sum _{t=1}^T\left|\frac{e_t}{\frac{1}{T-1}\times \sum _{i=2}^T|y_i-y_{i-1}|}\right|$	Measures the average ratio of error to average error of one-step Random Walk method	Yes	Fair	RMSSE	No	-
Percent Better (PB)	$\mathit{\text{PB}}=\frac{1}{T}\sum _{t=1}^T\left[I\right\{e_t,e_{R{W}_t}\left\}\right]$	Demonstrates average number of times that method overcomes the Random Walk method	Yes	Good	-	No	Not good for calibration and close competitive methods.
	$\left|e_{s,t}\right|\le \left|e_{R{W}_t}\right|\iff I\{e_t,e_{R{W}_t}\}=1$
Mean Arctangent Absolute Percentage Error (MAAPE)	$\mathit{\text{MAAPE}}=\frac{1}{T}\sum _{t=1}^T\mathit{\text{arctan}}\left|\frac{e_t}{y_t}\right|$	Calculates the average arctangent of absolute percentage error	Yes	Good	MdAAPE	No	Smooths large errors. Solve division by zero problem.
Normalized Mean Squared Error (NMSE)	$\mathit{\text{NMSE}}=\frac{\mathit{\text{MSE}}}{{\sigma }^2}=\frac{1}{{\sigma }^{2}T}\sum _{t=1}^T{e}_t^2$	Normalized version of MSE: value of error is balanced	No	Not Good	NA	No	Balanced error by dividing by variance of real data.

^aMd represent Median ^bRMS represent Root Mean Square