Table 8.

List of advanced error measures to aggregating the error values across multiple series

Measure name	Formula	Description
Absolute Percentage Error (A P E t,s)	$\mathit{\text{AP}}{E}_{t,s}=\left|\frac{y_t-x_{t,s}}{y_t}\right|$	where t is time horizon and s is the series index.
Mean Absolute Percentage Error (M A P E t)	$\mathit{\text{MAPE}}=\frac{1}{S}\sum _{s=1}^S\mathit{\text{AP}}{E}_{t,s}$	where t is time horizon, s is the series index S is the number of series for the method.
Median Absolute Percentage Error (M d A P E t)	Median Observation of A P E s	Obtaining median of APE errors over series.
Relative Absolute Error (R A E t,s)	$\mathit{\text{RA}}{E}_{t,s}=\frac{|y_t-x_{t,s}|}{|y_t-x_{R{W}_{t,s}}|}$	Measures the ratio of absolute error to Random walk error in time horizon t.
Geometric Mean Relative Absolute Error (G M R A E t)	$\mathit{\text{GMRA}}{E}_t={\left[\prod _{s=1}^S\right|\mathit{\text{RA}}{E}_{t,s}\left|\right]}^{1/S}$	Measures the Geometric average ratio of absolute error to Random walk error
Median Relative Absolute Error (M d R A E t)	Median Observation of R A E s	Measures the median observation of R A E s for time horizon t
Cumulative Relative Error (C u m R A E s)	$\mathit{\text{CumRA}}{E}_s=\frac{\sum _{t=1}^T|y_{t,s}-x_{t,s}|}{\sum _{t=1}^T|y_{t,s}-x_{R{W}_{t,s}}|}$	Ratio of accumulation of errors to cumulative error of Random walk Method
Geometric Mean Cumulative Relative Error (GMCumRAE)	$\mathit{\text{GMCumRAE}}={\left[\prod _{s=1}^S\right|\mathit{\text{CumRA}}{E}_s\left|\right]}^{1/S}$	Geometric Mean of Cumulative Relative Error across all series.
Median Cumulative Relative Error (MdCumRAE)	M d C u m R A E=M e d i a n(|C u m R A E s|)	Median of Cumulative Relative Error across all series.
Root Mean Squared Error (R M S E t)	$\mathit{\text{RMS}}{E}_t=\sqrt{\frac{\sum _{s=1}^S{(y_t-x_{t,s})}^2}{S}}$	Square root of average squared error across series in time horizon t
Percent Better (P B t)	$P{B}_t=\frac{1}{S}\sum _{s=1}^S\left[I\right\{e_{s,t},e_{\mathit{\text{WRt}}}\left\}\right]$	Demonstrates average number of times that method overcomes the Random Walk method in time horizon t.
	|e s,t|≤|e WRt|⇔I{e s,t,e WRt}=1