Measure name | Formula | Description |
---|---|---|
Absolute Percentage Error (A P E t,s ) | \( APE_{t,s}=|\frac {y_{t} - x_{t,s}}{y_{t}}| \) | where t is time horizon and s is the series index. |
Mean Absolute Percentage Error (M A P E t ) | \( MAPE=\frac {1}{S} \sum _{s=1}^{S} APE_{t,s} \) | 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 ) | \( RAE_{t,s}=\frac {|y_{t} - x_{t,s}|}{|y_{t} - x_{RW_{t,s}}|} \) | 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 ) | \( GMRAE_{t}= [\prod _{s=1}^{S} |RAE_{t,s}| ]^{1/S} \) | 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 ) | \( CumRAE_{s} =\frac {\sum _{t=1}^{T} |y_{t,s} - x_{t,s}|}{\sum _{t=1}^{T}|y_{t,s} - x_{RW_{t,s}}|} \) | Ratio of accumulation of errors to cumulative error of Random walk Method |
Geometric Mean Cumulative Relative Error (GMCumRAE) | \( GMCumRAE =[\prod _{s=1}^{S} |CumRAE_{s}| ]^{1/S} \) | 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 ) | \( RMSE_{t}= \sqrt {\frac {\sum _{s=1}^{S} (y_{t} - x_{t,s})^{2}}{S}} \) | Square root of average squared error across series in time horizon t |
Percent Better (P B t ) | \( PB_{t}=\frac {1}{S} \sum _{s=1}^{S} [I\{e_{s,t},e_{WRt}\}] \) | 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 |