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Table 8 List of advanced error measures to aggregating the error values across multiple series

From: A framework for evaluating epidemic forecasts

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