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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