Using the inverse of expected error variance to determine weights of individual ensemble members: Application to temperature prediction

The Impact of Applying Individually Perturbed Parametrization Tendency Scheme on the Simulated El Niño-Southern Oscillation in the Community Earth System Model

“…A simple average without any weighting of ensemble member status will result in ignoring the useful signals. This may call for proper weights, as Sun et al (2017) proposed, to be determined before summarizing such a collection of projections.…”

An Investigation Into the Variant‐Weight Multi‐Model Ensemble Forecasting Technique Based on the Analyses of Model Systematic Errors Calculation and Elimination

“…The computational formula for the variant‐weight multi‐model ensemble forecast $${F}_{jk}$$ at the k th leading time of the j th day can be expressed as $$${F}_{jk}=\sum\limits _{i=1}^{n}{\omega }_{ijk}\times {f}_{ijk},$$$ where $${\omega }_{ijk}$$ and $${f}_{ijk}$$ are the weight coefficient and forecasting value at the k th leading time of the j th day for the i th model, respectively; the weight $${\omega }_{ijk}$$ uses the form of reciprocal of forecasting error whose computational formula is denoted as follow (Sun et al., 2017; Wu et al., 2017): $$${\omega }_{ijk}=\frac{{e}_{ijk}^{-1}}{\sum\limits _{i=1}^{n}{e}_{ijk}^{-1}}.$$$ …”

“…In Equation , $${e}_{ijk}$$ is the reference forecasting error at the k th leading time of the j th day for the i th model, which generally adopts absolute forecasting error (Wu et al., 2017) or mean square forecasting error (Sun et al., 2017) over a period before the forecast day, and n is the number of member models.…”

“…Sun et al. (2017) proved that when taking mean square error as the reference forecasting error, incorporating an additional model with a poorer performance as a new member into the original ensemble would not reduce the ensemble forecasting accuracy. Thus, mean square error is adopted as the reference forecasting error in this paper, and under the conditions without systematic error taken into consideration, its mathematical formula can be written as $$${e}_{ijk}=\left(\sum\limits _{d=j}^{j-m+1}{\left({f}_{idk}-{O}_{dk}\right)}^{2}\right)/m,$$$ where m is the number of the days over which the average forecasting error is calculated; and $${O}_{dk}$$ and $${f}_{idk}$$ are the observation and the forecast for the i th model at the k th leading time of the d th day, respectively.…”

“…where 𝐴𝐴 𝐴𝐴𝑖𝑖𝑖𝑖𝑖𝑖 and 𝐴𝐴 𝐴𝐴𝑖𝑖𝑖𝑖𝑖𝑖 are the weight coefficient and forecasting value at the kth leading time of the jth day for the ith model, respectively; the weight 𝐴𝐴 𝐴𝐴𝑖𝑖𝑖𝑖𝑖𝑖 uses the form of reciprocal of forecasting error whose computational formula is denoted as follow (Sun et al, 2017;Wu et al, 2017):…”