Extracting Predictive Information from an Ensemble
The Bayesian Processor of Ensemble (BPE) produces a posterior distribution, which quantifies uncertainty about a predictand, by optimally fusing climatic data with an ensemble forecast output from a numerical weather prediction (NWP) model. In principle, the posterior distribution is conditional on all ensemble members, which enter the BPE via a likelihood function. In application, the challenge is to reduce the dimensionality of the conditioning, without losing any predictive information, by identifying sufficient statistics of the ensemble.
A methodology for identifying the sufficient statistics is presented along with an application to 20-member ensemble forecasts of surface temperature at 1200 UTC produced with lead times of 12–348 h (day 1 – day 15) during 2 years (March 2007 – February 2009) at the National Centers for Environmental Prediction. After a suitable standardization of ensemble and predictand time series to make them margin stationary (throughout the year), individual ensemble members, ensemble statistics, derived statistics, and their combinations are compared in terms of their informativeness (as predictors of either a conditional mean or a conditional variance).
The results support three hypotheses: (i) That the ensemble has a reduced conditional dependence structure albeit the members are not conditionally independent (viz, do not constitute a random sample). (ii) That a few statistics (about 2–5) are as informative as the entire ensemble; however, these sufficient statistics vary across lead times and seasons (viz, they are unstable). An adaptive BPE, which can identify these sufficient statistics in real time, may be attractive for sophisticated users with specific forecast needs, who can benefit from every bit of predictive information extracted from the ensemble. (iii) That there exist several pairs of fixed statistics which offer a reasonable trade-off: they are somewhat less informative than the sufficient statistics but stable (across lead times and seasons). A BPE with such fixed statistics as predictors may be preferable for mass-processing of ensemble forecasts at centers, which must perform the task in a limited time, for all predictands, lead times, and grid points of a global NWP model.