Scalar and vector partitions of the ranked probability score, RPS, are described and compared. These partitions are formulated in the same manner as the scalar and vector partitions of the probability score, PS, recently described by Murphy. However, since the RPS is defined in terms of cumulative probability distributions, the scalar and vector partitions of the RPS provide measures of the reliability and resolution of scalar and vector cumulative forecasts, respectively. The scalar and vector partitions of the RPS provide similar, but not equivalent (i.e., linearly related), measures of these attributes. Specifically, the reliability (resolution) of cumulative forecasts according to the scalar partition is equal to or greater (less) than their reliability (resolution) according to the vector partition. A sample collection of forecasts is used to illustrate the differences between the scalar and vector partitions of the RPS and between the vector partitions of the RPS and the PS.
Several questions related to the interpretation and use of the scalar and vector partitions of the RPS are briefly discussed, including the information that these partitions provide about the reliability and resolution of forecasts (as opposed to cumulative forecasts) and the relative merits of these partitions. These discussions indicate that, since a one-to-one correspondence exists between vector and vector cumulative forecasts, the vector partition of the RPS can also be considered to provide measures of the reliability and resolution of vector forecasts and that the vector partition is generally more appropriate than the scalar partition.
Now with the Advanced Study Program, National Center for Atmospheric Research, Boulder, Colo.
Publication No. 203 from the Department of Meteorology and Oceanography, University of Michigan, Ann Arbor