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Allan H. Murphy

Abstract

Probability of precipitation (PoP) forecasts can often be interpreted as average point probability forecasts. Since the latter are equivalent to (unconditional) expected areal coverage forecasts, PoP forecasts can be evaluated in terms of observed areal coverages in those situations in which observations of precipitation occurrence are available from a network of points in the forecast area. The purpose of this paper is to describe a partition of the average Brier, or probability, score—a measure of the average accuracy of average point probability forecasts over the network of points of concern—that facilitates such an evaluation. The partition consists of two terms: 1) a term that represents the average squared error of the average point probability forecasts interpreted as areal coverage forecasts and 2) a term that represents the average variance of the observations of precipitation occurrence in the forecast area. The relative magnitudes of the terms in this partition are examined, and it is concluded (party on the basis of experimental data) that the variance term generally makes a significant contribution to the overall probability score. This result, together with the fact that the variance term does not depend on the forecasts, suggests that the squared error term (rather than the overall score) should be used to evaluate PoP forecasts in many situations. The basis for the interpretation of PoP forecasts as average point probability forecasts and some implications of the results presented in this paper for the evaluation of PoP forecasts are briefly discussed.

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Allan H. Murphy

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Allan H. Murphy

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Allan H. Murphy

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Allan H. Murphy

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Allan H. Murphy

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Allan H. Murphy

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Allan H. Murphy

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ALLAN H. MURPHY

Abstract

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.

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Allan H. Murphy

Abstract

An individual skill score (SS) and a collective skill score (CSS) are examined to determine whether these scoring or improper. The SS and the CSS are both standardized versions of the Brier, or probability, score (PS) and have been used to measure the “skill” of probability forecasts. The SS is defined in terms of individual forecasts, while the CSS is defined in terms of collections of forecasts. The SS and the CSS are shown to be improper scoring rules, and, as a result, both the SS and the CSS encourage hedging on the part of forecasters.

The results of a preliminary, investigation of the nature of the hedging produced by. the SS and the CSS indicate that, while the SS may encourage a considerable amount of hedging, the CSS, in general, encourages only a modest amount of hedging, and even this hedging decreases as the sample size K of the collection forecasts increases. In fact, the CSS is approximately strictly Proper for large collections of forecasts (K ≥ 100).

Finally, we briefly consider two questions related to the standardization of scoring rules: 1) the use of different scoring rules in the assessment and evaluation tasks, and 2) the transformation of strictly proper scoring rules. With regard to the latter, we identify standardized versions of the PS which are strictly proper scoring rules and which, as a result, appear to be appropriate scoring rules to use to measure the “skill” of probability forecasts.

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