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David B. Stephenson

Abstract

This study investigates ways of quantifying the skill in forecasts of dichotomous weather events. The odds ratio, widely used in medical studies, can provide a powerful way of testing the association between categorical forecasts and observations. A skill score can be constructed from the odds ratio that is less sensitive to hedging than previously used scores. Furthermore, significance tests can easily be performed on the logarithm of the odds ratio to test whether the skill is purely due to chance sampling. Functions of the odds ratio and the Peirce skill score define a general class of skill scores that are symmetric with respect to taking the complement of the event. The study illustrates the ideas using Finley’s classic set of tornado forecasts.

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Ian T. Jolliffe
and
David B. Stephenson
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Christopher A. T. Ferro
and
David B. Stephenson

Abstract

Verifying forecasts of rare events is challenging, in part because traditional performance measures degenerate to trivial values as events become rarer. The extreme dependency score was proposed recently as a nondegenerating measure for the quality of deterministic forecasts of rare binary events. This measure has some undesirable properties, including being both easy to hedge and dependent on the base rate. A symmetric extreme dependency score was also proposed recently, but this too is dependent on the base rate. These two scores and their properties are reviewed and the meanings of several properties, such as base-rate dependence and complement symmetry that have caused confusion are clarified. Two modified versions of the extreme dependency score, the extremal dependence index, and the symmetric extremal dependence index, are then proposed and are shown to overcome all of its shortcomings. The new measures are nondegenerating, base-rate independent, asymptotically equitable, harder to hedge, and have regular isopleths that correspond to symmetric and asymmetric relative operating characteristic curves.

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Robin J. Hogan
,
Christopher A. T. Ferro
,
Ian T. Jolliffe
, and
David B. Stephenson

Abstract

In the forecasting of binary events, verification measures that are “equitable” were defined by Gandin and Murphy to satisfy two requirements: 1) they award all random forecasting systems, including those that always issue the same forecast, the same expected score (typically zero), and 2) they are expressible as the linear weighted sum of the elements of the contingency table, where the weights are independent of the entries in the table, apart from the base rate. The authors demonstrate that the widely used “equitable threat score” (ETS), as well as numerous others, satisfies neither of these requirements and only satisfies the first requirement in the limit of an infinite sample size. Such measures are referred to as “asymptotically equitable.” In the case of ETS, the expected score of a random forecasting system is always positive and only falls below 0.01 when the number of samples is greater than around 30. Two other asymptotically equitable measures are the odds ratio skill score and the symmetric extreme dependency score, which are more strongly inequitable than ETS, particularly for rare events; for example, when the base rate is 2% and the sample size is 1000, random but unbiased forecasting systems yield an expected score of around −0.5, reducing in magnitude to −0.01 or smaller only for sample sizes exceeding 25 000. This presents a problem since these nonlinear measures have other desirable properties, in particular being reliable indicators of skill for rare events (provided that the sample size is large enough). A potential way to reconcile these properties with equitability is to recognize that Gandin and Murphy’s two requirements are independent, and the second can be safely discarded without losing the key advantages of equitability that are embodied in the first. This enables inequitable and asymptotically equitable measures to be scaled to make them equitable, while retaining their nonlinearity and other properties such as being reliable indicators of skill for rare events. It also opens up the possibility of designing new equitable verification measures.

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