A Note on Gandin and Murphy's Equitable Skill Score

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Abstract
Gandin and Murphy introduced an “equitable skill score” for use in evaluating categorical forecasts. For forecasts involving more than two categories, the elements of the scoring matrix are not defined uniquely. In this note, a specific formula for the general multiple-category scoring matrix is presented and proven to satisfy the necessary conditions for “equitability.” It is shown that, while it is not the only possible scoring matrix satisfying these necessary conditions, it is compatible with a logical condensation of the general K-category problem into a set of K−1 two-category problems. Each of the two-category problems is associated with one of the K−1 partitions defining the categories of the original problem. Abstract Gandin and Murphy introduced an “equitable skill score” for use in evaluating categorical forecasts. For forecasts involving more than two categories, the elements of the scoring matrix are not defined uniquely. In this note, a specific formula for the general multiple-category scoring matrix is presented and proven to satisfy the necessary conditions for “equitability.” It is shown that, while it is not the only possible scoring matrix satisfying these necessary conditions, it is compatible with a logical condensation of the general K-category problem into a set of K−1 two-category problems. Each of the two-category problems is associated with one of the K−1 partitions defining the categories of the original problem.

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