Spherical score for binary and categorical forecasts

Computes the positively oriented spherical score. Vector probability input is treated as binary; matrix probability input is treated as categorical.

Usage

spherical_score(p, y)

Arguments

p

Numeric vector in [0, 1] for binary forecasts, or a numeric matrix whose rows are probability vectors for categorical forecasts.

y

For binary vector input, numeric vector in {0, 1}. For categorical matrix input, integer vector in {1, ..., K}, where K = ncol(p).

Value

Numeric vector of scores in [0, 1]. Higher is better.

Details

For binary forecasts, this computes $$S(p, y) = \frac{py + (1-p)(1-y)}{\sqrt{p^2 + (1-p)^2}}.$$ For categorical forecasts, this computes $$S(\mathbf{p}, y) = \frac{p_y}{\|\mathbf{p}\|_2},$$ where p_y is the forecast probability assigned to the realised category.

Score differences lie in [-1, 1], so use c = 1 for Theorem 1 and c = 2 for Theorems 2 and 3.

Examples

p <- c(0.2, 0.7, 0.9)
y <- c(0, 1, 1)
spherical_score(p, y)
#> [1] 0.9701425 0.9191450 0.9938837