Quantile score (pinball loss)

Computes the asymmetric absolute loss commonly used to assess quantile forecasts. Lower scores indicate a better match between the estimated quantile and the observed value at level tau.

Usage

quantile_score(x, xhat, tau)

Arguments

x

Numeric vector of observed values.

xhat

Numeric vector of estimated quantiles.

tau

Numeric vector of quantile levels in (0, 1).

Value

Numeric vector of quantile scores corresponding to each element of the recycled inputs.

Details

The score minimises to zero when the observation equals the estimated quantile, so that smaller scores indicate a better fitting model. Positive residuals are penalised by a factor of tau, and negative residuals by tau - 1. This loss is also known as the stick function, check loss, asymmetric absolute deviation, or pinball loss.

For observation x, estimate x_hat, and level tau, the score (c.f. Gneiting, 2011) is

$$ S_\tau(x, \hat{x}) = \begin{cases} \tau |x - \hat{x})|, & x \ge \hat{x}, \\ (1 - \tau)|x - \hat{x}|, & x < \hat{x}. \end{cases} $$

Vector recycling of all three arguments follows the rules in vctrs::vec_recycle_common().

References

Gneiting, T. (2011). Making and evaluating point forecasts. Journal of the American Statistical Association, 106(494), 746–762.

Examples

quantile_score(c(5, 15, 10), xhat = 7, tau = 0.8)
#> [1] 0.4 6.4 2.4
quantile_score(c(5, 15, 10), xhat = c(6, 19, 12), tau = c(0.2, 0.9, 0.5))
#> [1] 0.8 0.4 1.0