interval_sf: Interval scoring function (Winkler scoring function)

In scoringfunctions: A Collection of Loss Functions for Assessing Point Forecasts

Interval scoring function (Winkler scoring function)

Description

The function interval_sf computes the interval scoring function (Winkler scoring function) when y materialises and [x_1, x_2] is the central 1 - p prediction interval.

The interval scoring function is defined by eq. (43) in Gneiting and Raftery (2007).

Usage

interval_sf(x1, x2, y, p)

Arguments

x1	Predictive quantile (prediction) at level p/2. It can be a vector of length n (must have the same length as y).
x2	Predictive quantile (prediction) at level 1 - p/2. It can be a vector of length n (must have the same length as y).
y	Realisation (true value) of process. It can be a vector of length n (must have the same length as x_1).
p	It can be a vector of length n (must have the same length as y).

Details

The interval scoring function is defined by:

S(x_1, x_2, y, p) := (x_2 - x_1) + (2/p) (x_1 - y) \textbf{1} \lbrace y < x_1 \rbrace + (2/p) (y - x_2) \textbf{1} \lbrace y > x_2 \rbrace

Domain of function:

x_1 \in \mathbb{R}

x_2 \in \mathbb{R}

x_1 < x_2

y \in \mathbb{R}

0 < p < 1

Range of function:

S(x_1, x_2, y, p) \geq 0, \forall x_1, x_2, y \in \mathbb{R}, x_1 < x_2, p \in (0, 1)

Value

Vector of interval losses.

Note

For the definition of quantiles, see Koenker and Bassett Jr (1978).

The interval scoring function is negatively oriented (i.e. the smaller, the better).

The interval scoring function is strictly \mathbb{F}-consistent for the central 1 - p prediction interval [x_1, x_2]. x_1 and x_2 are quantile functionals at levels p/2 and 1 - p/2 respectively.

\mathbb{F} is the family of probability distributions F for which \textnormal{E}_F[Y] exists and is finite (Dunsmore 1968; Winkler 1972; Gneiting and Raftery 2007; Winkler and Murphy 1979; Fissler and Ziegel 2016; Brehmer and Gneiting 2021).

References

Brehmer JR, Gneiting T (2021) Scoring interval forecasts: Equal-tailed, shortest, and modal interval. Bernoulli 27(3):1993–2010. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3150/20-BEJ1298")}.

Dunsmore IR (1968) A Bayesian approach to calibration. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 30(2):396–405. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.2517-6161.1968.tb00740.x")}.

Fissler T, Ziegel JF (2016) Higher order elicitability and Osband's principle. The Annals of Statistics 44(4):1680–1707. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/16-AOS1439")}.

Gneiting T, Raftery AE (2007) Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association 102(477):359–378. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1198/016214506000001437")}.

Koenker R, Bassett Jr G (1978) Regression quantiles. Econometrica 46(1):33–50. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/1913643")}.

Winkler RL (1972) A decision-theoretic approach to interval estimation. Journal of the American Statistical Association 67(337):187–191. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.1972.10481224")}.

Winkler RL, Murphy AH (1979) The use of probabilities in forecasts of maximum and minimum temperatures.Meteorological Magazine 108(1288):317–329.

Examples

# Compute the interval scoring function (Winkler scoring function).

df <- data.frame(
    y = rep(x = 0, times = 6),
    x1 = c(-3, -2, -1, 0, 1, 2),
    x2 = c(1, 2, 3, 4, 5, 6),
    p = rep(x = c(0.05, 0.95), times = 3)
)

df$interval_penalty <- interval_sf(x1 = df$x1, x2 = df$x2, y = df$y, p = df$p)

print(df)

scoringfunctions documentation built on April 4, 2025, 12:28 a.m.