interval_sf {scoringfunctions} | R Documentation |
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 |
x2 |
Predictive quantile (prediction) at level |
y |
Realisation (true value) of process. It can be a vector of length
|
p |
It can be a vector of length |
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. 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. 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. 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. doi:10.1198/016214506000001437.
Koenker R, Bassett Jr G (1978) Regression quantiles. Econometrica 46(1):33–50. doi:10.2307/1913643.
Winkler RL (1972) A decision-theoretic approach to interval estimation. Journal of the American Statistical Association 67(337):187–191. 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)