errorspread_sf: Error - spread scoring function

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

Error - spread scoring function

Description

The function errorspread_sf computes the error - spread scoring function, when y materialises, x_1 is the predictive mean, x_2 is the predictive variance and x_3 is the predictive skewness.

The error - spread scoring function is defined by eq. (14) in Christensen et al. (2015).

Usage

errorspread_sf(x1, x2, x3, y)

Arguments

x1	Predictive mean (prediction). It can be a vector of length n (must have the same length as y).
x2	Predictive variance (prediction). It can be a vector of length n (must have the same length as y).
x3	Predictive skewness (prediction). 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).

Details

The error - spread scoring function is defined by:

S(x_1, x_2, x_3, y) := (x_2 - (x_1 - y)^2 - (x_1 - y) x_2^{1/2} x_3)^2

Domain of function:

x_1 \in \mathbb{R}

x_2 > 0

x_3 \in \mathbb{R}

y \in \mathbb{R}

Value

Vector of error - spread losses.

Note

The mean functional is the mean \textnormal{E}_F[Y] of the probability distribution F of y (Christensen et al. 2015).

The variance functional is the variance \textnormal{Var}_F[Y] := \textnormal{E}_F[Y^2] - (\textnormal{E}_F[Y])^{2} of the probability distribution F of y (Christensen et al. 2015).

The skewness functional is the skewness \textnormal{Sk}_F[Y] := \textnormal{E}_F[((Y - \textnormal{E}_F[Y])/(\textnormal{Var}_F[Y])^{1/2})^3] (Christensen et al. 2015).

The error - spread scoring function is negatively oriented (i.e. the smaller, the better).

The error - spread scoring function is strictly consistent for the triple (mean, variance, skewness) functional (Christensen et al. 2015).

References

Christensen HM, Moroz IM, Palmer TN (2015) Evaluation of ensemble forecast uncertainty using a new proper score: Application to medium-range and seasonal forecasts. Quarterly Journal of the Royal Meteorological Society 141(687)(Part B):538–549. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/qj.2375")}.

Examples

# Compute the error - spread scoring function.

df <- data.frame(
    y = rep(x = 0, times = 6),
    x1 = c(2, 2, -2, -2, 0, 0),
    x2 = c(1, 2, 1, 2, 1, 2),
    x3 = c(3, 3, -3, -3, 0, 0)
)

df$errorspread_penalty <- errorspread_sf(x1 = df$x1, x2 = df$x2, x3 = df$x3,
    y = df$y)

print(df)

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