obsweighted_sf: Observation-weighted scoring function

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Observation-weighted scoring function

Description

The function obsweighted_sf computes the observation-weighted scoring function when y materialises and x is the predictive \dfrac{\textnormal{E}_F [Y^{2}]}{\textnormal{E}_F [Y]} functional.

The observation-weighted scoring function is defined on p. 752 in Gneiting (2011).

Usage

obsweighted_sf(x, y)

Arguments

x	Predictive \dfrac{\textnormal{E}_F [Y^{2}]}{\textnormal{E}_F [Y]} functional (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).

Details

The observation-weighted scoring function is defined by:

S(x, y) := y (x - y)^{2}

Domain of function:

x > 0

y > 0

Range of function:

S(x, y) \geq 0, \forall x, y > 0

Value

Vector of observation-weighted errors.

Note

For details on the observation-weighted scoring function, see Gneiting (2011).

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

The observation-weighted scoring function is strictly \mathbb{F}-consistent for the \dfrac{\textnormal{E}_F [Y^{2}]}{\textnormal{E}_F [Y]} functional (Theorem 5 and eq. (12) in Gneiting 2011). \mathbb{F} is the family of probability distributions F concentrated on (0, \infty) for which \textnormal{E}_F[Y], \textnormal{E}_F[Y^{2}] and \textnormal{E}_F[Y^{3}] exist and are finite (Theorem 5 in Gneiting 2011).

References

Gneiting T (2011) Making and evaluating point forecasts. Journal of the American Statistical Association 106(494):746–762. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1198/jasa.2011.r10138")}.

See Also

obsweighted_rs

Examples

# Compute the observation-weighted scoring function.

df <- data.frame(
    y = rep(x = 2, times = 3),
    x = 1:3
)

df$obsweighted_penalty <- obsweighted_sf(x = df$x, y = df$y)

print(df)

scoringfunctions documentation built on Aug. 30, 2026, 5:07 p.m.