srelerr_sf: Squared relative error scoring function

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Squared relative error scoring function

Description

The function srelerr_sf computes the squared relative error scoring function when y materialises and x is the predictive \dfrac{\textnormal{E}_F [Y^{2}]}{\textnormal{E}_F [Y]} functional.

The squared relative error scoring function is defined on p. 752 in Gneiting (2011).

Usage

srelerr_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 squared relative error scoring function is defined by:

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

Domain of function:

x > 0

y > 0

Range of function:

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

Value

Vector of squared relative errors.

Note

For details on the squared relative error scoring function, see Gneiting (2011).

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

The squared relative error scoring function is strictly \mathbb{F}-consistent for the \dfrac{\textnormal{E}_F [Y^{2}]}{\textnormal{E}_F [Y]} functional (eq. (12) on p. 752 in Gneiting 2011). \mathbb{F} is the family of probability distributions F concentrated on (0, \infty) for which \textnormal{E}_F[Y^{2}] exists and is finite (Gneiting 2011).

The squared relative error scoring function weights with the prediction x rather than with the realisation y, so it is not a member of the weighted family of Theorem 5 in Gneiting (2011); it targets the same functional as obsweighted_sf, which is \dfrac{\textnormal{E}_F [Y^{2}]}{\textnormal{E}_F [Y]} (eq. (12) on p. 752 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

msre

Examples

# Compute the squared relative error scoring function.

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

df$squared_relative_error <- srelerr_sf(x = df$x, y = df$y)

print(df)

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