powerweighted_sf: Power-weighted squared error scoring function

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Power-weighted squared error scoring function

Description

The function powerweighted_sf computes the power-weighted squared error scoring function with parameter a, when y materialises and x is the predictive \dfrac{\textnormal{E}_F [Y^{a + 1}]}{\textnormal{E}_F [Y^a]} functional.

The power-weighted squared error scoring function is defined by eqs. (10) and (11) in Gneiting (2011), applied to the weight function w(y) = y^a and to the squared error scoring function.

Usage

powerweighted_sf(x, y, a)

Arguments

x	Predictive \dfrac{\textnormal{E}_F [Y^{a + 1}]}{\textnormal{E}_F [Y^a]} 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).
a	It can be a vector of length n (must have the same length as y).

Details

The power-weighted squared error scoring function is defined by:

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

Domain of function:

x > 0

y > 0

a \in \mathbb{R}

Range of function:

S(x, y, a) \geq 0, \forall x, y > 0, a \in \mathbb{R}

Value

Vector of power-weighted squared errors.

Note

The \dfrac{\textnormal{E}_F [Y^{a + 1}]}{\textnormal{E}_F [Y^a]} functional is the ratio of expectations formed by weighting the probability distribution F of Y with w(y) = y^a (eq. (11) in Gneiting 2011).

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

The power-weighted squared error scoring function is strictly \mathbb{F}-consistent for the \dfrac{\textnormal{E}_F [Y^{a + 1}]}{\textnormal{E}_F [Y^a]} functional (Theorem 5 in Gneiting 2011). \mathbb{F} is the family of probability distributions F for which \textnormal{E}_F[Y^a], \textnormal{E}_F[Y^{a + 1}] and \textnormal{E}_F[Y^{a + 2}] exist and are finite (Theorem 5 in Gneiting 2011).

At a = 0 the power-weighted squared error scoring function reduces to serr_sf and the functional to the mean \textnormal{E}_F[Y], at a = 1 it reduces to obsweighted_sf, and at a = -2 it reduces to sperr_sf.

srelerr_sf targets the same functional as the case a = 1, but is not a member of this family, because it weights with the prediction x rather than with the realisation y (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

powerweighted_if

Examples

# Compute the power-weighted squared error scoring function.

df <- data.frame(
    y = rep(x = 2, times = 6),
    x = c(1, 2, 3, 1, 2, 3),
    a = rep(x = c(1, -2), each = 3)
)

df$powerweighted_penalty <- powerweighted_sf(x = df$x, y = df$y, a = df$a)

print(df)

# The power-weighted squared error scoring function reduces to the squared
# error scoring function at a = 0, to the observation-weighted scoring function
# at a = 1 and to the squared percentage error scoring function at a = -2.

set.seed(12345)

n <- 10

x <- runif(n = n, min = 0, max = 2)
y <- runif(n = n, min = 0, max = 2)

max(abs(powerweighted_sf(x = x, y = y, a = 0) - serr_sf(x = x, y = y)))

max(abs(powerweighted_sf(x = x, y = y, a = 1) - obsweighted_sf(x = x, y = y)))

max(abs(powerweighted_sf(x = x, y = y, a = -2) - sperr_sf(x = x, y = y)))

# values are slightly higher than 0 due to rounding error

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