serrpower_sf: Squared error of power transformations scoring function

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Squared error of power transformations scoring function

Description

The function serrpower_sf computes the squared error of power transformations scoring function when y materialises and x is the (\textnormal{E}_F[Y^a])^{(1/a)} predictive functional.

The squared error of power transformations scoring function is defined in Tyralis and Papacharalampous (2026).

Usage

serrpower_sf(x, y, a)

Arguments

x	Predictive (\textnormal{E}_F[Y^a])^{(1/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 squared error of power transformations scoring function is defined by:

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

Domain of function:

Case #1

a > 0

x \geq 0

y \geq 0

Case #2

a \neq 0

x > 0

y > 0

Range of function:

Case #1

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

Case #2

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

Value

Vector of squared errors of power-transformed variables.

Note

For details on the squared error of power transformations scoring function, see Tyralis and Papacharalampous (2026).

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

The squared error of power transformations scoring function is strictly \mathbb{F}-consistent for the (\textnormal{E}_F[Y^a])^{(1/a)} functional. \mathbb{F} is the family of probability distributions F for which \textnormal{E}_F[Y^{2 a}] exists and is finite (Tyralis and Papacharalampous 2026).

References

Tyralis H, Papacharalampous G (2026) Variable transformations in consistent loss functions. Knowledge-Based Systems 336:115202. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.knosys.2025.115202")}.

See Also

serrpower_rs, meanpower_if

Examples

# Compute the squared error of power transformations scoring function.

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

df$squaredpower_error <- serrpower_sf(x = df$x, y = df$y, a = df$a)

print(df)

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