scoringfunctions (version 1.2)

meanpower_if: Power-transformed identification function

Description

The function meanpower_if computes the power-transformed identification function with parameter \(a\), when \(y\) materialises and \((\textnormal{E}_F[Y^a])^{(1/a)}\) is the predictive functional.

The power-transformed identification function is defined by Remark 1 in Tyralis and Papacharalampous (2026), applied to \(g(t) = t^a\).

Usage

meanpower_if(x, y, a)

Value

Vector of values of the power-transformed identification function.

Arguments

x

Predictive \((\textnormal{E}_F[Y^a])^{(1/a)}\) functional. 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-transformed identification function is defined by:

$$V(x, y, a) := x^a - y^a$$

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:

$$V(x, y, a) \in \mathbb{R}$$

References

Tyralis H, Papacharalampous G (2026) Variable transformations in consistent loss functions. Knowledge-Based Systems 336:115202. tools:::Rd_expr_doi("10.1016/j.knosys.2025.115202").

See Also

serrpower_sf, serrpower_rs

Examples

# Compute the power-transformed identification function.

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

df$meanpower_if <- meanpower_if(x = df$x, y = df$y, a = df$a)

print(df)