serrpower_rs: Realised squared error of power transformations score

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Realised squared error of power transformations score

Description

The function serrpower_rs computes the realised squared error of power transformations score with parameter a, when \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the prediction.

Realised squared error of power transformations score is a realised score corresponding to the squared error of power transformations scoring function serrpower_sf.

Usage

serrpower_rs(x, y, a)

Arguments

x	Prediction. It can be a vector of length n (must have the same length as \textbf{\textit{y}}).
y	Realisation (true value) of process. It can be a vector of length n (must have the same length as \textbf{\textit{x}}).
a	It can be a scalar.

Details

The realised squared error of power transformations score is defined by:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, a) := (1/n) \sum_{i = 1}^{n} L(x_i, y_i, a)

where

\textbf{\textit{x}} = (x_1, ..., x_n)^\mathsf{T}

\textbf{\textit{y}} = (y_1, ..., y_n)^\mathsf{T}

and

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

Domain of function:

Case #1

a > 0

\textbf{\textit{x}} \geq \textbf{0}

\textbf{\textit{y}} \geq \textbf{0}

Case #2

a \neq 0

\textbf{\textit{x}} > \textbf{0}

\textbf{\textit{y}} > \textbf{0}

where

\textbf{0} = (0, ..., 0)^\mathsf{T}

is the zero vector of length n and the symbols \geq and > indicate pairwise inequality.

Range of function:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, a) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}}, a

Value

Value of the realised squared error of power transformations score.

Note

For details on the squared error of power transformations scoring function, see serrpower_sf.

The concept of realised (average) scores is defined by Gneiting (2011) and Fissler and Ziegel (2019).

The realised squared error of power transformations score is the realised (average) score corresponding to the squared error of power transformations scoring function.

References

Fissler T, Ziegel JF (2019) Order-sensitivity and equivariance of scoring functions. Electronic Journal of Statistics 13(1):1166–1211. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/19-EJS1552")}.

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

serrpower_sf, meanpower_if

Examples

# Compute the realised squared error of power transformations score.

set.seed(12345)

a <- 2

x <- 0.5

y <- rlnorm(n = 100, meanlog = 0, sdlog = 1)

print(serrpower_rs(x = x, y = y, a = a))

print(serrpower_rs(x = rep(x = x, times = 100), y = y, a = a))

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