gpl1_rs: Realised generalized piecewise linear power score (type 1)

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Realised generalized piecewise linear power score (type 1)

Description

The function gpl1_rs computes the realised generalized piecewise linear power score (type 1) at a specific level p and parameter b, when \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the prediction.

Realised generalized piecewise linear power score (type 1) is a realised score corresponding to the generalized piecewise linear power scoring function (type 1) gpl1_sf.

Usage

gpl1_rs(x, y, p, b)

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}}).
p	It can be a scalar.
b	It can be a scalar.

Details

The realised generalized piecewise linear power score (type 1) is defined by:

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

where

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

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

and

L(x, y, p, b) := (1/b) (\textbf{1} \lbrace x \geq y \rbrace - p) (x^b - y^b)

Domain of function:

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

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

0 < p < 1

b > 0

where

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

is the zero vector of length n and the symbol > indicates pairwise inequality.

Range of function:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, p, b) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}} > \textbf{0}, p \in (0, 1), b > 0

Value

Value of the realised generalized piecewise linear power score (type 1).

Note

For details on the generalized piecewise linear power scoring function (type 1), see gpl1_sf.

The parameter is restricted to b > 0, where |b| = b, because g(x) = x^b/|b| is strictly increasing only for b > 0, as required by Theorem 9(b) in Gneiting (2011). See gpl1_sf.

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

The realised generalized piecewise linear power score (type 1) is the realised (average) score corresponding to the generalized piecewise linear power scoring function (type 1).

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

gpl1_sf, quantile_if

Examples

# Compute the realised generalized piecewise linear power score (type 1).

set.seed(12345)

p <- 0.7

b <- 2

x <- qlnorm(p = p, meanlog = 0, sdlog = 1)

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

print(gpl1_rs(x = x, y = y, p = p, b = b))

print(gpl1_rs(x = rep(x = x, times = 100), y = y, p = p, b = b))

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