ghuber_rs: Realised generalized Huber score

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Realised generalized Huber score

Description

The function ghuber_rs computes the realised generalized Huber score at a specific level p and parameters a and b, when \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the prediction.

Realised generalized Huber score is a realised score corresponding to the generalized Huber scoring function ghuber_sf.

Usage

ghuber_rs(x, y, p, a, 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.
a	It can be a value.
b	It can be a value.

Details

The realised generalized Huber score is defined by:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, p, a, b) := (1/n) \sum_{i = 1}^{n} L(x_i, y_i, p, a, 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, a, b) := |\textbf{1} \lbrace x \geq y \rbrace - p| f_{a, b}(x - y)

where

f_{a, b}(t) := \kappa_{a, b}(t) (2 t - \kappa_{a, b}(t))

and \kappa_{a, b}(t) is the capping function defined by:

\kappa_{a, b}(t) := \max \lbrace \min \lbrace t, b \rbrace, -a \rbrace

Domain of function:

\textbf{\textit{x}} \in \mathbb{R}^n

\textbf{\textit{y}} \in \mathbb{R}^n

0 < p < 1

a > 0

b > 0

Range of function:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, p, a, b) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n, p \in (0, 1), a, b > 0

Value

Value of the realised generalized Huber score.

Note

For details on the generalized Huber scoring function, see ghuber_sf.

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

The realised generalized Huber score is the realised (average) score corresponding to the generalized Huber 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

ghuber_sf, huberquantile_if

Examples

# Compute the realised generalized Huber score.

set.seed(12345)

p <- 0.7

a <- 0.5

b <- 0.5

x <- 0

y <- rnorm(n = 100, mean = 0, sd = 1)

print(ghuber_rs(x = x, y = y, p = p, a = a, b = b))

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

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