bregman1_rs: Realised Bregman score (type 1)

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Realised Bregman score (type 1)

Description

The function bregman1_rs computes the realised Bregman score (type 1) with parameter a, when \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the prediction.

Realised Bregman score (type 1) is a realised score corresponding to the Bregman scoring function (type 1) bregman1_sf.

Usage

bregman1_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 Bregman score (type 1) 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) := |y|^a - |x|^a - a \textnormal{sign}(x) |x|^{a - 1} (y - x)

Domain of function:

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

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

a > 1

Range of function:

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

Value

Value of the realised Bregman score (type 1).

Note

For details on the Bregman scoring function (type 1), see bregman1_sf.

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

The realised Bregman score (type 1) is the realised (average) score corresponding to the Bregman 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

bregman1_sf, mean_if

Examples

# Compute the realised Bregman score (type 1).

set.seed(12345)

a <- 3

x <- 0

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

print(bregman1_rs(x = x, y = y, a = a))

print(bregman1_rs(x = rep(x = x, times = 100), y = y, a = a))

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