scoringfunctions (version 1.2)

bregman1_rs: 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)

Value

Value of the realised Bregman score (type 1).

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:

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

References

Fissler T, Ziegel JF (2019) Order-sensitivity and equivariance of scoring functions. Electronic Journal of Statistics 13(1):1166--1211. 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. 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))