scoringfunctions (version 1.1)

huber_rs: Mean Huber score

Description

The function huber_rs computes the mean Huber score with parameter $a$, when $\mathbf{\text{textit{y}}}$ materialises and $\mathbf{\text{textit{x}}}$ is the prediction.

Mean Huber score is a realised score corresponding to the Huber scoring function huber_sf.

Usage

huber_rs(x, y, a)

Value

Value of the mean Huber score.

Arguments

x

Prediction. It can be a vector of length $n$ (must have the same length as $\mathbf{\text{textit{y}}}$).

y

Realisation (true value) of process. It can be a vector of length $n$ (must have the same length as $\mathbf{\text{textit{x}}}$).

a

It can be a vector of length $n$ (must have the same length as $y$) or a scalar.

Details

The mean Huber score is defined by:

$$S(\mathbf{\text{textit{x}}},\mathbf{\text{textit{y}}},a):=(1/n)\sum _{i=1}^{n}L(x_i,y_i,a)$$

where

$$\mathbf{\text{textit{x}}}=(x_1,...,x_n{)}^{\mathsf{T}}$$

$$\mathbf{\text{textit{y}}}=(y_1,...,y_n{)}^{\mathsf{T}}$$

and

$$L(x,y,a):=\{\begin{array}{ll}{\displaystyle \frac{1}{2}}(x-y{)}^2,& |x-y|\le a\\ a|x-y|-{\displaystyle \frac{1}{2}}{a}^2,& |x-y|>a\end{array}$$

Domain of function:

$$\mathbf{\text{textit{x}}}\in {\mathbb{R}}^n$$

$$\mathbf{\text{textit{y}}}\in {\mathbb{R}}^n$$

$$a>0$$

Range of function:

$$S(\mathbf{\text{textit{x}}},\mathbf{\text{textit{y}}},a)\ge 0,\mathrm{\forall }\mathbf{\text{textit{x}}},\mathbf{\text{textit{y}}}\in {\mathbb{R}}^n,a>0$$

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").

Examples

# Compute the Huber mean score.

set.seed(12345)

a <- 0.5

x <- 0

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

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

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