scoringfunctions (version 1.2)

serrsq_rs: Realised squared error of squares score

Description

The function serrsq_rs computes the realised squared error of squares score when \(\textbf{\textit{y}}\) materialises and \(\textbf{\textit{x}}\) is the prediction.

Realised squared error of squares score is a realised score corresponding to the squared error of squares scoring function serrsq_sf.

Usage

serrsq_rs(x, y)

Value

Value of the realised squared error of squares score.

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 \(\textbf{\textit{x}}\)).

Details

The realised squared error of squares score is defined by:

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

where

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

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

and

$$L(x, y) := (x^2 - y^2)^2$$

Domain of function:

$$\textbf{\textit{x}} \geq \textbf{0}$$

$$\textbf{\textit{y}} \geq \textbf{0}$$

where

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

is the zero vector of length \(n\) and the symbol \(\geq\) indicates pairwise inequality.

Range of function:

$$S(\textbf{\textit{x}}, \textbf{\textit{y}}) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \geq \textbf{0}$$

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").

See Also

serrsq_sf

Examples

# Compute the realised squared error of squares score.

set.seed(12345)

x <- 0.5

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

print(serrsq_rs(x = x, y = y))

print(serrsq_rs(x = rep(x = x, times = 100), y = y))