scoringfunctions (version 1.2)

lqquantile_rs: Realised \(L_q\)-quantile score

Description

The function lqquantile_rs computes the realised \(L_q\)-quantile score at a specific level \(p\) and parameter \(q\), when \(\textbf{\textit{y}}\) materialises and \(\textbf{\textit{x}}\) is the prediction.

Realised \(L_q\)-quantile score is a realised score corresponding to the \(L_q\)-quantile scoring function lqquantile_sf.

Usage

lqquantile_rs(x, y, p, q)

Value

Value of the realised \(L_q\)-quantile 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 \(\textbf{\textit{x}}\)).

p

It can be a scalar.

q

It can be a scalar.

Details

The realised \(L_q\)-quantile score is defined by:

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

where

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

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

and

$$ L(x, y, p, q) := |\textbf{1} \lbrace x \geq y \rbrace - p| |x - y|^q $$

Domain of function:

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

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

$$0 < p < 1$$

$$q > 1$$

Range of function:

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

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

lqquantile_sf

Examples

# Compute the realised Lq-quantile score.

set.seed(12345)

p <- 0.7

q <- 2

x <- qnorm(p = p, mean = 0, sd = 1)

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

print(lqquantile_rs(x = x, y = y, p = p, q = q))

print(lqquantile_rs(x = rep(x = x, times = 100), y = y, p = p, q = q))