quantile_rs: Realised quantile score

In scoringfunctions: A Collection of Loss Functions for Assessing Point Forecasts

Realised quantile score

Description

The function quantile_rs computes the realised quantile score at a specific level p when \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the prediction.

Realised quantile score is a realised score corresponding to the quantile scoring function quantile_sf.

Usage

quantile_rs(x, y, p)

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 vector of length n (must have the same length as \textbf{\textit{y}}) or a scalar value.

Details

The realized quantile score is defined by:

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

where

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

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

and

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

Domain of function:

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

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

0 < p < 1

Range of function:

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

Value

Value of the realised quantile score.

Note

For details on the quantile scoring function, see quantile_sf.

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

The realised quantile score is the realised (average) score corresponding to the quantile scoring function.

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")}.

Examples

# Compute the realised quantile score.

set.seed(12345)

x <- qnorm(p = 0.7, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

y <- rnorm(n = 1000, mean = 0, sd = 1)

print(quantile_rs(x = x, y = y, p = 0.7))

print(quantile_rs(x = rep(x = x, times = 1000), y = y, p = 0.7))

print(quantile_rs(x = rep(x = x, times = 1000) - 0.1, y = y, p = 0.7))

scoringfunctions documentation built on April 4, 2025, 12:28 a.m.