quantile_rs {scoringfunctions} | R Documentation |
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 |
y |
Realisation (true value) of process. It can be a vector of length
|
p |
It can be a vector of length |
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. doi:10.1214/19-EJS1552.
Gneiting T (2011) Making and evaluating point forecasts. Journal of the American Statistical Association 106(494):746–762. 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))