expectile_rs {scoringfunctions} | R Documentation |
Realised expectile score
Description
The function expectile_rs computes the realised expectile score at a specific
level p
when \textbf{\textit{y}}
materialises and
\textbf{\textit{x}}
is the prediction.
Realised expectile score is a realised score corresponding to the expectile scoring function expectile_sf.
Usage
expectile_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 expectile 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)^2
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 expectile score.
Note
For details on the expectile scoring function, see expectile_sf.
The concept of realised (average) scores is defined by Gneiting (2011) and Fissler and Ziegel (2019).
The realised expectile score is the realised (average) score corresponding to the expectile 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 expectile score.
set.seed(12345)
x <- 0.5
y <- rnorm(n = 100, mean = 0, sd = 1)
print(expectile_rs(x = x, y = y, p = 0.7))
print(expectile_rs(x = rep(x = x, times = 100), y = y, p = 0.7))