bmedian_sf: beta-median scoring function
| bmedian_sf | R Documentation |
\beta-median scoring function
Description
The function bmedian_sf computes the \beta-median scoring function
when y materialises and x is the predictive
\textnormal{med}^{(\beta)}(F) functional.
The \beta-median scoring function is defined in eq. (4) in Gneiting
(2011).
Usage
bmedian_sf(x, y, b)
Arguments
x |
Predictive |
y |
Realisation (true value) of process. It can be a vector of length
|
b |
It can be a vector of length |
Details
The \beta-median scoring function is defined by:
S(x, y, b) := |1 - (y/x)^b|
Domain of function:
x > 0
y > 0
b \neq 0
Range of function:
S(x, y, b) \geq 0, \forall x, y > 0, b \neq 0
Value
Vector of \beta-median losses.
Note
For details on the \beta-median scoring function, see Gneiting (2011).
The \beta-median functional, \textnormal{med}^{(\beta)}(F) is the
median of a probability distribution whose density is proportional to
y^\beta f(y), where f is the density of the probability distribution
F of y (Gneiting 2011).
The \beta-median scoring function is negatively oriented (i.e. the
smaller, the better).
The \beta-median scoring function is strictly
\mathbb{F}^{(w)}-consistent for the \textnormal{med}^{(\beta)}(F)
functional. \mathbb{F} is the family of probability distributions for
which \textnormal{E}_F[Y] exists and is finite. \mathbb{F}^{(w)} is
the subclass of probability distributions in \mathbb{F}, which are such
that w(y) f(y), w(y) = y^\beta has finite integral over
(0, \infty), and the probability distribution F^{(w)} with density
proportional to w(y) f(y) belongs to \mathbb{F} (see Theorems 5 and
9 in Gneiting 2011)
References
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 bmedian scoring function.
df <- data.frame(
y = rep(x = 2, times = 3),
x = 1:3,
b = c(-1, 1, 2)
)
df$bmedian_error <- bmedian_sf(x = df$x, y = df$y, b = df$b)
print(df)
