scoringfunctions (version 1.1)

bmedian_sf: \(\beta\)-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)

Value

Vector of \(\beta\)-median losses.

Arguments

x

Predictive \(\textnormal{med}^{(\beta)}(F)\) functional (prediction). It can be a vector of length \(n\) (must have the same length as \(y\)).

y

Realisation (true value) of process. It can be a vector of length \(n\) (must have the same length as \(x\)).

b

It can be a vector of length \(n\) (must have the same length as \(y\)).

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:

$$S(x, y, b) \geq 0, \forall x, y > 0, b \neq 0$$

References

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").

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)