aerr_sf: Absolute error scoring function

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

Absolute error scoring function

Description

The function aerr_sf computes the absolute error scoring function when y materialises and x is the predictive median functional.

The absolute error scoring function is defined in Table 1 in Gneiting (2011).

Usage

aerr_sf(x, y)

Arguments

x	Predictive median 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).

Details

The absolute error scoring function is defined by:

S(x, y) := |x - y|

Domain of function:

x \in \mathbb{R}

y \in \mathbb{R}

Range of function:

S(x, y) \geq 0, \forall x, y \in \mathbb{R}

Value

Vector of absolute errors.

Note

For details on the absolute error scoring function, see Gneiting (2011).

The median functional is the median of the probability distribution F of y (Gneiting 2011).

The absolute error scoring function is negatively oriented (i.e. the smaller, the better).

The absolute error scoring function is strictly \mathbb{F}-consistent for the median functional. \mathbb{F} is the family of probability distributions F for which \textnormal{E}_F[Y] exists and is finite (Raiffa and Schlaifer 1961, p.196; Ferguson 1967, p.51; Thomson 1979; Saerens 2000; Gneiting 2011).

References

Ferguson TS (1967) Mathematical Statistics: A Decision-Theoretic Approach. Academic Press, New York.

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")}.

Raiffa H,Schlaifer R (1961) Applied Statistical Decision Theory. Colonial Press, Clinton.

Saerens M (2000) Building cost functions minimizing to some summary statistics. IEEE Transactions on Neural Networks 11(6):1263–1271. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1109/72.883416")}.

Thomson W (1979) Eliciting production possibilities from a well-informed manager. Journal of Economic Theory 20(3):360–380. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/0022-0531(79)90042-5")}.

Examples

# Compute the absolute error scoring function.

df <- data.frame(
    y = rep(x = 0, times = 5),
    x = -2:2
)

df$absolute_error <- aerr_sf(x = df$x, y = df$y)

print(df)

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