quantile_sf: Asymmetric piecewise linear scoring function (quantile...

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

Asymmetric piecewise linear scoring function (quantile scoring function, quantile loss function)

Description

The function quantile_sf computes the asymmetric piecewise linear scoring function (quantile scoring function) at a specific level p, when y materialises and x is the predictive quantile at level p.

The asymmetric piecewise linear scoring function is defined by eq. (24) in Gneiting (2011).

Usage

quantile_sf(x, y, p)

Arguments

x	Predictive quantile (prediction) at level p. 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).
p	It can be a vector of length n (must have the same length as y).

Details

The assymetric piecewise linear scoring function is defined by:

S(x, y, p) := (\textbf{1} \lbrace x \geq y \rbrace - p) (x - y)

or equivalently,

S(x, y, p) := p | \max \lbrace -(x - y), 0 \rbrace | + (1 - p) | \max \lbrace x - y, 0 \rbrace |

Domain of function:

x \in \mathbb{R}

y \in \mathbb{R}

0 < p < 1

Range of function:

S(x, y, p) \geq 0, \forall x, y \in \mathbb{R}, p \in (0, 1)

Value

Vector of quantile losses.

Note

For the definition of quantiles, see Koenker and Bassett Jr (1978).

The asymmetric piecewise linear scoring function is negatively oriented (i.e. the smaller, the better).

The asymmetric piecewise linear scoring function is strictly \mathbb{F}-consistent for the p-quantile functional. \mathbb{F} is the family of probability distributions F for which \textnormal{E}_F[Y] exists and is finite (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")}.

Koenker R, Bassett Jr G (1978) Regression quantiles. Econometrica 46(1):33–50. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/1913643")}.

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 asymmetric piecewise linear scoring function (quantile scoring
# function).

df <- data.frame(
    y = rep(x = 0, times = 6),
    x = c(2, 2, -2, -2, 0, 0),
    p = rep(x = c(0.05, 0.95), times = 3)
)

df$quantile_penalty <- quantile_sf(x = df$x, y = df$y, p = df$p)

print(df)

# The absolute error scoring function is twice the asymmetric piecewise linear
# scoring function (quantile scoring function) at level p = 0.5.

df <- data.frame(
    y = rep(x = 0, times = 3),
    x = c(-2, 0, 2),
    p = rep(x = c(0.5), times = 3)
)

df$quantile_penalty <- quantile_sf(x = df$x, y = df$y, p = df$p)

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

print(df)

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