quantile_sf {scoringfunctions}R Documentation

Asymmetric piecewise linear scoring function (quantile scoring function)

Description

The function quantile_sf computes the asymmetric piecewise linear scoring function (quantile scoring function) at a specific level pp, when yy materializes and xx is the predictive quantile at level pp.

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 pp. It can be a vector of length nn (must have the same length as yy).

y

Realization (true value) of process. It can be a vector of length nn (must have the same length as xx).

p

It can be a vector of length nn (must have the same length as yy).

Details

The assymetric piecewise linear scoring function is defined by:

S(x,y,p):=(1(xy)p)(xy)S(x, y, p) := (1(x \geq y) - p) (x - y)

Domain of function:

xRx \in \R

yRy \in \R

0<p<10 < p < 1

Range of function:

S(x,y,p)0,x,yR,p(0,1)S(x, y, p) \geq 0, \forall x, y \in \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 consistent for the pp-quantile functional relative to the family F\mathbb{F} of potential probability distributions FF for the future yy for which EF[Y]E_F[Y] exists and is finite (Thomson 1979, Saerens 2000, Gneiting 2011).

References

Gneiting T (2011) Making and evaluating point forecasts. Journal of the American Statistical Association 106(494):746–762. doi:10.1198/jasa.2011.r10138.

Koenker R, Bassett Jr G (1978) Regression quantiles. Econometrica 46(1):33–50. doi:10.2307/1913643.

Saerens M (2000) Building cost functions minimizing to some summary statistics. IEEE Transactions on Neural Networks 11(6):1263–1271. doi:10.1109/72.883416.

Thomson W (1979) Eliciting production possibilities from a well-informed manager. Journal of Economic Theory 20(3):360–380. 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)

[Package scoringfunctions version 0.0.6 Index]