scoringfunctions (version 1.1)

gpl1_sf: Generalized piecewise linear power scoring function (type 1)

Description

The function gpl1_sf computes the generalized piecewise linear power scoring function at a specific level \(p\) for \(g(x) = x^b/|b|\), \(b > 0\), when \(y\) materialises and \(x\) is the predictive quantile at level \(p\).

The generalized piecewise linear power scoring function is defined by eq. (25) in Gneiting (2011) and the form implemented here for the specific \(g(x)\) is defined by eq. (26) in Gneiting (2011).

Usage

gpl1_sf(x, y, p, b)

Value

Vector of generalized piecewise linear power losses.

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 \(y\)).

b

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

Details

The generalized piecewise linear power scoring function (type 1) is defined by:

$$ S(x, y, p, b) := (1/|b|) (\textbf{1} \lbrace x \geq y \rbrace - p) (x^b - y^b) $$

or equivalently

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

Domain of function:

$$x > 0$$

$$y > 0$$

$$0 < p < 1$$

$$b > 0$$

Range of function:

$$S(x, y, p, b) \geq 0, \forall x, y > 0, p \in (0, 1), b > 0$$

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").

Koenker R, Bassett Jr G (1978) Regression quantiles. Econometrica 46(1):33--50. tools:::Rd_expr_doi("10.2307/1913643").

Saerens M (2000) Building cost functions minimizing to some summary statistics. IEEE Transactions on Neural Networks 11(6):1263--1271. 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. tools:::Rd_expr_doi("10.1016/0022-0531(79)90042-5").

Examples

# Compute the generalized piecewise linear scoring function (type 1).

df <- data.frame(
    y = rep(x = 2, times = 6),
    x = c(1, 2, 3, 1, 2, 3),
    p = c(rep(x = 0.05, times = 3), rep(x = 0.95, times = 3)),
    b = rep(x = 2, times = 6)
)

df$gpl1_penalty <- gpl1_sf(x = df$x, y = df$y, p = df$p, b = df$b)

print(df)

# Equivalence of generalized piecewise linear scoring function (type 1) and
# asymmetric piecewise linear scoring function (quantile scoring function), when
# b = 1.

set.seed(12345)

n <- 100

x <- runif(n, 0, 20)
y <- runif(n, 0, 20)
p <- runif(n, 0, 1)
b <- rep(x = 1, times = n)

u <- gpl1_sf(x = x, y = y, p = p, b = b)
v <- quantile_sf(x = x, y = y, p = p)

max(abs(u - v))

# Equivalence of generalized piecewise linear scoring function (type 1) and
# MAE-SD scoring function, when p = 1/2 and b = 1/2.

set.seed(12345)

n <- 100

x <- runif(n, 0, 20)
y <- runif(n, 0, 20)
p <- rep(x = 0.5, times = n)
b <- rep(x = 1/2, times = n)

u <- gpl1_sf(x = x, y = y, p = p, b = b)
v <- maesd_sf(x = x, y = y)

max(abs(u - v))