scoringfunctions (version 1.1)

ghuber_sf: Generalized Huber scoring function

Description

The function ghuber_sf computes the generalized Huber scoring function at a specific level \(p\) and parameters \(a\) and \(b\), when \(y\) materialises and \(x\) is the predictive Huber functional at level \(p\).

The generalized Huber scoring function is defined by eq. (4.7) in Taggart (2022) for \(\phi(t) = t^2\).

Usage

ghuber_sf(x, y, p, a, b)

Value

Vector of generalized Huber losses.

Arguments

x

Predictive Huber functional (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 \(y\)).

a

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 Huber scoring function is defined by:

$$ S(x, y, p, a, b) := |\textbf{1} \lbrace x \geq y \rbrace - p| (y^2 - (\kappa_{a,b}(x - y) + y)^2 + 2 x \kappa_{a,b}(x - y)) $$

or equivalently

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

or

$$ S(x, y, p, a, b) := p f_{a,b}(- \max \lbrace -(x - y), 0 \rbrace) + (1 - p) f_{a,b}(\max \lbrace x - y, 0 \rbrace) $$

where

$$f_{a,b}(t) := \kappa_{a,b}(t) (2 t - \kappa_{a,b}(t))$$

and \(\kappa_{a,b}(t)\) is the capping function defined by:

$$\kappa_{a,b}(t) := \max \lbrace \min \lbrace t,b \rbrace, -a \rbrace$$

Domain of function:

$$x \in \mathbb{R}$$

$$y \in \mathbb{R}$$

$$0 < p < 1$$

$$a > 0$$

$$b > 0$$

Range of function:

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

References

Taggart RJ (2022) Point forecasting and forecast evaluation with generalized Huber loss. Electronic Journal of Statistics 16:201--231. tools:::Rd_expr_doi("10.1214/21-EJS1957").

Examples

# Compute the generalized Huber scoring function.

set.seed(12345)

n <- 10

df <- data.frame(
    x = runif(n, -2, 2),
    y = runif(n, -2, 2),
    p = runif(n, 0, 1),
    a = runif(n, 0, 1),
    b = runif(n, 0, 1)
)

df$ghuber_penalty <- ghuber_sf(x = df$x, y = df$y, p = df$p, a = df$a, b = df$b)

print(df)

# Equivalence of the generalized Huber scoring function and the asymmetric
# piecewise quadratic scoring function (expectile scoring function), when
# a = Inf and b = Inf.

set.seed(12345)

n <- 100

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

u <- ghuber_sf(x = x, y = y, p = p, a = a, b = b)
v <- expectile_sf(x = x, y = y, p = p)

max(abs(u - v)) # values are slightly higher than 0 due to rounding error
min(abs(u - v))

# Equivalence of the generalized Huber scoring function and the Huber scoring
# function when p = 1/2 and a = b.

set.seed(12345)

n <- 100

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

u <- ghuber_sf(x = x, y = y, p = p, a = a, b = a)
v <- huber_sf(x = x, y = y, a = a)

max(abs(u - v)) # values are slightly higher than 0 due to rounding error
min(abs(u - v))