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 $\varphi (t)=t^2$.

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:

$$S(x,y,p,a,b):=|\mathbf{\text{1}}\{x\ge y\}-p|(y^2-({\kappa }_{a,b}(x-y)+y{)}^2+2x{\kappa }_{a,b}(x-y))$$

or equivalently

$$S(x,y,p,a,b):=|\mathbf{\text{1}}\{x\ge y\}-p|f_{a,b}(x-y)$$

or

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

where

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

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

$${\kappa }_{a,b}(t):=max\{min\{t,b\},-a\}$$

Domain of function:

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

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

$$0<p<1$$

$$a>0$$

$$b>0$$

Range of function:

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

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))