Package 'scoringfunctions'

Overview of the functions in the scoringfunctions package

Description

The scoringfunctions package implements consistent scoring (loss) functions and identification functions

Details

The package functions are categorized into the following classes:

• 1. Scoring functions

• 1.1. Consistent scoring functions for one-dimensional functionals

• 1.2. Consistent scoring functions for two-dimensional functionals

• 1.3. Consistent scoring functions for multi-dimensional functionals

• 2. Realised (average) score functions

• 2.1 Realised (average) score functions for one-dimensional functionals

• 3. Skill score functions

• 3.1 Skill score functions for one-dimensional functionals

• 4. Identification functions

• 4.1. Identification functions for one-dimensional functionals

• 4.2. Identification functions for two-dimensional functionals

• 5. Functions for sample levels

• 6. Supporting functions

.

1. Scoring functions

1.1. Consistent scoring functions for one-dimensional functionals

$\textit{1.1.1. Consistent scoring functions for the mean}$

bregman1_sf: Bregman scoring function (type 1)

bregman2_sf: Bregman scoring function (type 2, Patton scoring function)

bregman3_sf: Bregman scoring function (type 3, QLIKE scoring function)

bregman4_sf: Bregman scoring function (type 4, Patton scoring function)

serr_sf: Squared error scoring function

$\textit{1.1.2. Consistent scoring functions for expectiles}$

expectile_sf: Asymmetric piecewise quadratic scoring function (expectile scoring function, expectile loss function)

$\textit{1.1.3. Consistent scoring functions for the median}$

aerr_sf: Absolute error scoring function

maelog_sf: MAE-LOG scoring function

maesd_sf: MAE-SD scoring function

$\textit{1.1.4. Consistent scoring functions for quantiles}$

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

gpl2_sf: Generalized piecewise linear power scoring function (type 2)

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

$\textit{1.1.5. Consistent scoring functions for Huber functionals}$

ghuber_sf: Generalized Huber scoring function

huber_sf: Huber scoring function

$\textit{1.1.6. Consistent scoring functions for other functionals}$

aperr_sf: Absolute percentage error scoring function

bmedian_sf: $\beta$-median scoring function

linex_sf: LINEX scoring function

lqmean_sf: $L_q$-mean scoring function

lqquantile_sf: $L_q$-quantile scoring function

nmoment_sf: $n$-th moment scoring function

obsweighted_sf: Observation-weighted scoring function

relerr_sf: Relative error scoring function (MAE-PROP scoring function)

serrexp_sf: Squared error exp scoring function

serrlog_sf: Squared error log scoring function

serrpower_sf: Squared error of power transformations scoring function

serrsq_sf: Squared error of squares scoring function

sperr_sf: Squared percentage error scoring function

srelerr_sf: Squared relative error scoring function

1.2. Consistent scoring functions for two-dimensional functionals

interval_sf: Interval scoring function (Winkler scoring function)

mv_sf: Mean - variance scoring function

1.3. Consistent scoring functions for multi-dimensional functionals

errorspread_sf: Error - spread scoring function

2. Realised (average) score functions

2.1. Realised (average) score functions for one-dimensional functionals

$\textit{2.1.1. Realised (average) score functions for the mean}$

mse: Mean squared error (MSE)

$\textit{2.1.2. Realised (average) score functions for expectiles}$

expectile_rs: Realised expectile score

$\textit{2.1.3. Realised (average) score functions for the median}$

mae: Mean absolute error (MAE)

$\textit{2.1.4. Realised (average) score functions for quantiles}$

quantile_rs: Realised quantile score

$\textit{2.1.5. Realised (average) score functions for Huber functionals}$

huber_rs: Mean Huber score

$\textit{2.1.6. Realised (average) score functions for other functionals}$

mape: Mean absolute percentage error (MAPE)

mre: Mean relative error (MRE)

mspe: Mean squared percentage error (MSPE)

msre: Mean squared relative error (MSRE)

3. Skill score functions

3.1. Skill score functions for one-dimensional functionals

$\textit{3.1.1. Skill score functions for the mean}$

nse: Nash-Sutcliffe efficiency (NSE)

4. Identification functions

4.1. Identification functions for one-dimensional functionals

expectile_if: Expectile identification function

hubermean_if: Huber mean identification function

huberquantile_if: Huber quantile identification function

mean_if: Mean identification function

meanlog_if: Log-transformed identification function

nmoment_if: $n$-th moment identification function

quantile_if: Quantile identification function

4.2. Identification functions for two-dimensional functionals

mv_if: Mean - variance identification function

5. Functions for sample levels

quantile_level: Sample quantile level function

6. Supporting functions

capping_function: Capping function

---

Absolute error scoring function

Description

The function aerr_sf computes the absolute error scoring function when $y$ materialises and $x$ is the predictive median functional.

The absolute error scoring function is defined in Table 1 in Gneiting (2011).

Usage

aerr_sf(x, y)

Arguments

x	Predictive median functional (prediction). 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 $x$).

Details

The absolute error scoring function is defined by:

$S(x, y) := |x - y|$

Domain of function:

$x \in \mathbb{R}$

$y \in \mathbb{R}$

Range of function:

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

Value

Vector of absolute errors.

Note

For details on the absolute error scoring function, see Gneiting (2011).

The median functional is the median of the probability distribution $F$ of $y$ (Gneiting 2011).

The absolute error scoring function is negatively oriented (i.e. the smaller, the better).

The absolute error scoring function is strictly $\mathbb{F}$-consistent for the median functional. $\mathbb{F}$ is the family of probability distributions $F$ for which $\textnormal{E}_F[Y]$ exists and is finite (Raiffa and 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. doi:10.1198/jasa.2011.r10138.

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. 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 absolute error scoring function.

df <- data.frame(
    y = rep(x = 0, times = 5),
    x = -2:2
)

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

print(df)

---

Absolute percentage error scoring function

Description

The function aperr_sf computes the absolute percentage error scoring function when $y$ materialises and $x$ is the predictive $\textnormal{med}^{(-1)}(F)$ functional.

The absolute percentage error scoring function is defined in Table 1 in Gneiting (2011).

Usage

aperr_sf(x, y)

Arguments

x	Predictive $\textnormal{med}^{(-1)}(F)$ functional (prediction). 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 $x$).

Details

The absolute percentage error scoring function is defined by:

$S(x, y) := |(x - y)/y|$

Domain of function:

$x > 0$

$y > 0$

Range of function:

$S(x, y) \geq 0, \forall x, y > 0$

Value

Vector of absolute percentage errors.

Note

For details on the absolute percentage error scoring function, see Gneiting (2011).

The $\beta$-median functional, $\textnormal{med}^{(\beta)}(F)$ is the median of a probability distribution whose density is proportional to $y^\beta f(y)$, where $f$ is the density of the probability distribution $F$ of $y$ (Gneiting 2011).

The absolute percentage error scoring function is negatively oriented (i.e. the smaller, the better).

The absolute percentage error scoring function is strictly $\mathbb{F}^{(w)}$-consistent for the $\textnormal{med}^{(-1)}(F)$ functional. $\mathbb{F}$ is the family of probability distributions for which $\textnormal{E}_F[Y]$ exists and is finite. $\mathbb{F}^{(w)}$ is the subclass of probability distributions in $\mathbb{F}$, which are such that $w(y) f(y)$, $w(y) = 1/y$ has finite integral over $(0, \infty)$, and the probability distribution $F^{(w)}$ with density proportional to $w(y) f(y)$ belongs to $\mathbb{F}$ (see Theorems 5 and 9 in 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.

Examples

# Compute the absolute percentage error scoring function.

df <- data.frame(
    y = rep(x = 2, times = 3),
    x = 1:3
)

df$absolute_percentage_error <- aperr_sf(x = df$x, y = df$y)

print(df)

---

$\beta$-median scoring function

Description

The function bmedian_sf computes the $\beta$-median scoring function when $y$ materialises and $x$ is the predictive $\textnormal{med}^{(\beta)}(F)$ functional.

The $\beta$-median scoring function is defined in eq. (4) in Gneiting (2011).

Usage

bmedian_sf(x, y, b)

Arguments

x	Predictive $\textnormal{med}^{(\beta)}(F)$ functional (prediction). 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 $x$).
b	It can be a vector of length $n$ (must have the same length as $y$).

Details

The $\beta$-median scoring function is defined by:

$S(x, y, b) := |1 - (y/x)^b|$

Domain of function:

$x > 0$

$y > 0$

$b \neq 0$

Range of function:

$S(x, y, b) \geq 0, \forall x, y > 0, b \neq 0$

Value

Vector of $\beta$-median losses.

Note

For details on the $\beta$-median scoring function, see Gneiting (2011).

The $\beta$-median functional, $\textnormal{med}^{(\beta)}(F)$ is the median of a probability distribution whose density is proportional to $y^\beta f(y)$, where $f$ is the density of the probability distribution $F$ of $y$ (Gneiting 2011).

The $\beta$-median scoring function is negatively oriented (i.e. the smaller, the better).

The $\beta$-median scoring function is strictly $\mathbb{F}^{(w)}$-consistent for the $\textnormal{med}^{(\beta)}(F)$ functional. $\mathbb{F}$ is the family of probability distributions for which $\textnormal{E}_F[Y]$ exists and is finite. $\mathbb{F}^{(w)}$ is the subclass of probability distributions in $\mathbb{F}$, which are such that $w(y) f(y)$, $w(y) = y^\beta$ has finite integral over $(0, \infty)$, and the probability distribution $F^{(w)}$ with density proportional to $w(y) f(y)$ belongs to $\mathbb{F}$ (see Theorems 5 and 9 in 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.

Examples

# Compute the bmedian scoring function.

df <- data.frame(
    y = rep(x = 2, times = 3),
    x = 1:3,
    b = c(-1, 1, 2)
)

df$bmedian_error <- bmedian_sf(x = df$x, y = df$y, b = df$b)

print(df)

---

Bregman scoring function (type 1)

Description

The function bregman1_sf computes the Bregman scoring function when $y$ materialises and $x$ is the predictive mean functional.

The Bregman scoring function is defined by eq. (18) in Gneiting (2011) and the form implemented here for $\phi(x) = |x|^a$ is defined by eq. (19) in Gneiting (2011).

Usage

bregman1_sf(x, y, a)

Arguments

x	Predictive mean functional (prediction). 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 $x$).
a	It can be a vector of length $n$ (must have the same length as $y$).

Details

The Bregman scoring function (type 1) is defined by:

$S(x, y, a) := |y|^a - |x|^a - a \textnormal{sign}(x) |x|^{a - 1} (y - x)$

Domain of function:

$x \in \mathbb{R}$

$y \in \mathbb{R}$

$a > 1$

Range of function:

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

Value

Vector of Bregman losses.

Note

The implemented function is denoted as type 1 since it corresponds to a specific type of $\phi(x)$ of the general form of the Bregman scoring function defined by eq. (18) in Gneiting (2011).

For details on the Bregman scoring function, see Savage (1971), Banerjee et al. (2005) and Gneiting (2011).

The mean functional is the mean $\textnormal{E}_F[Y]$ of the probability distribution $F$ of $y$ (Gneiting 2011).

The Bregman scoring function is negatively oriented (i.e. the smaller, the better).

The herein implemented Bregman scoring function is strictly $\mathbb{F}$-consistent for the mean functional. $\mathbb{F}$ is the family of probability distributions for which $\textnormal{E}_F[Y]$ and $\textnormal{E}_F[|Y|^a]$ exist and are finite (Savage 1971; Gneiting 2011).

References

Banerjee A, Guo X, Wang H (2005) On the optimality of conditional expectation as a Bregman predictor. IEEE Transactions on Information Theory 51(7):2664–2669. doi:10.1109/TIT.2005.850145.

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

Savage LJ (1971) Elicitation of personal probabilities and expectations. Journal of the American Statistical Association 66(337):783–810. doi:10.1080/01621459.1971.10482346.

Examples

# Compute the Bregman scoring function (type 1).

df <- data.frame(
    y = rep(x = 0, times = 7),
    x = c(-3, -2, -1, 0, 1, 2, 3),
    a = rep(x = 3, times = 7)
)

df$bregman1_penalty <- bregman1_sf(x = df$x, y = df$y, a = df$a)

print(df)

# Equivalence of Bregman scoring function (type 1) and squared error scoring
# function, when a = 2.

set.seed(12345)

n <- 100

x <- runif(n, -20, 20)
y <- runif(n, -20, 20)
a <- rep(x = 2, times = n)

u <- bregman1_sf(x = x, y = y, a = a)

v <- serr_sf(x = x, y = y)

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

---

Bregman scoring function (type 2, Patton scoring function)

Description

The function bregman2_sf computes the Bregman scoring function when $y$ materialises and $x$ is the predictive mean functional.

The Bregman scoring function is defined by eq. (18) in Gneiting (2011) and the form implemented here for $\phi(x) = \dfrac{1}{b (b - 1)} x^b$, $b \in \R \setminus \lbrace 0, 1 \rbrace$ is defined by eq. (20) in Gneiting (2011).

Usage

bregman2_sf(x, y, b)

Arguments

x	Predictive mean functional (prediction). 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 $x$).
b	It can be a vector of length $n$ (must have the same length as $y$).

Details

The Bregman scoring function (type 2) is defined by:

$S(x, y, b) := \dfrac{1}{b (b - 1)} (y^b - x^b) - \dfrac{1}{b - 1} x^{b - 1} (y - x)$

Domain of function:

$x > 0$

$y > 0$

$b \in \mathbb{R} \setminus \lbrace 0, 1 \rbrace$

Range of function:

$S(x, y, b) \geq 0, \forall x, y > 0, b \in \mathbb{R} \setminus \lbrace 0, 1 \rbrace$

Value

Vector of Bregman losses.

Note

The implemented function is denoted as type 2 since it corresponds to a specific type of $\phi(x)$ of the general form of the Bregman scoring function defined by eq. (18) in Gneiting (2011).

For details on the Bregman scoring function, see Savage (1971), Banerjee et al. (2005) and Gneiting (2011). For details on the specific form implemented here, see Patton (2011).

The mean functional is the mean $\textnormal{E}_F[Y]$ of the probability distribution $F$ of $y$ (Gneiting 2011).

The Bregman scoring function is negatively oriented (i.e. the smaller, the better).

The herein implemented Bregman scoring function is strictly $\mathbb{F}$-consistent for the mean functional. $\mathbb{F}$ is the family of probability distributions $F$ for which $\textnormal{E}_F[Y]$ and $\textnormal{E}_F[\dfrac{1}{b (b - 1)} Y^b]$ exist and are finite (Savage 1971; Gneiting 2011).

References

Banerjee A, Guo X, Wang H (2005) On the optimality of conditional expectation as a Bregman predictor. IEEE Transactions on Information Theory 51(7):2664–2669. doi:10.1109/TIT.2005.850145.

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

Patton AJ (2011) Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics 160(1):246–256. doi:10.1016/j.jeconom.2010.03.034.

Savage LJ (1971) Elicitation of personal probabilities and expectations. Journal of the American Statistical Association 66(337):783–810. doi:10.1080/01621459.1971.10482346.

Examples

# Compute the Bregman scoring function (type 2).

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

df$bregman2_penalty <- bregman2_sf(x = df$x, y = df$y, b = df$b)

print(df)

# The Bregman scoring function (type 2) is half the squared error scoring
# function, when b = 2.

df <- data.frame(
    y = rep(x = 5.5, times = 10),
    x = 1:10,
    b = rep(x = 2, times = 10)
)

df$bregman2_penalty <- bregman2_sf(x = df$x, y = df$y, b = df$b)

df$squared_error <- serr_sf(x = df$x, y = df$y)

df$ratio <- df$bregman2_penalty/df$squared_error

print(df)


# When a = b > 1 the Bregman scoring function (type 2) coincides with the
# Bregman scoring function (type 1) up to a multiplicative constant.

df <- data.frame(
    y = rep(x = 5.5, times = 10),
    x = 1:10,
    b = rep(x = c(3, 4), each = 5)
)

df$bregman2_penalty <- bregman2_sf(x = df$x, y = df$y, b = df$b)

df$bregman1_penalty <- bregman1_sf(x = df$x, y = df$y, a = df$b)

df$ratio <- df$bregman2_penalty/df$bregman1_penalty

print(df)

---

Bregman scoring function (type 3, QLIKE scoring function)

Description

The function bregman3_sf computes the Bregman scoring function when $y$ materialises and $x$ is the predictive mean functional.

The Bregman scoring function is defined by eq. (18) in Gneiting (2011) and the form implemented here for $\phi(x) = -\log(x)$ is defined by eq. (20) in Gneiting (2011).

Usage

bregman3_sf(x, y)

Arguments

x	Predictive mean functional (prediction). 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 $x$).

Details

The Bregman scoring function (type 3) is defined by:

$S(x, y) := (y/x) - \log(y/x) - 1$

Domain of function:

$x > 0$

$y > 0$

Range of function:

$S(x, y) \geq 0, \forall x, y > 0$

Value

Vector of Bregman losses.

Note

The implemented function is denoted as type 3 since it corresponds to a specific type of $\phi(x)$ of the general form of the Bregman scoring function defined by eq. (18) in Gneiting (2011).

For details on the Bregman scoring function, see Savage (1971), Banerjee et al. (2005) and Gneiting (2011). For details on the specific form implemented here, see the QLIKE scoring function in Patton (2011).

The mean functional is the mean $\textnormal{E}_F[Y]$ of the probability distribution $F$ of $y$ (Gneiting 2011).

The Bregman scoring function is negatively oriented (i.e. the smaller, the better).

The herein implemented Bregman scoring function is strictly $\mathbb{F}$-consistent for the mean functional. $\mathbb{F}$ is the family of probability distributions $F$ for which $\textnormal{E}_F[Y]$ and $\textnormal{E}_F[\log(Y)]$ exist and are finite (Savage 1971; Gneiting 2011).

References

Banerjee A, Guo X, Wang H (2005) On the optimality of conditional expectation as a Bregman predictor. IEEE Transactions on Information Theory 51(7):2664–2669. doi:10.1109/TIT.2005.850145.

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

Patton AJ (2011) Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics 160(1):246–256. doi:10.1016/j.jeconom.2010.03.034.

Savage LJ (1971) Elicitation of personal probabilities and expectations. Journal of the American Statistical Association 66(337):783–810. doi:10.1080/01621459.1971.10482346.

Examples

# Compute the Bregman scoring function (type 3, QLIKE scoring function).

df <- data.frame(
    y = rep(x = 2, times = 3),
    x = 1:3
)

df$bregman3_penalty <- bregman3_sf(x = df$x, y = df$y)

print(df)

---

Bregman scoring function (type 4, Patton scoring function)

Description

The function bregman4_sf computes the Bregman scoring function when $y$ materialises and $x$ is the predictive mean functional.

The Bregman scoring function is defined by eq. (18) in Gneiting (2011) and the form implemented here for $\phi(x) = x \log(x)$ is defined by eq. (20) in Gneiting (2011).

Usage

bregman4_sf(x, y)

Arguments

x	Predictive mean functional (prediction). 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 $x$).

Details

The Bregman scoring function (type 4) is defined by:

$S(x, y) := y \log(y/x) - y + x$

Domain of function:

$x > 0$

$y > 0$

Range of function:

$S(x, y) \geq 0, \forall x, y > 0$

Value

Vector of Bregman losses.

Note

The implemented function is denoted as type 4 since it corresponds to a specific type of $\phi(x)$ of the general form of the Bregman scoring function defined by eq. (18) in Gneiting (2011).

For details on the Bregman scoring function, see Savage (1971), Banerjee et al. (2005) and Gneiting (2011). For details on the specific form implemented here, see Patton (2011).

The mean functional is the mean $\textnormal{E}_F[Y]$ of the probability distribution $F$ of $y$ (Gneiting 2011).

The Bregman scoring function is negatively oriented (i.e. the smaller, the better).

The herein implemented Bregman scoring function is strictly $\mathbb{F}$-consistent for the mean functional. $\mathbb{F}$ is the family of probability distributions $F$ for which $\textnormal{E}_F[Y]$ and $\textnormal{E}_F[Y \log(Y)]$ exist and are finite (Savage 1971; Gneiting 2011).

References

Banerjee A, Guo X, Wang H (2005) On the optimality of conditional expectation as a Bregman predictor. IEEE Transactions on Information Theory 51(7):2664–2669. doi:10.1109/TIT.2005.850145.

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

Patton AJ (2011) Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics 160(1):246–256. doi:10.1016/j.jeconom.2010.03.034.

Savage LJ (1971) Elicitation of personal probabilities and expectations. Journal of the American Statistical Association 66(337):783–810. doi:10.1080/01621459.1971.10482346.

Examples

# Compute the Bregman scoring function (type 4).

df <- data.frame(
    y = rep(x = 2, times = 3),
    x = 1:3
)

df$bregman4_penalty <- bregman4_sf(x = df$x, y = df$y)

print(df)

---

Capping function

Description

The function capping_function computes the value of the capping function, defined in Taggart (2022), p.205.

It is used by the generalized Huber loss function among others (see Taggart 2022).

Usage

capping_function(t, a, b)

Arguments

t	It can be a vector of length $n$.
a	It can be a vector of length $n$ (must have the same length as $t$).
b	It can be a vector of length $n$ (must have the same length as $t$).

Details

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

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

or equivalently,

$\kappa_{a,b}(t) := \left\{ \begin{array}{ll} -a, & t \leq -a\\ t, & -a < t \leq b\\ b, & t > b \end{array} \right.$

Domain of function:

$t \in \mathbb{R}$

$a \geq 0$

$b \geq 0$

Value

Vector of values of the capping function.

Note

For the definition of the capping function, see Taggart (2022), p.205.

References

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

Examples

# Compute the capping function.

df <- data.frame(
    t = c(1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 2.5, 2.5, 3.5, 3.5),
    a = c(0, 0, 0, 0, Inf, Inf, Inf, Inf, 2, 3, 2, 3, 2, 3),
    b = c(0, 0, Inf, Inf, 0, 0, Inf, Inf, 3, 2, 3, 2, 3, 2)
)

df$cf <- capping_function(t = df$t, a = df$a, b = df$b)

print(df)

---

Error - spread scoring function

Description

The function errorspread_sf computes the error - spread scoring function, when $y$ materialises, $x_1$ is the predictive mean, $x_2$ is the predictive variance and $x_3$ is the predictive skewness.

The error - spread scoring function is defined by eq. (14) in Christensen et al. (2015).

Usage

errorspread_sf(x1, x2, x3, y)

Arguments

x1	Predictive mean (prediction). It can be a vector of length $n$ (must have the same length as $y$).
x2	Predictive variance (prediction). It can be a vector of length $n$ (must have the same length as $y$).
x3	Predictive skewness (prediction). 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 $x_1$).

Details

The error - spread scoring function is defined by:

$S(x_1, x_2, x_3, y) := (x_2 - (x_1 - y)^2 - (x_1 - y) x_2^{1/2} x_3)^2$

Domain of function:

$x_1 \in \mathbb{R}$

$x_2 > 0$

$x_3 \in \mathbb{R}$

$y \in \mathbb{R}$

Value

Vector of error - spread losses.

Note

The mean functional is the mean $\textnormal{E}_F[Y]$ of the probability distribution $F$ of $y$ (Christensen et al. 2015).

The variance functional is the variance $\textnormal{Var}_F[Y] := \textnormal{E}_F[Y^2] - (\textnormal{E}_F[Y])^{2}$ of the probability distribution $F$ of $y$ (Christensen et al. 2015).

The skewness functional is the skewness $\textnormal{Sk}_F[Y] := \textnormal{E}_F[((Y - \textnormal{E}_F[Y])/(\textnormal{Var}_F[Y])^{1/2})^3]$ (Christensen et al. 2015).

The error - spread scoring function is negatively oriented (i.e. the smaller, the better).

The error - spread scoring function is strictly consistent for the triple (mean, variance, skewness) functional (Christensen et al. 2015).

References

Christensen HM, Moroz IM, Palmer TN (2015) Evaluation of ensemble forecast uncertainty using a new proper score: Application to medium-range and seasonal forecasts. Quarterly Journal of the Royal Meteorological Society 141(687)(Part B):538–549. doi:10.1002/qj.2375.

Examples

# Compute the error - spread scoring function.

df <- data.frame(
    y = rep(x = 0, times = 6),
    x1 = c(2, 2, -2, -2, 0, 0),
    x2 = c(1, 2, 1, 2, 1, 2),
    x3 = c(3, 3, -3, -3, 0, 0)
)

df$errorspread_penalty <- errorspread_sf(x1 = df$x1, x2 = df$x2, x3 = df$x3,
    y = df$y)

print(df)

---

Expectile identification function

Description

The function expectile_if computes the expectile identification function at a specific level $p$, when $y$ materialises and $x$ is the predictive expectile at level $p$.

The expectile identification function is defined in Table 9 in Gneiting (2011).

Usage

expectile_if(x, y, p)

Arguments

x	Predictive expectile (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 expectile identification function is defined by:

$V(x, y, p) := 2 |\textbf{1} \lbrace x \geq y \rbrace - p| (x - y)$

Domain of function:

$x \in \mathbb{R}$

$y \in \mathbb{R}$

$0 < p < 1$

Range of function:

$V(x, y, p) \in \mathbb{R}$

Value

Vector of values of the expectile identification function.

Note

For the definition of expectiles, see Newey and Powell (1987).

The expectile identification function is a strict $\mathbb{F}$-identification function for the $p$-expectile functional (Gneiting 2011; Fissler and Ziegel 2016; Dimitriadis et al. 2024).

$\mathbb{F}$ is the family of probability distributions $F$ for which $\textnormal{E}_F[Y]$ exists and is finite (Gneiting 2011; Fissler and Ziegel 2016; Dimitriadis et al. 2024).

References

Dimitriadis T, Fissler T, Ziegel JF (2024) Osband's principle for identification functions. Statistical Papers 65:1125–1132. doi:10.1007/s00362-023-01428-x.

Fissler T, Ziegel JF (2016) Higher order elicitability and Osband's principle. The Annals of Statistics 44(4):1680–1707. doi:10.1214/16-AOS1439.

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

Newey WK, Powell JL (1987) Asymmetric least squares estimation and testing. Econometrica 55(4):819–847. doi:10.2307/1911031.

Examples

# Compute the expectile identification 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$expectile_if <- expectile_if(x = df$x, y = df$y, p = df$p)

---

Realised expectile score

Description

The function expectile_rs computes the realised expectile score at a specific level $p$ when $\textbf{\textit{y}}$ materialises and $\textbf{\textit{x}}$ is the prediction.

Realised expectile score is a realised score corresponding to the expectile scoring function expectile_sf.

Usage

expectile_rs(x, y, p)

Arguments

x	Prediction. It can be a vector of length $n$ (must have the same length as $\textbf{\textit{y}}$).
y	Realisation (true value) of process. It can be a vector of length $n$ (must have the same length as $\textbf{\textit{x}}$).
p	It can be a vector of length $n$ (must have the same length as $\textbf{\textit{y}}$) or a scalar value.

Details

The realized expectile score is defined by:

$S(\textbf{\textit{x}}, \textbf{\textit{y}}, p) := (1/n) \sum_{i = 1}^{n} L(x_i, y_i, p)$

where

$\textbf{\textit{x}} = (x_1, ..., x_n)^\mathsf{T}$

$\textbf{\textit{y}} = (y_1, ..., y_n)^\mathsf{T}$

and

$L(x, y, p) := |\textbf{1} \lbrace x \geq y \rbrace - p| (x - y)^2$

Domain of function:

$\textbf{\textit{x}} \in \mathbb{R}^n$

$\textbf{\textit{y}} \in \mathbb{R}^n$

$0 < p < 1$

Range of function:

$S(\textbf{\textit{x}}, \textbf{\textit{y}}, p) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n, p \in (0, 1)$

Value

Value of the realised expectile score.

Note

For details on the expectile scoring function, see expectile_sf.

The concept of realised (average) scores is defined by Gneiting (2011) and Fissler and Ziegel (2019).

The realised expectile score is the realised (average) score corresponding to the expectile scoring function.

References

Fissler T, Ziegel JF (2019) Order-sensitivity and equivariance of scoring functions. Electronic Journal of Statistics 13(1):1166–1211. doi:10.1214/19-EJS1552.

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

Examples

# Compute the realised expectile score.

set.seed(12345)

x <- 0.5

y <- rnorm(n = 100, mean = 0, sd = 1)

print(expectile_rs(x = x, y = y, p = 0.7))

print(expectile_rs(x = rep(x = x, times = 100), y = y, p = 0.7))

---

Asymmetric piecewise quadratic scoring function (expectile scoring function, expectile loss function)

Description

The function expectile_sf computes the asymmetric piecewise quadratic scoring function (expectile scoring function) at a specific level $p$, when $y$ materialises and $x$ is the predictive expectile at level $p$.

The asymmetric piecewise quadratic scoring function is defined by eq. (27) in Gneiting (2011).

Usage

expectile_sf(x, y, p)

Arguments

x	Predictive expectile (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 asymmetric piecewise quadratic scoring function is defined by:

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

or equivalently,

$S(x, y, p) := p (\max \lbrace -(x - y), 0 \rbrace)^2 + (1 - p) (\max \lbrace x - y, 0 \rbrace)^2$

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 expectile losses.

Note

For the definition of expectiles, see Newey and Powell (1987).

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

The asymmetric piecewise quadratic scoring function is strictly $\mathbb{F}$-consistent for the $p$-expectile functional. $\mathbb{F}$ is the family of probability distributions $F$ for which $\textnormal{E}_F[Y^2]$ exists and is finite (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.

Newey WK, Powell JL (1987) Asymmetric least squares estimation and testing. Econometrica 55(4):819–847. doi:10.2307/1911031.

Examples

# Compute the asymmetric piecewise quadratic scoring function (expectile 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$expectile_penalty <- expectile_sf(x = df$x, y = df$y, p = df$p)

print(df)

# The asymmetric piecewise quadratic scoring function (expectile scoring
# function) at level p = 0.5 is half the squared error scoring function.

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

df$expectile_penalty <- expectile_sf(x = df$x, y = df$y, p = df$p)

df$squared_error <- serr_sf(x = df$x, y = df$y)

print(df)

---

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)

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:

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

Value

Vector of generalized Huber losses.

Note

For the definition of Huber functionals, see definition 3.3 in Taggart (2022). The value of eq. (4.7) is twice the value of the equation in definition 4.2 in Taggart (2002).

The generalized Huber scoring function is negatively oriented (i.e. the smaller, the better).

The generalized Huber scoring function is strictly $\mathbb{F}$-consistent for the $p$-Huber functional. $\mathbb{F}$ is the family of probability distributions $F$ for which $\textnormal{E}_F[Y^2 - (Y - a)^2]$ and $\textnormal{E}_F[Y^2 - (Y + b)^2]$ (or equivalently $\textnormal{E}_F[Y]$) exist and are finite (Taggart 2022).

References

Taggart RJ (2022) Point forecasting and forecast evaluation with generalized Huber loss. Electronic Journal of Statistics 16:201–231. 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))

---

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)

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:

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

Value

Vector of generalized piecewise linear power losses.

Note

The implemented function is denoted as type 1 since it corresponds to a specific type of $g(x)$ of the general form of the generalized piecewise linear power scoring function defined by eq. (25) in Gneiting (2011).

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

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

The herein implemented generalized piecewise linear power scoring function is strictly $\mathbb{F}$-consistent for the $p$-quantile functional. $\mathbb{F}$ is the family of probability distributions $F$ for which $\textnormal{E}_F[Y^b]$ 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 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))

---

Generalized piecewise linear power scoring function (type 2)

Description

The function gpl2_sf computes the generalized piecewise linea power scoring function at a specific level $p$ for $g(x) = \log(x)$, when $y$ materialises and $x$ is the predictive quantile at level $p$.

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

The generalized piecewise linear 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) for $b = 0$.

Usage

gpl2_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 generalized piecewise linear power scoring function (type 2) is defined by:

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

or equivalently

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

Domain of function:

$x > 0$

$y > 0$

$0 < p < 1$

Range of function:

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

Value

Vector of generalized piecewise linear losses.

Note

The implemented function is denoted as type 2 since it corresponds to a specific type of $g(x)$ of the general form of the generalized piecewise linear power scoring function defined by eq. (25) in Gneiting (2011).

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

The herein implemented generalized piecewise linear power scoring function is strictly $\mathbb{F}$-consistent for the $p$-quantile functional. $\mathbb{F}$ is the family of probability distributions $F$ for which $\textnormal{E}_F[\log(Y)]$ 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 generalized piecewise linear scoring function (type 2).

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))
)

df$gpl2_penalty <- gpl2_sf(x = df$x, y = df$y, p = df$p)

print(df)

# The generalized piecewise linear scoring function (type 2) is half the MAE-LOG
# scoring function.

df <- data.frame(
    y = rep(x = 5.5, times = 10),
    x = 1:10,
    p = rep(x = 0.5, times = 10)
)

df$gpl2_penalty <- gpl2_sf(x = df$x, y = df$y, p = df$p)

df$mae_log_penalty <- maelog_sf(x = df$x, y = df$y)

df$ratio <- df$gpl2_penalty/df$mae_log_penalty

print(df)

---

Mean Huber score

Description

The function huber_rs computes the mean Huber score with parameter $a$, when $\textbf{\textit{y}}$ materialises and $\textbf{\textit{x}}$ is the prediction.

Mean Huber score is a realised score corresponding to the Huber scoring function huber_sf.

Usage

huber_rs(x, y, a)

Arguments

x	Prediction. It can be a vector of length $n$ (must have the same length as $\textbf{\textit{y}}$).
y	Realisation (true value) of process. It can be a vector of length $n$ (must have the same length as $\textbf{\textit{x}}$).
a	It can be a vector of length $n$ (must have the same length as $y$) or a scalar.

Details

The mean Huber score is defined by:

$S(\textbf{\textit{x}}, \textbf{\textit{y}}, a) := (1/n) \sum_{i = 1}^{n} L(x_i, y_i, a)$

where

$\textbf{\textit{x}} = (x_1, ..., x_n)^\mathsf{T}$

$\textbf{\textit{y}} = (y_1, ..., y_n)^\mathsf{T}$

and

$L(x, y, a) := \left\{ \begin{array}{ll} \dfrac{1}{2} (x - y)^2, & |x - y| \leq a\\ a |x - y| - \dfrac{1}{2} a^2, & |x - y| > a \end{array} \right.$

Domain of function:

$\textbf{\textit{x}} \in \mathbb{R}^n$

$\textbf{\textit{y}} \in \mathbb{R}^n$

$a > 0$

Range of function:

$S(\textbf{\textit{x}}, \textbf{\textit{y}}, a) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n, a > 0$

Value

Value of the mean Huber score.

Note

For details on the Huber scoring function, see huber_sf.

The concept of realised (average) scores is defined by Gneiting (2011) and Fissler and Ziegel (2019).

The mean Huber score is the realised (average) score corresponding to the Huber scoring function.

References

Fissler T, Ziegel JF (2019) Order-sensitivity and equivariance of scoring functions. Electronic Journal of Statistics 13(1):1166–1211. doi:10.1214/19-EJS1552.

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

Examples

# Compute the Huber mean score.

set.seed(12345)

a <- 0.5

x <- 0

y <- rnorm(n = 100, mean = 0, sd = 1)

print(huber_rs(x = x, y = y, a = a))

print(huber_rs(x = rep(x = x, times = 100), y = y, a = a))

---