gpl1_sf {scoringfunctions}

Generalized piecewise linear scoring function (type 1)

Description

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

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).

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

Realization (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 scoring function (type 1) is defined by:

S(x, y, p, b) := (1/|b|) (1(x \geq y) - p) (x^b - y^b)

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 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 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 scoring function is strictly consistent for the p-quantile functional relative to the family \mathbb{F} of potential probability distributions F for the future y for which 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))