bregman4_sf {scoringfunctions} | R Documentation |
Bregman scoring function (type 4, Patton scoring function)
Description
The function bregman4_sf computes the Bregman scoring function when
materializes and
is the predictive mean functional.
The Bregman scoring function is defined by eq. (18) in Gneiting (2011) and the
form implemented here for 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
|
y |
Realization (true value) of process. It can be a vector of length
|
Details
The Bregman scoring function (type 4) is defined by:
Domain of function:
Range of function:
Value
Vector of Bregman losses.
Note
The implemented function is denoted as type 4 since it corresponds to a specific
type of 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 of the probability distribution
of
(Gneiting 2011).
The Bregman scoring function is negatively oriented (i.e. the smaller, the better).
The herein implemented Bregman scoring function is strictly consistent for the
mean functional relative to the family of potential probability
distributions
for the future
for which
and
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)