Returns predictions and weights calculated by numeric optimization. The optimization is done in hindsight. This means all observations are used.

## Usage

oracle(y, experts, tau, affine = FALSE,
positive = FALSE, intercept = FALSE, debias = TRUE,
loss_function = "quantile", loss_parameter = 1, forget = 0)

## Arguments

y

A numeric matrix of realizations. In probabilistic settings a matrix of dimension Tx1, in multivariate settings a TxP matrix. In the latter case, each slice of the expert's array gets evaluated using the corresponding column of the y matrix.

experts

An array of predictions with dimension (Observations, Quantiles, Experts).

tau

A numeric vector of probabilities.

affine

Defines whether weights are summing to 1 or not. Defaults to FALSE.

positive

Defines if a positivity constraint is applied to the weights. Defaults to FALSE.

intercept

Determines if an intercept is added, defaults to FALSE. If true, a new first expert is added, always predicting 1.

debias

Defines whether the intercepts weight is constrained or not. If TRUE (the default), the intercept weight is unconstrained. Only affects the results if affine and or positive is set to TRUE. If FALSE, the intercept is treated as an expert.

loss_function

Either "quantile", "expectile" or "percentage".

loss_parameter

Optional parameter scaling the power of the loss function.

forget

Adds an exponential forgetting to the optimization. Past observations will get less influence on the optimization. Defaults to 0, which corresponds to no forgetting.

## Value

Returns weights and corresponding predictions. It is possible to calculate the best convex combination of weights by setting affine and positive to TRUE.

## Examples

if (FALSE) {
T <- 50 # Observations
N <- 2 # Experts
P <- 9 # Quantiles
prob_grid <- 1:P / (P + 1)

y <- rnorm(n = T) # Realized
experts <- array(dim = c(T, P, N)) # Predictions
for (t in 1:T) {
experts[t, , 1] <- qnorm(prob_grid, mean = -1, sd = 1)
experts[t, , 2] <- qnorm(prob_grid, mean = 3, sd = sqrt(4))
}

model <- oracle(
y = matrix(y),
experts = experts
)
}