oracle: Compute oracle predictions

Description

The function oracle performs a strategie that cannot be defined online (in contrast to mixture). It requires in advance the knowledge of the whole data set Y and the expert advice to be well defined. Examples of oracles are the best fixed expert, the best fixed convex combination rule, the best linear combination rule, or the best expert that can shift a few times.

Usage

oracle(
  Y,
  experts,
  model = "convex",
  loss.type = "square",
  awake = NULL,
  lambda = NULL,
  niter = NULL,
  ...
)

Arguments

A vector containing the observations to be predicted.

experts

A matrix containing the experts forecasts. Each column corresponds to the predictions proposed by an expert to predict Y. It has as many columns as there are experts.

model

A character string specifying the oracle to use or a list with a component name specifying the oracle and any additional parameter needed. Currently available oracles are:

'expert': The best fixed (constant over time) expert oracle.
'convex': The best fixed convex combination (vector of non-negative weights that sum to 1)
'linear': The best fixed linear combination of expert
'shifting': It computes for all number $m$ of stwitches the sequence of experts with at most $m$ shifts that would have performed the best to predict the sequence of observations in Y.

loss.type

character, list or function.

character: Name of the loss to be applied ('square', 'absolute', 'percentage', or 'pinball');
list: When using pinball loss: list with field name equal to 'pinball' and field tau equal to the required quantile in [0,1];
function: A custom loss as a function of two parameters.

awake

A matrix specifying the activation coefficients of the experts. Its entries lie in [0,1]. Possible if some experts are specialists and do not always form and suggest prediction. If the expert number k at instance t does not form any prediction of observation Y_t, we can put awake[t,k]=0 so that the mixture does not consider expert k in the mixture to predict Y_t. Remark that to compute the best expert oracle, the performance of unactive (or partially active) experts is computed by using the prediction of the uniform average of active experts.

lambda

A positive number used by the 'linear' oracle only. A possible $L_2$ regularization parameter for computing the linear oracle (if the design matrix is not identifiable)

niter

A positive integer for 'convex' and 'linear' oracles if direct computation of the oracle is not implemented. It defines the number of optimization steps to perform in order to approximate the oracle (default value is 3).

...

Additional parameters that are passed to optim function is order to perform convex optimization (see parameter niter).

Value

An object of class 'oracle' that contains:

loss

The average loss suffered by the oracle. For the 'shifting' oracle, it is a vector of length T where T is the number of instance to be predicted (i.e., the length of the sequence Y). The value of $loss(m)$ is the loss (determined by the parameter loss.type) suffered by the best sequence of expert with at most $m-1$ shifts.

coefficients

Not for the 'shifting' oracle. A vector containing the best weight vector corresponding to the oracle.

prediction

Not for the 'shifting' oracle. A vector containing the predictions of the oracle.

rmse

If loss.type is the square loss (default) only. The root mean square error (i.e., it is the square root of loss.