Returns predictions and weights calculated by online-learning algorithms using CRPS Learning.

```
online(
y,
experts,
tau,
lead_time = 0,
loss_function = "quantile",
loss_parameter = 1,
loss_gradient = TRUE,
method = "bewa",
b_smooth_pr = list(knots = P, mu = 0.5, sigma = 1, nonc = 0, tailweight = 1, deg = 1),
p_smooth_pr = list(knots = P, mu = 0.5, sigma = 1, nonc = 0, tailweight = 1, deg = 1,
ndiff = 1.5, lambda = -Inf),
b_smooth_mv = list(knots = D, mu = 0.5, sigma = 1, nonc = 0, tailweight = 1, deg = 1),
p_smooth_mv = list(knots = D, mu = 0.5, sigma = 1, nonc = 0, tailweight = 1, deg = 1,
ndiff = 1.5, lambda = -Inf),
forget_regret = 0,
soft_threshold = -Inf,
hard_threshold = -Inf,
fixed_share = 0,
gamma = 1,
parametergrid_max_combinations = 100,
parametergrid = NULL,
forget_past_performance = 0,
allow_quantile_crossing = FALSE,
init = NULL,
loss = NULL,
regret = NULL,
trace = TRUE
)
```

- 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 T x D x P x K (Observations x Variables x Quantiles x Experts) or T x D x K or T x P x K.

- tau
A numeric vector of probabilities.

- lead_time
offset for expert forecasts. Defaults to 0, which means that experts forecast t+1 at t. Setting this to h means experts predictions refer to t+1+h at time t. The weight updates delay accordingly.

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

- loss_parameter
Optional parameter scaling the power of the loss function.

- loss_gradient
Determines if a linearized version of the loss is used.

- method
One of "boa", "bewa", "ml_poly" or "ewa". Where "bewa" refers to a mixture of boa and ewa, including the second order refinement of boa, but updating weights with the simple exponential weighting.

- b_smooth_pr
A named list determining how the B-Spline matrices for probabilistic smoothing are created. Default corresponds to no probabilistic smoothing. See details.

- p_smooth_pr
A named list determining how the hat matrices for probabilistic P-Spline smoothing are created. Default corresponds to no smoothing. See details.

- b_smooth_mv
A named list determining how the B-Spline matrices for multivariate smoothing are created. Default corresponds to no probabilistic smoothing. See details.

- p_smooth_mv
A named list determining how the hat matrices for probabilistic P-Spline smoothing are created. Default corresponds to no smoothing. See details.

- forget_regret
Share of past regret not to be considered, resp. to be forgotten in every iteration of the algorithm. Defaults to 0.

- soft_threshold
If specified, the following soft threshold will be applied to the weights: w = sgn(w)*max(abs(w)-t,0) where t is the soft_threshold parameter. Defaults to -inf, which means that no threshold will be applied. If all expert weights are thresholded to 0, a weight of 1 will be assigned to the expert with the highest weights prior to thresholding. Thus soft_threshold = 1 leads to the 'follow the leader' strategy if method is set to "ewa".

- hard_threshold
If specified, the following hard thresholding will be applied to the weights: w = w*(abs(w)>t) where t is the threshold_hard parameter. Defaults to -inf, which means that no threshold will be applied. If all expert weights are thresholded to 0, a weight of 1 will be assigned to the expert with the highest weight prior to thresholding. Thus hard_threshold = 1 leads to the 'follow the leader' strategy if method is set to "ewa".

- fixed_share
Amount of fixed share to be added to the weights. Defaults to 0. 1 leads to uniform weights.

- gamma
Scaling parameter for the learning rate.

- parametergrid_max_combinations
Integer specifying the maximum number of parameter combinations that should be considered. If the number of possible combinations exceeds this threshold, the maximum allowed number is randomly sampled. Defaults to 100.

- parametergrid
User supplied grid of parameters. Can be used if not all combinations of the input vectors should be considered. Must be a matrix with 13 columns (online) or 12 columns batch with the following order: basis_knot_distance, basis_knot_distance_power, basis_deg, forget_regret, soft_threshold, hard_threshold, fixed_share, p_smooth_lambda, p_smooth_knot_distance, p_smooth_knot_distance_power, p_smooth_deg, p_smooth_ndiff, gamma.

- forget_past_performance
Share of past performance not to be considered, resp. to be forgotten in every iteration of the algorithm when selecting the best parameter combination. Defaults to 0.

- allow_quantile_crossing
Shall quantile crossing be allowed? Defaults to false, which means that predictions are sorted in ascending order.

- init
A named list containing "init_weights": Array of dimension DxPxK used as starting weights. "R0" a matrix of dimension PxK or 1xK used as starting regret.

- loss
User specified loss array. Can also be a list with elements "loss_array" and "share", share mixes the provided loss with the loss calculated by profoc. 1 means, only the provided loss will be used. share can also be vector of shares to consider.

- regret
User specified regret array. If specific, the regret will not be calculated by profoc. Can also be a list with elements "regret_array" and "share", share mixes the provided regret with the regret calculated by profoc. 1 means, only the provided regret will be used. share can also be vector of shares to consider.

- trace
Print a progress bar to the console? Defaults to TRUE.

Returns weights and corresponding predictions.

online selects various parameters automatically based on the past loss. For this, lambda, forget, fixed_share, gamma, and the smoothing parameters (see below) can be specified as numeric vectors containing values to consider.

This package offers two options for smoothing (Basis Smoothing
and P-Splines). Both options can be used to smooth the weights
over dimension D (covariates) or P (quantiles) or both.
Parameters `b_smooth_pr`

and `b_smooth_mv`

take named lists to
create the corresponding basis matrices. The arguments include are:
`knots`

which determines the number of knots to be created, `mu`

,
`sigma`

, `sigma`

, `nonc`

, `tailweight`

correspond to
to parameters of the beta distribution, which defines how the knots are
#distributed (see `?make_knots2`

for details) the defaults will create
an equidistant knot sequence, `deg`

sets the degree of the spline
function and also influences how many outer knots will be used.
It's possible to provide vectors of values for each of these parameters.
In that case, all parameter combinations will be used to create the
respective matrices and all candidates will be considered during
online-learning.
Parameters `p_smooth_pr`

and `p_smooth_mv`

determine the hat-matrix
creation for P-Spline smoothing. In addition to the inputs mentioned
before, they require to provide `ndiff`

which determines the degree
of differentiation applied to the basis-matrix (can take any value
between and including 1 and 2), `lambda`

which determines the degree
of penalization applied to the smoothing, higher values will give
smoother weight functions. As for the other parameters, it is possible
to provide multiple values.

```
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 <- online(
y = matrix(y),
experts = experts,
tau = prob_grid,
p_smooth_pr = list(lambda = 10)
)
print(model)
plot(model)
new_y <- matrix(rnorm(1)) # Realized
new_experts <- experts[T, , , drop = FALSE]
# Update will update the models weights etc if you provide new realizations
model <- update(model, new_y = new_y, new_experts = new_experts)
# Predict will expand \code{model$predictions} by default
model <- predict(model, new_experts = new_experts, update_model = TRUE)
}
```