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

online(
y,
experts,
tau,
loss_function = "quantile",
loss_parameter = 1,
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
)

## 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 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.

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.

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.

## Value

Returns weights and corresponding predictions.

## Details

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.

## 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 <- 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)
}