Returns predictions and weights calculated by sequential numeric optimization. The optimization is done stepwise, always calculating a one-step-ahead forecast.

## Usage

```
batch(
y,
experts,
tau = 1:dim(experts)[2]/(dim(experts)[2] + 1),
affine = FALSE,
positive = FALSE,
intercept = FALSE,
debias = TRUE,
lead_time = 0,
initial_window = 30,
rolling_window = initial_window,
loss_function = "quantile",
loss_parameter = 1,
qw_crps = FALSE,
b_smooth = list(knots = length(tau), mu = 0.5, sigma = 1, nonc = 0, tailweight = 1, deg
= 1, periodic = FALSE),
p_smooth = list(knots = length(tau), mu = 0.5, sigma = 1, nonc = 0, tailweight = 1, deg
= 1, ndiff = 1.5, lambda = -Inf, periodic = FALSE),
forget = 0,
soft_threshold = -Inf,
hard_threshold = -Inf,
fixed_share = 0,
parametergrid_max_combinations = 100,
parametergrid = NULL,
forget_past_performance = 0,
allow_quantile_crossing = FALSE,
trace = TRUE
)
```

## Arguments

- y
A numeric matrix of realizations. In probabilistic settings a matrix of dimension Tx1, in multivariate settings a TxD 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.

- 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.

- initial_window
Defines the size of the initial estimation window.

- rolling_window
Defines the size of the rolling window. Defaults to the value of initial_window. Set it to the number of observations to receive an expanding window.

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

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

- qw_crps
Decides whether the sum of quantile scores (FALSE) or the quantile weighted CRPS (TRUE) should be minimized. Defaults to FALSE. Which corresponds to Berrisch & Ziel (2021)

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

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

- 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.

- 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.

- 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.

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

## Value

Returns weights and corresponding predictions. It is possible to impose a convexity constraint to the weights by setting affine and positive to TRUE.

## Details

batch selects various parameters automatically based on the past loss. For this, the parameters 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).
Parameters `b_smooth`

and `p_smooth`

take named lists to
create the corresponding basis and hat matrices. The arguments 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_knots`

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 and
`periodic`

which determines whether the spline basis will be periodic.
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. In addition to the inputs mentioned
before `p_smooth`

requires `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 <- batch(
y = matrix(y),
experts = experts,
p_smooth = list(lambda = 10)
)
print(model)
plot(model)
autoplot(model)
}
```