Probabilistic Forecast Combination - Batch

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 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.
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)
}