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Temporal Gaussian probabilistic reconciliation

Source: R/reco_gauss.R
ctgauss.Rd

Temporal Gaussian probabilistic reconciliation

Usage

ctgauss(
  base,
  agg_mat,
  cons_mat,
  agg_order,
  comb = "ols",
  comb_base = comb,
  res = NULL,
  tew = "sum",
  approach = "proj",
  reduce_form = FALSE,
  ...
)

Arguments

base

A (\(n \times h(k^\ast+m)\)) numeric matrix containing the base forecasts to be reconciled; \(n\) is the total number of variables, \(m\) is the maximum aggregation order, and \(k^\ast\) is the sum of a chosen subset of the \(p - 1\) factors of \(m\) (excluding \(m\) itself), and \(h\) is the forecast horizon for the lowest frequency time series. The row identifies a time series, and the forecasts in each row are ordered from the lowest frequency (most temporally aggregated) to the highest frequency.

agg_mat

A (\(n_a \times n_b\)) numeric matrix representing the cross-sectional aggregation matrix. It maps the \(n_b\) bottom-level (free) variables into the \(n_a\) upper (constrained) variables.

cons_mat

A (\(n_a \times n\)) numeric matrix representing the cross-sectional zero constraints: each row represents a constraint equation, and each column represents a variable. The matrix can be of full rank, meaning the rows are linearly independent, but this is not a strict requirement, as the function allows for redundancy in the constraints.

agg_order

Highest available sampling frequency per seasonal cycle (max. order of temporal aggregation, \(m\)), or a vector representing a subset of \(p\) factors of \(m\).

comb

A string specifying the reconciliation method. For a complete list, see ctcov.

comb_base

A string specifying the reconciliation method. For a complete list, see ctcov.

res

A (\(n \times N(k^\ast+m)\)) optional numeric matrix containing the in-sample residuals ordered from the lowest frequency to the highest frequency (columns) for each variable (rows). This matrix is used to compute some covariance matrices.

tew

A string specifying the type of temporal aggregation. Options include: "sum" (simple summation, default), "avg" (average), "first" (first value of the period), and "last" (last value of the period).

approach

A string specifying the approach used to compute the reconciled forecasts. Options include:

  • "proj" (default): Projection approach according to Byron (1978, 1979).

  • "strc": Structural approach as proposed by Hyndman et al. (2011).

  • "proj_osqp": Numerical solution using osqp for projection approach.

  • "strc_osqp": Numerical solution using osqp for structural approach.

reduce_form

A logical parameter indicating whether the function should return the full distribution (FALSE, default) or only the distribution corresponding to the high-frequency bottom time series (TRUE).

...

Arguments passed on to ctcov

mse

If TRUE (default) the residuals used to compute the covariance matrix are not mean-corrected.

shrink_fun

Shrinkage function of the covariance matrix, shrink_estim (default).

Value

A distributional::dist_multivariate_normal object.

References

Byron, R.P. (1978), The estimation of large social account matrices, Journal of the Royal Statistical Society, Series A, 141, 3, 359-367. doi:10.2307/2344807

Byron, R.P. (1979), Corrigenda: The estimation of large social account matrices, Journal of the Royal Statistical Society, Series A, 142(3), 405. doi:10.2307/2982515

Girolimetto, D., Athanasopoulos, G., Di Fonzo, T. and Hyndman, R.J. (2024), Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues. International Journal of Forecasting, 40, 3, 1134-1151. doi:10.1016/j.ijforecast.2023.10.003

Hyndman, R.J., Ahmed, R.A., Athanasopoulos, G. and Shang, H.L. (2011), Optimal combination forecasts for hierarchical time series, Computational Statistics & Data Analysis, 55, 9, 2579-2589. doi:10.1016/j.csda.2011.03.006

Panagiotelis, A., Gamakumara, P., Athanasopoulos, G. and Hyndman, R.J. (2023), Probabilistic forecast reconciliation: Properties, evaluation and score optimisation, European Journal of Operational Research 306(2), 693–706. doi:10.1016/j.ejor.2022.07.040

See also

Probabilistic reconciliation: csgauss(), cssample(), ctsample(), tegauss(), tesample()

Cross-temporal framework: ctboot(), ctbu(), ctcov(), ctlcc(), ctmo(), ctrec(), ctsample(), cttd(), cttools(), iterec(), tcsrec()

Examples

set.seed(123)
# (3 x 7) base forecasts matrix (simulated), Z = X + Y and m = 4
base <- rbind(rnorm(7, rep(c(20, 10, 5), c(1, 2, 4))),
              rnorm(7, rep(c(10, 5, 2.5), c(1, 2, 4))),
              rnorm(7, rep(c(10, 5, 2.5), c(1, 2, 4))))
# (3 x 70) in-sample residuals matrix (simulated)
res <- rbind(rnorm(70), rnorm(70), rnorm(70))
A <- t(c(1,1))
reco_dist <- ctgauss(base = base, res = res, agg_mat = A, agg_order = 4)