ctmvn: Cross-temporal Gaussian probabilistic reconciliation

In FoReco: Forecast Reconciliation

Cross-temporal Gaussian probabilistic reconciliation

Description

This function performs cross-temporal probabilistic forecast reconciliation assuming a multivariate normal base forecast distribution (Girolimetto et al., 2024) for linearly constrained multiple time series observed across both cross-sectional and temporal dimensions (Di Fonzo and Girolimetto, 2023).

Usage

ctmvn(base, agg_mat, cons_mat, agg_order, tew = "sum", comb = "ols",
      res = NULL, approach = "proj", comb_base = comb,
      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.
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).
comb	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 or validation errors ordered from the lowest frequency to the highest frequency (columns) for each variable (rows). This matrix is used to compute some covariance matrices.
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.
comb_base	A string specifying the base covariance matrix approach. For a complete list, see ctcov. Default is the equal to comb.
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 errors 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. \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.ejor.2022.07.040")}

See Also

Probabilistic reconciliation: csmvn(), cssmp(), ctsmp(), temvn(), tesmp()

Cross-temporal framework: ctboot(), ctbu(), ctcov(), ctlcc(), ctmo(), ctrec(), ctsmp(), 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 <- ctmvn(base = base, res = res, agg_mat = A, agg_order = 4)

FoReco documentation built on March 12, 2026, 5:07 p.m.