Projection matrix for optimal combination temporal reconciliation

This function computes the projection or the mapping matrix \(\mathbf{M}\) and \(\mathbf{G}\), respectively, such that \(\widetilde{\mathbf{y}} = \mathbf{M}\widehat{\mathbf{y}} = \mathbf{S}_{te}\mathbf{G}\widehat{\mathbf{y}}\), where \(\widetilde{\mathbf{y}}\) is the vector of the reconciled forecasts, \(\widehat{\mathbf{y}}\) is the vector of the base forecasts, \(\mathbf{S}_{te}\) is the temporal structural matrix, and \(\mathbf{M} = \mathbf{S}_{te}\mathbf{G}\). For further information regarding on the structure of these matrices, refer to Girolimetto et al. (2023).

Usage

teprojmat(agg_order, comb = "ols", res = NULL, mat = "M", tew = "sum", ...)

Arguments

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 tecov.

res

A (\(N(k^\ast+m) \times 1\)) optional numeric vector containing the in-sample residuals at all the temporal frequencies ordered from the lowest frequency to the highest frequency. This vector is used to compute come covariance matrices.

mat

A string specifying which matrix to return: "M" (default) for \(\mathbf{M}\) and "G" for \(\mathbf{G}\).

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).

...

Arguments passed on to tecov

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

The projection matrix \(\mathbf{M}\) (mat = "M") or the mapping matrix \(\mathbf{G}\) (mat = "G").

References

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

See also

Utilities: FoReco2matrix(), aggts(), balance_hierarchy(), commat(), csprojmat(), cstools(), ctprojmat(), cttools(), df2aggmat(), lcmat(), recoinfo(), res2matrix(), shrink_estim(), tetools(), unbalance_hierarchy()

Examples

# Temporal framework (annual-quarterly)
Mte <- teprojmat(agg_order = 4, comb = "ols")
Gte <- teprojmat(agg_order = 4, comb = "ols", mat = "G")