teprojmat function

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).

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)

Returns

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

Examples

# Temporal framework (annual-quarterly)
Mte <- teprojmat(agg_order = 4, comb = "ols")
Gte <- teprojmat(agg_order = 4, comb = "ols", 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. tools:::Rd_expr_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()