Linear combination (aggregation) matrix for a general linearly constrained multiple time series

This function transforms a general (possibly redundant) zero constraints matrix into a linear combination (aggregation) matrix \(\mathbf{A}_{cs}\). When working with a general linearly constrained multiple (\(n\)-variate) time series, getting a linear combination matrix \(\mathbf{A}_{cs}\) is a critical step to obtain a structural-like representation such that $$\mathbf{C}_{cs} = [\mathbf{I} \quad -\mathbf{A}],$$ where \(\mathbf{C}_{cs}\) is the full rank zero constraints matrix (Girolimetto and Di Fonzo, 2023).

Usage

lcmat(cons_mat, method = "rref", tol = sqrt(.Machine$double.eps),
       verbose = FALSE, sparse = TRUE)

Arguments

cons_mat

A (\(r \times n\)) numeric matrix representing the cross-sectional zero constraints.

method

Method to use: "rref" for the Reduced Row Echelon Form through Gauss-Jordan elimination (default), or "qr" for the (pivoting) QR decomposition (Strang, 2019).

tol

Tolerance for the "rref" or "qr" method.

verbose

If TRUE, intermediate steps are printed (default is FALSE).

sparse

Option to return a sparse matrix (default is TRUE).

Value

A list with two elements: (i) the linear combination (aggregation) matrix (agg_mat) and (ii) the vector of the column permutations (pivot).

References

Girolimetto, D. and Di Fonzo, T. (2023), Point and probabilistic forecast reconciliation for general linearly constrained multiple time series, Statistical Methods & Applications, in press. doi:10.1007/s10260-023-00738-6 .

Strang, G. (2019), Linear algebra and learning from data, Wellesley, Cambridge Press.

See also

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

Examples

## Two hierarchy sharing the same top-level variable, but not sharing the bottom variables
#        X            X
#    |-------|    |-------|
#    A       B    C       D
#  |---|
# A1   A2
# 1) X = C + D,
# 2) X = A + B,
# 3) A = A1 + A2.
cons_mat <- matrix(c(1,-1,-1,0,0,0,0,
               1,0,0,-1,-1,0,0,
               0,0,0,1,0,-1,-1), nrow = 3, byrow = TRUE)
obj <- lcmat(cons_mat = cons_mat, verbose = TRUE)
#> ! A pivot is performed. Remember to apply the pivot also to the base forecast.
#> ℹ E.g. `base[, pivot]` in cross-sectional or `base[pivot, ]` in cross-temporal.
agg_mat <- obj$agg_mat # linear combination matrix
pivot <- obj$pivot # Pivot vector