lcmat function

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 [REMOVE_ME]$\mathbf{C}_{cs} = [\mathbf{I} \quad -\mathbf{A}], [REMOVE_ME_2]$

where $\mathbf{C}_{cs}$ is the full rank zero constraints matrix (Girolimetto and Di Fonzo, 2023).

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).

Returns

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

Description

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

where $\mathbf{C}_{cs}$ is the full rank zero constraints matrix (Girolimetto and Di Fonzo, 2023).

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)
agg_mat <- obj$agg_mat # linear combination matrix
pivot <- obj$pivot # Pivot vector

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. tools:::Rd_expr_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()