coherent_cmat: Zero-constraint matrix for coherent time series

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Zero-constraint matrix for coherent time series

Description

Constructs the n_a \times n zero-constraint matrix \mathbf{C} for a set of linearly related time series, where n_a = n - n_b is the number of aggregated series.

Usage

coherent_cmat(data, sparse = FALSE)

Arguments

data	A data object which contains linearly related coherent structures.
sparse	If TRUE, a sparse matrix (class "dgCMatrix" from the Matrix package) is returned. Defaults to FALSE.

Details

Given the aggregation matrix \mathbf{A} (see coherent_smat()), the constraint matrix is defined as

\mathbf{C} = [\mathbf{I}_{n_a} \;\; {-\mathbf{A}}],

so that \mathbf{C}\boldsymbol{y}_t = \boldsymbol{0}_{n_a} for all coherent vectors \boldsymbol{y}_t. This zero-constrained representation yields the mapping matrix

\mathbf{M} = \mathbf{I}_n - \mathbf{W}\mathbf{C}'(\mathbf{C}\mathbf{W}\mathbf{C}')^{-1}\mathbf{C},

which requires inverting an n_a \times n_a matrix rather than the n_b \times n_b matrix in the structural form, and is therefore more efficient when n_a < n_b.

Value

An n_a \times n matrix (dense or sparse) whose rows encode each aggregation constraint as \mathbf{C}\boldsymbol{y}_t = \boldsymbol{0}_{n_a}.

References

Hyndman, R. J., & Athanasopoulos, G. (2022). Notation for forecast reconciliation. https://robjhyndman.com/hyndsight/reconciliation-notation.html

Di Fonzo, T., & Girolimetto, D. (2021). Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives. International Journal of Forecasting. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.ijforecast.2021.08.004")}

See Also

coherent_smat() for the corresponding structural matrix \mathbf{S} and aggregate_key() for computing cross-sectional aggregations with tsibble data sets.

fabletools documentation built on April 30, 2026, 9:07 a.m.