coherent_smat: Structural (summing) matrix for coherent time series

In fabletools: Core Tools for Packages in the 'fable' Framework

Structural (summing) matrix for coherent time series

Description

Constructs the n \times n_b structural matrix \mathbf{S} for a set of linearly related time series, where n is the total number of series and n_b is the number of bottom-level series.

Usage

coherent_smat(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

Let \boldsymbol{b}_t be the n_b-vector of bottom-level series at time t, and \boldsymbol{a}_t = \mathbf{A}\boldsymbol{b}_t be the n_a = n - n_b aggregated series, where \mathbf{A} is the n_a \times n_b aggregation matrix. The full n-vector of series is \boldsymbol{y}_t = [\boldsymbol{a}_t', \boldsymbol{b}_t']', and the structural matrix satisfies

\boldsymbol{y}_t = \mathbf{S}\boldsymbol{b}_t, \quad \mathbf{S} = \begin{bmatrix} \mathbf{A} \\ \mathbf{I}_{n_b} \end{bmatrix}.

Any linear reconciliation method can be written as \tilde{\boldsymbol{y}}_h = \mathbf{S}\mathbf{G}\hat{\boldsymbol{y}}_h, where \hat{\boldsymbol{y}}_h are the h-step base forecasts. Optimal reconciled forecasts use \mathbf{G} = (\mathbf{S}'\mathbf{W}^{-1}\mathbf{S})^{-1}\mathbf{S}'\mathbf{W}^{-1}, with \mathbf{W} a positive definite weight matrix (see min_trace()).

Value

An n \times n_b matrix (dense or sparse) encoding the structural relationships among all series.

References

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

See Also

coherent_cmat() for the corresponding zero-constraint matrix \mathbf{C} and aggregate_key() for computing cross-sectional aggregations with tsibble data sets.

Examples

tsibble::tourism %>% 
  aggregate_key(Purpose, Trips = sum(Trips)) %>%
  coherent_smat()

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