Cross-sectional Gaussian probabilistic reconciliation

This function performs cross-sectional probabilistic forecast reconciliation assuming a multivariate normal base forecast distribution (Panagiotelis et al., 2023; Girolimetto et al., 2024; Wickramasuriya, 2024) for linearly constrained (e.g. hierarchical or grouped) multiple time series.

Usage

csmvn(base, agg_mat, cons_mat, comb = "ols", comb_base = comb,
      res = NULL, approach = "proj", reduce_form = FALSE, ...)

Arguments

base

A (\(h \times n\)) numeric matrix or multivariate time series (mts class) containing the base forecasts to be reconciled; \(h\) is the forecast horizon, and \(n\) is the total number of time series (\(n = n_a + n_b\)).

agg_mat

A (\(n_a \times n_b\)) numeric matrix representing the cross-sectional aggregation matrix. It maps the \(n_b\) bottom-level (free) variables into the \(n_a\) upper (constrained) variables.

cons_mat

A (\(n_a \times n\)) numeric matrix representing the cross-sectional zero constraints: each row represents a constraint equation, and each column represents a variable. The matrix can be of full rank, meaning the rows are linearly independent, but this is not a strict requirement, as the function allows for redundancy in the constraints.

comb

A string specifying the reconciliation method. For a complete list, see cscov.

comb_base

A string specifying the reconciliation method. For a complete list, see cscov.

res

An (\(N \times n\)) optional numeric matrix containing the in-sample residuals or validation errors. This matrix is used to compute some covariance matrices.

approach

A string specifying the approach used to compute the reconciled mean and covariance matrix. Options include:

"proj" (default): Projection approach according to Byron (1978, 1979).
"strc": Structural approach as proposed by Hyndman et al. (2011).

reduce_form

A logical parameter indicating whether the function should return the full distribution (FALSE, default) or only the distribution corresponding to the bottom-level time series (TRUE).

...

Arguments passed on to cscov

mse: If TRUE (default) the errors used to compute the covariance matrix are not mean-corrected.
shrink_fun: Shrinkage function of the covariance matrix, shrink_estim (default).

Value

A distributional::dist_multivariate_normal object.

References

Byron, R.P. (1978), The estimation of large social account matrices, Journal of the Royal Statistical Society, Series A, 141, 3, 359-367. doi:10.2307/2344807

Byron, R.P. (1979), Corrigenda: The estimation of large social account matrices, Journal of the Royal Statistical Society, Series A, 142(3), 405. doi:10.2307/2982515

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. doi:10.1016/j.ijforecast.2023.10.003

Hyndman, R.J., Ahmed, R.A., Athanasopoulos, G. and Shang, H.L. (2011), Optimal combination forecasts for hierarchical time series, Computational Statistics & Data Analysis, 55, 9, 2579-2589. doi:10.1016/j.csda.2011.03.006

Panagiotelis, A., Gamakumara, P., Athanasopoulos, G. and Hyndman, R.J. (2023), Probabilistic forecast reconciliation: Properties, evaluation and score optimisation, European Journal of Operational Research 306(2), 693–706. doi:10.1016/j.ejor.2022.07.040

Wickramasuriya, S. L. (2024). Probabilistic Forecast Reconciliation under the Gaussian Framework. Journal of Business & Economic Statistics, 42(1), 272–285. doi:10.1080/07350015.2023.2181176

Examples

set.seed(123)
# (2 x 3) base forecasts matrix (simulated), Z = X + Y
base <- matrix(rnorm(6, mean = c(20, 10, 10)), 2, byrow = TRUE)
# (10 x 3) in-sample residuals matrix (simulated)
res <- t(matrix(rnorm(n = 30), nrow = 3))

# Aggregation matrix for Z = X + Y
A <- t(c(1,1))
reco_dist <- csmvn(base = base, agg_mat = A, comb = "shr", res = res)