Cross-sectional top-down reconciliation

Top-down forecast reconciliation for genuine hierarchical/grouped time series (Gross and Sohl, 1990), where the forecast of a `Total' (top-level series, expected to be positive) is disaggregated according to a proportional scheme (weights). Besides fulfilling any aggregation constraint, the top-down reconciled forecasts should respect two main properties:

the top-level value remains unchanged;
all the bottom time series reconciled forecasts are non-negative.

Usage

cstd(base, agg_mat, weights, normalize = TRUE)

Arguments

base: A (\(h \times 1\)) numeric vector containing the top-level base forecast; \(h\) is the forecast horizon.
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.
weights: A (\(h \times n_b\)) numeric matrix containing the proportions for the bottom time series; \(h\) is the forecast horizon, and \(n_b\) is the total number of bottom variables.
normalize: If TRUE (default), the weights will sum to 1.

Value

A (\(h \times n\)) numeric matrix of cross-sectional reconciled forecasts.

References

Gross, C.W. and Sohl, J.E. (1990), Disaggregation methods to expedite product line forecasting. Journal of Forecasting 9(3), 233–254. doi:10.1002/for.3980090304

Examples

set.seed(123)
# Aggregation matrix for Z = X + Y, X = XX + XY and Y = YX + YY
A <- matrix(c(1,1,1,1,1,1,0,0,0,0,1,1), 3, byrow = TRUE)
# (3 x 1) top base forecasts vector (simulated), forecast horizon = 3
topf <- rnorm(3, 10)
# Same weights for different forecast horizons
fix_weights <- runif(4)
reco <- cstd(base = topf, agg_mat = A, weights = fix_weights)

# Different weights for different forecast horizons
h_weights <- matrix(runif(4*3), 3, 4)
recoh <- cstd(base = topf, agg_mat = A, weights = h_weights)