Closed form computation of the reconciled forecasts in case of Gaussian base forecasts.
Value
A list containing the bottom reconciled forecasts. The list has the following named elements:
bottom_reconciled_mean: reconciled mean for the bottom forecasts;bottom_reconciled_covariance: reconciled covariance for the bottom forecasts.
Details
In the vector of the means of the base forecasts the order must be: first the upper, then the bottom; the order within the uppers is given by the rows of A, the order within the bottoms by the columns of A. The order of the rows of the covariance matrix of the base forecasts is the same.
The function returns only the reconciled parameters of the bottom variables. The reconciled upper parameters and the reconciled samples for the entire hierarchy can be obtained from the reconciled bottom parameters. See the example section.
References
Corani, G., Azzimonti, D., Augusto, J.P.S.C., Zaffalon, M. (2021). Probabilistic Reconciliation of Hierarchical Forecast via Bayes' Rule. ECML PKDD 2020. Lecture Notes in Computer Science, vol 12459. doi:10.1007/978-3-030-67664-3_13 .
Zambon, L., Agosto, A., Giudici, P., Corani, G. (2024). Properties of the reconciled distributions for Gaussian and count forecasts. International Journal of Forecasting (in press). doi:10.1016/j.ijforecast.2023.12.004 .
Examples
library(bayesRecon)
# Create a minimal hierarchy with 2 bottom and 1 upper variable
A <- get_reconc_matrices(agg_levels=c(1,2), h=2)$A
#Set the parameters of the Gaussian base forecast distributions
mu1 <- 2
mu2 <- 4
muY <- 9
mus <- c(muY,mu1,mu2)
sigma1 <- 2
sigma2 <- 2
sigmaY <- 3
sigmas <- c(sigmaY,sigma1,sigma2)
Sigma <- diag(sigmas^2) # need to transform into covariance matrix
analytic_rec <- reconc_gaussian(A, base_forecasts.mu = mus,
base_forecasts.Sigma = Sigma)
bottom_mu_reconc <- analytic_rec$bottom_reconciled_mean
bottom_Sigma_reconc <- analytic_rec$bottom_reconciled_covariance
# Obtain reconciled mu and Sigma for the upper variable
upper_mu_reconc <- A %*% bottom_mu_reconc
upper_Sigma_reconc <- A %*% bottom_Sigma_reconc %*% t(A)
# Obtain reconciled mu and Sigma for the entire hierarchy
S <- rbind(A, diag(2)) # first, get summing matrix S
Y_mu_reconc <- S %*% bottom_mu_reconc
Y_Sigma_reconc <- S %*% bottom_Sigma_reconc %*% t(S) # note that this is a singular matrix
# Obtain reconciled samples for the entire hierarchy:
# i.e., sample from the reconciled bottoms and multiply by S
chol_decomp = chol(bottom_Sigma_reconc) # Compute the Cholesky Decomposition
Z = matrix(stats::rnorm(n = 2000), nrow = 2) # Sample from standard normal
B = t(chol_decomp) %*% Z + matrix(rep(bottom_mu_reconc, 1000), nrow=2) # Apply the transformation
U = S %*% B
Y_reconc = rbind(U, B)