# Probabilistic forecast reconciliation of mixed hierarchies via top-down conditioning Uses the top-down conditioning algorithm to draw samples from the reconciled forecast distribution. Reconciliation is performed in two steps: first, the upper base forecasts are reconciled via conditioning, using only the hierarchical constraints between the upper variables; then, the bottom distributions are updated via a probabilistic top-down procedure. ```r reconc_TDcond( A, fc_bottom, fc_upper, bottom_in_type = "pmf", distr = NULL, num_samples = 20000, return_type = "pmf", suppress_warnings = FALSE, seed = NULL ) ``` ## Arguments - `A`: aggregation matrix (n_upper x n_bottom). - `fc_bottom`: A list containing the bottom base forecasts, see details. - `fc_upper`: A list containing the upper base forecasts, see details. - `bottom_in_type`: A string with three possible values: * 'pmf' if the bottom base forecasts are in the form of pmf, see details; * 'samples' if the bottom base forecasts are in the form of samples; * 'params' if the bottom base forecasts are in the form of estimated parameters. - `distr`: A string describing the type of bottom base forecasts ('poisson' or 'nbinom'). This is only used if `bottom_in_type=='params'`. - `num_samples`: Number of samples drawn from the reconciled distribution. This is ignored if `bottom_in_type='samples'`; in this case, the number of reconciled samples is equal to the number of samples of the base forecasts. - `return_type`: The return type of the reconciled distributions. A string with three possible values: * 'pmf' returns a list containing the reconciled marginal pmf objects; * 'samples' returns a list containing the reconciled multivariate samples; * 'all' returns a list with both pmf objects and samples. - `suppress_warnings`: Logical. If `TRUE`, no warnings about samples are triggered. If `FALSE`, warnings are generated. Default is `FALSE`. See Details. - `seed`: Seed for reproducibility. ## Returns A list containing the reconciled forecasts. The list has the following named elements: * `bottom_reconciled`: a list containing the pmf, the samples (matrix n_bottom x `num_samples`) or both, depending on the value of `return_type`; * `upper_reconciled`: a list containing the pmf, the samples (matrix n_upper x `num_samples`) or both, depending on the value of `return_type`. ## Details The base bottom forecasts `fc_bottom` must be a list of length n_bottom, where each element is either * a PMF object (see details below), if `bottom_in_type='pmf'`; * a vector of samples, if `bottom_in_type='samples'`; * a list of parameters, if `bottom_in_type='params'`: * lambda for the Poisson base forecast if `distr`='poisson', see Poisson ; * size and prob (or mu) for the negative binomial base forecast if `distr`='nbinom', see NegBinomial . The base upper forecasts `fc_upper` must be a list containing the parameters of the multivariate Gaussian distribution of the upper forecasts. The list must contain only the named elements `mu` (vector of length n_upper) and `Sigma` (n_upper x n_upper matrix). The order of the upper and bottom base forecasts must match the order of (respectively) the rows and the columns of A. A PMF object is a numerical vector containing the probability mass function of a discrete distribution. Each element corresponds to the probability of the integers from 0 to the last value of the support. See also PMF.get_mean , PMF.get_var , PMF.sample , PMF.get_quantile , PMF.summary for functions that handle PMF objects. If some of the reconciled upper samples lie outside the support of the bottom-up distribution, those samples are discarded and a warning is triggered. The warning reports the percentage of samples kept. ## Examples ```r library(bayesRecon) # Consider a simple hierarchy with two bottom and one upper A <- matrix(c(1,1),nrow=1) # The bottom forecasts are Poisson with lambda=15 lambda <- 15 n_tot <- 60 fc_bottom <- list() fc_bottom[[1]] <- apply(matrix(seq(0,n_tot)),MARGIN=1,FUN=function(x) dpois(x,lambda=lambda)) fc_bottom[[2]] <- apply(matrix(seq(0,n_tot)),MARGIN=1,FUN=function(x) dpois(x,lambda=lambda)) # The upper forecast is a Normal with mean 40 and std 5 fc_upper<- list(mu=40, Sigma=matrix(c(5^2))) # We can reconcile with reconc_TDcond res.TDcond <- reconc_TDcond(A, fc_bottom, fc_upper) # Note that the bottom distributions are shifted to the right PMF.summary(res.TDcond$bottom_reconciled$pmf[[1]]) PMF.summary(fc_bottom[[1]]) PMF.summary(res.TDcond$bottom_reconciled$pmf[[2]]) PMF.summary(fc_bottom[[2]]) # The upper distribution remains similar PMF.summary(res.TDcond$upper_reconciled$pmf[[1]]) PMF.get_var(res.TDcond$upper_reconciled$pmf[[1]]) ## Example 2: reconciliation with unbalanced hierarchy # We consider the example in Fig. 9 of Zambon et al. (2024). # The hierarchy has 5 bottoms and 3 uppers A <- matrix(c(1,1,1,1,1, 1,1,0,0,0, 0,0,1,1,0),nrow=3,byrow = TRUE) # Note that the 5th bottom only appears in the highest level, this is an unbalanced hierarchy. n_upper = nrow(A) n_bottom = ncol(A) # The bottom forecasts are Poisson with lambda=15 lambda <- 15 n_tot <- 60 fc_bottom <- list() for(i in seq(n_bottom)){ fc_bottom[[i]] <- apply(matrix(seq(0,n_tot)),MARGIN=1,FUN=function(x) dpois(x,lambda=lambda)) } # The upper forecasts are a multivariate Gaussian mu = c(75, 30, 30) Sigma = matrix(c(5^2,5,5, 5, 10, 0, 5, 0,10), nrow=3, byrow = TRUE) fc_upper<- list(mu=mu, Sigma=Sigma) ## Not run: # If we reconcile with reconc_TDcond it won't work res.TDcond <- reconc_TDcond(A, fc_bottom, fc_upper) ## End(Not run) # We can balance the hierarchy with by duplicating the node b5 # In practice this means: # i) consider the time series observations for b5 as the upper u4, # ii) fit the multivariate ts model for u1, u2, u3, u4. # In this example we simply assume that the forecast for u1-u4 is # Gaussian with the mean and variance of u4 given by the parameters in b5. mean_b5 <- lambda var_b5 <- lambda mu = c(75, 30, 30,mean_b5) Sigma = matrix(c(5^2,5,5,5, 5, 10, 0, 0, 5, 0, 10, 0, 5, 0, 0, var_b5), nrow=4, byrow = TRUE) fc_upper<- list(mu=mu, Sigma=Sigma) # We also need to update the aggregation matrix A <- matrix(c(1,1,1,1,1, 1,1,0,0,0, 0,0,1,1,0, 0,0,0,0,1),nrow=4,byrow = TRUE) # We can now reconcile with TDcond res.TDcond <- reconc_TDcond(A, fc_bottom, fc_upper) # Note that the reconciled distribution of b5 and u4 are identical, # keep this in mind when using the results of your reconciliation! max(abs(res.TDcond$bottom_reconciled$pmf[[5]]- res.TDcond$upper_reconciled$pmf[[4]])) ``` ## References Zambon, L., Azzimonti, D., Rubattu, N., Corani, G. (2024). **Probabilistic reconciliation of mixed-type hierarchical time series**. Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence, PMLR 244:4078-4095. [https://proceedings.mlr.press/v244/zambon24a.html](https://proceedings.mlr.press/v244/zambon24a.html). ## See Also `reconc_MixCond()`, `reconc_BUIS()`

reconc_TDcond function