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Probabilistic forecast reconciliation of mixed hierarchies via top-down conditioning

Source: R/reconc_TDcond.R
reconc_TDcond.Rd

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.

Usage

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.

Value

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.

References

Zambon, L., Azzimonti, D., Rubattu, N., Corani, G. (2024). Probabilistic reconciliation of mixed-type hierarchical time series. The 40th Conference on Uncertainty in Artificial Intelligence, accepted.

See also

reconc_MixCond(), reconc_BUIS()

Examples


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]])
#>   Min. 1st Qu. Median    Mean 3rd Qu. Max
#> 1    5      17     20 20.0201      23  37
PMF.summary(fc_bottom[[1]])
#>   Min. 1st Qu. Median Mean 3rd Qu. Max
#> 1    0      12     15   15      18  43

PMF.summary(res.TDcond$bottom_reconciled$pmf[[2]])
#>   Min. 1st Qu. Median     Mean 3rd Qu. Max
#> 1    4      17     20 19.96865      23  37
PMF.summary(fc_bottom[[2]])
#>   Min. 1st Qu. Median Mean 3rd Qu. Max
#> 1    0      12     15   15      18  43

# The upper distribution remains similar
PMF.summary(res.TDcond$upper_reconciled$pmf[[1]])
#>   Min. 1st Qu. Median     Mean 3rd Qu. Max
#> 1   20      37     40 39.98875      43  60
PMF.get_var(res.TDcond$upper_reconciled$pmf[[1]])
#>          [,1]
#> [1,] 24.77582

## 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)
if (FALSE) { # \dontrun{
# If we reconcile with reconc_TDcond it won't work
res.TDcond <- reconc_TDcond(A, fc_bottom, fc_upper)
} # }

# 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]]))
#> [1] 0