Cross-temporal bottom-up reconciled forecasts for all series at any temporal aggregation level are computed by appropriate summation of the high-frequency bottom base forecasts \(\widehat{\mathbf{B}^{[1]}}\): $$\widetilde{\mathbf{X}} = \mathbf{S}_{cs}\widehat{\mathbf{B}^{[1]}}\mathbf{S}'_{te},$$ where \(\mathbf{S}_{cs}\) and \(\mathbf{S}_{te}\) are the cross-sectional and temporal structural matrices, respectively.

## Arguments

- base
A (\(n_b \times hm\)) numeric matrix containing high-frequency bottom base forecasts; \(n_b\) is the total number of high-frequency bottom variables, \(m\) is the max aggregation order, and \(h\) is the forecast horizon for the lowest frequency time series.

- 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.

- agg_order
Highest available sampling frequency per seasonal cycle (max. order of temporal aggregation, \(m\)), or a vector representing a subset of \(p\) factors of \(m\).

- tew
A string specifying the type of temporal aggregation. Options include: "

`sum`

" (simple summation,*default*), "`avg`

" (average), "`first`

" (first value of the period), and "`last`

" (last value of the period).- sntz
If

`TRUE`

, the negative base forecasts are set to zero before applying bottom-up.

## Examples

```
set.seed(123)
# Aggregation matrix for Z = X + Y
A <- t(c(1,1))
# (2 x 4) high frequency bottom base forecasts matrix (simulated),
# agg_order = 4 (annual-quarterly)
hfbts <- matrix(rnorm(4*2, 2.5), 2, 4)
reco <- ctbu(base = hfbts, agg_mat = A, agg_order = 4)
# Non negative reconciliation
hfbts[1,4] <- -hfbts[1,4] # Making negative one of the quarterly base forecasts for variable X
nnreco <- ctbu(base = hfbts, agg_mat = A, agg_order = 4, sntz = TRUE)
```