# Level conditional coherent reconciliation for cross-temporal hierarchies

Source:`R/reco_lcc.R`

`ctlcc.Rd`

This function implements a forecast reconciliation procedure inspired by the original proposal by Hollyman et al. (2021) for cross-temporal hierarchies. Level conditional coherent reconciled forecasts are conditional on (i.e., constrained by) the base forecasts of a specific upper level in the hierarchy (exogenous constraints). It also allows handling the linear constraints linking the variables endogenously (Di Fonzo and Girolimetto, 2022). The function can calculate Combined Conditional Coherent (CCC) forecasts as simple averages of Level-Conditional Coherent (LCC) and bottom-up reconciled forecasts, with either endogenous or exogenous constraints.

## Usage

```
ctlcc(base, agg_mat, nodes = "auto", agg_order, comb = "ols", res = NULL,
CCC = TRUE, const = "exogenous", hfbts = NULL, tew = "sum",
approach = "proj", nn = NULL, settings = NULL, ...)
```

## Arguments

- base
A (\(n \times h(k^\ast+m)\)) numeric matrix containing the base forecasts to be reconciled; \(n\) is the total number of variables, \(m\) is the max. order of temporal aggregation, \(k^\ast\) is the sum of (a subset of) (\(p-1\)) factors of \(m\), excluding \(m\), and \(h\) is the forecast horizon for the lowest frequency time series. The row identifies a time series, and the forecasts in each row are ordered from the lowest frequency (most temporally aggregated) to the highest frequency.

- 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.

- nodes
A (\(L \times 1\)) numeric vector indicating the number of variables in each of the upper \(L\) levels of the hierarchy. The

*default*value is the string "`auto`

" which calculates the number of variables in each level.- 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 \(m\).

- comb
A string specifying the reconciliation method. For a complete list, see ctcov.

- res
A (\(n \times N(k^\ast+m)\)) optional numeric matrix containing the in-sample residuals at all the temporal frequencies ordered from the lowest frequency to the highest frequency (columns) for each variable (rows). This matrix is used to compute some covariance matrices.

- CCC
A logical value indicating whether the Combined Conditional Coherent reconciled forecasts reconciliation should include bottom-up forecasts (

`TRUE`

,*default*), or not.- const
A string specifying the reconciliation constraints:

"

`exogenous`

" (*default*): Fixes the top level of each sub-hierarchy."

`endogenous`

": Coherently revises both the top and bottom levels.

- hfbts
A (\(n \times mh\)) numeric matrix containing high frequency bottom base forecasts defined by the user. This parameter can be omitted if only base forecasts are used (see Di Fonzo and Girolimetto, 2024).

- 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).- approach
A string specifying the approach used to compute the reconciled forecasts. Options include:

- nn
A string specifying the algorithm to compute non-negative reconciled forecasts:

"

`osqp`

": quadratic programming optimization (osqp solver)."

`sntz`

": heuristic "set-negative-to-zero" (Di Fonzo and Girolimetto, 2023).

- settings
An object of class

`osqpSettings`

specifying settings for the osqp solver. For details, refer to the osqp documentation (Stellato et al., 2020).- ...
Arguments passed on to

`ctcov`

`mse`

If

`TRUE`

(*default*) the residuals used to compute the covariance matrix are not mean-corrected.`shrink_fun`

Shrinkage function of the covariance matrix, shrink_estim (

*default*).

## References

Byron, R.P. (1978), The estimation of large social account matrices,
*Journal of the Royal Statistical Society, Series A*, 141, 3, 359-367.
doi:10.2307/2344807

Byron, R.P. (1979), Corrigenda: The estimation of large social account matrices,
*Journal of the Royal Statistical Society, Series A*, 142(3), 405.
doi:10.2307/2982515

Di Fonzo, T. and Girolimetto, D. (2024), Forecast combination-based forecast reconciliation:
Insights and extensions, *International Journal of Forecasting*, 40(2), 490–514.
doi:10.1016/j.ijforecast.2022.07.001

Di Fonzo, T. and Girolimetto, D. (2023b) Spatio-temporal reconciliation of solar forecasts.
*Solar Energy* 251, 13–29. doi:10.1016/j.solener.2023.01.003

Hyndman, R.J., Ahmed, R.A., Athanasopoulos, G. and Shang, H.L. (2011),
Optimal combination forecasts for hierarchical time series,
*Computational Statistics & Data Analysis*, 55, 9, 2579-2589.
doi:10.1016/j.csda.2011.03.006

Hollyman, R., Petropoulos, F. and Tipping, M.E. (2021), Understanding forecast reconciliation.
*European Journal of Operational Research*, 294, 149–160. doi:10.1016/j.ejor.2021.01.017

Stellato, B., Banjac, G., Goulart, P., Bemporad, A. and Boyd, S. (2020), OSQP:
An Operator Splitting solver for Quadratic Programs,
*Mathematical Programming Computation*, 12, 4, 637-672.
doi:10.1007/s12532-020-00179-2

## 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)
# (7 x 7) base forecasts matrix (simulated), agg_order = 4
base <- rbind(rnorm(7, rep(c(40, 20, 10), c(1, 2, 4))),
rnorm(7, rep(c(20, 10, 5), c(1, 2, 4))),
rnorm(7, rep(c(20, 10, 5), c(1, 2, 4))),
rnorm(7, rep(c(10, 5, 2.5), c(1, 2, 4))),
rnorm(7, rep(c(10, 5, 2.5), c(1, 2, 4))),
rnorm(7, rep(c(10, 5, 2.5), c(1, 2, 4))),
rnorm(7, rep(c(10, 5, 2.5), c(1, 2, 4))))
# (7 x 70) in-sample residuals matrix (simulated)
res <- matrix(rnorm(70*7), nrow = 7)
# (4 x 4) Naive high frequency bottom base forecasts vector:
# all forecasts are set equal to 2.5
naive <- matrix(2.5, 4, 4)
## EXOGENOUS CONSTRAINTS (Hollyman et al., 2021)
# Level Conditional Coherent (LCC) reconciled forecasts
exo_LC <- ctlcc(base = base, agg_mat = A, agg_order = 4, comb = "wlsh", nn = "osqp",
hfbts = naive, res = res, nodes = "auto", CCC = FALSE)
# Combined Conditional Coherent (CCC) reconciled forecasts
exo_CCC <- ctlcc(base = base, agg_mat = A, agg_order = 4, comb = "wlsh",
hfbts = naive, res = res, nodes = "auto", CCC = TRUE)
# Results detailed by level:
info_exo <- recoinfo(exo_CCC, verbose = FALSE)
# info_exo$lcc
## ENDOGENOUS CONSTRAINTS (Di Fonzo and Girolimetto, 2024)
# Level Conditional Coherent (LCC) reconciled forecasts
endo_LC <- ctlcc(base = base, agg_mat = A, agg_order = 4, comb = "wlsh",
res = res, nodes = "auto", CCC = FALSE,
const = "endogenous")
# Combined Conditional Coherent (CCC) reconciled forecasts
endo_CCC <- ctlcc(base = base, agg_mat = A, agg_order = 4, comb = "wlsh",
res = res, nodes = "auto", CCC = TRUE,
const = "endogenous")
# Results detailed by level:
info_endo <- recoinfo(endo_CCC, verbose = FALSE)
# info_endo$lcc
```