This function implements a forecast reconciliation procedure inspired by the original proposal by Hollyman et al. (2021) for 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

telcc(base, agg_order, comb = "ols", res = NULL, CCC = TRUE,
const = "exogenous", hfts = NULL, tew = "sum",
approach = "proj", nn = NULL, settings = NULL, ...)

## Arguments

base

A ($$h(k^\ast + m) \times 1$$) numeric vector containing base forecasts to be reconciled ordered from the lowest frequency to the highest frequency; $$m$$ is the max aggregation order, $$k^\ast$$ 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.

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 [tecov].

res

A ($$N(k^\ast+m) \times 1$$) optional numeric vector containing the in-sample residuals at all the temporal frequencies ordered from the lowest frequency to the highest frequency. This vector is used to compute come 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.

hfts

A ($$mh \times 1$$) numeric vector containing high frequency base forecasts defined by the user. This parameter can be omitted if only base forecasts in base 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:

• "proj" (default): Projection approach according to Byron (1978, 1979).

• "strc": Structural approach as proposed by Hyndman et al. (2011).

• "proj_osqp": Numerical solution using osqp for projection approach.

• "strc_osqp": Numerical solution using osqp for structural approach.

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 tecov

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)

## Value

A ($$h(k^\ast+m) \times 1$$) numeric vector of temporal reconciled forecasts.

## 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

Level conditional coherent reconciliation: cslcc(), ctlcc()

Temporal framework: teboot(), tebu(), tecov(), temo(), terec(), tetd(), tetools()

## Examples

set.seed(123)
# (7 x 1) base forecasts vector (simulated), agg_order = 4
base <- rnorm(7, rep(c(20, 10, 5), c(1, 2, 4)))
# (70 x 1) in-sample residuals vector (simulated)
res <- rnorm(70)
# (4 x 1) Naive high frequency base forecasts vector: all forecasts are set equal to 2.5
naive <- rep(2.5, 4)

## EXOGENOUS CONSTRAINTS
# Level Conditional Coherent (LCC) reconciled forecasts
exo_LC <- telcc(base = base, agg_order = 4, comb = "wlsh", hfts = naive,
res = res, nodes = "auto", CCC = FALSE)

# Combined Conditional Coherent (CCC) reconciled forecasts
exo_CCC <- telcc(base = base, agg_order = 4, comb = "wlsh", hfts = naive,
res = res, nodes = "auto", CCC = TRUE)

# Results detailed by level:
info_exo <- recoinfo(exo_CCC, verbose = FALSE)
# info_exo$lcc ## ENDOGENOUS CONSTRAINTS # Level Conditional Coherent (LCC) reconciled forecasts endo_LC <- telcc(base = base, agg_order = 4, comb = "wlsh", res = res, nodes = "auto", CCC = FALSE, const = "endogenous") # Combined Conditional Coherent (CCC) reconciled forecasts endo_CCC <- telcc(base = base, agg_order = 4, comb = "wlsh", res = res, nodes = "auto", CCC = TRUE, const = "endogenous") # Results detailed by level: info_endo <- recoinfo(endo_CCC, verbose = FALSE) # info_endo$lcc