This function performs optimal (in least squares sense) combination cross-temporal forecast reconciliation (Di Fonzo and Girolimetto 2023a, Girolimetto et al. 2023). The reconciled forecasts are calculated using either a projection approach (Byron, 1978, 1979) or the equivalent structural approach by Hyndman et al. (2011). Non-negative (Di Fonzo and Girolimetto, 2023) and immutable reconciled forecasts can be considered.

## Usage

ctrec(base, agg_mat, cons_mat, agg_order, comb = "ols", res = NULL,
tew = "sum", approach = "proj", nn = NULL, settings = NULL,
bounds = NULL, immutable = 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.

cons_mat

A ($$n_a \times n$$) numeric matrix representing the cross-sectional zero constraints. It spans the null space for the reconciled forecasts.

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.

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).

bounds

A ($$n(k^\ast + m) \times 2$$) numeric matrix specifying the cross-temporal bounds. The first column represents the lower bound, and the second column represents the upper bound.

immutable

A matrix with three columns ($$i,k,j$$), such that

Column 1

Represents the cross-sectional series ($$i = 1, \dots, n$$).

Column 2

Denotes the temporal aggregation order ($$k = m,\dots,1$$).

Column 3

Indicates the temporal forecast horizon ($$j = 1,\dots,m/k$$).

For example, when working with a quarterly multivariate time series ($$n = 3$$):

• t(c(1, 4, 1)) - Fix the one step ahead annual forecast of the first time series.

• t(c(2, 1, 2)) - Fix the two step ahead quarterly forecast of the second time series.

...

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).

## Value

A ($$n \times h(k^\ast+m)$$) numeric matrix of cross-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. (2023a), Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives, International Journal of Forecasting, 39, 1, 39-57. doi:10.1016/j.ijforecast.2021.08.004

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

Girolimetto, D., Athanasopoulos, G., Di Fonzo, T. and Hyndman, R.J. (2024), Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues. International Journal of Forecasting, 40, 3, 1134-1151. doi:10.1016/j.ijforecast.2023. 10.003

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

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

Regression-based reconciliation: csrec(), terec()

Cross-temporal framework: ctboot(), ctbu(), ctcov(), ctlcc(), ctmo(), cttd(), cttools(), iterec(), tcsrec()

## Examples

set.seed(123)
# (3 x 7) base forecasts matrix (simulated), Z = X + Y and m = 4
base <- rbind(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))))
# (3 x 70) in-sample residuals matrix (simulated)
res <- rbind(rnorm(70), rnorm(70), rnorm(70))

A <- t(c(1,1)) # Aggregation matrix for Z = X + Y
m <- 4 # from quarterly to annual temporal aggregation
reco <- ctrec(base = base, agg_mat = A, agg_order = m, comb = "wlsv", res = res)

C <- t(c(1, -1, -1)) # Zero constraints matrix for Z - X - Y = 0
reco <- ctrec(base = base, cons_mat = C, agg_order = m, comb = "wlsv", res = res)

# Immutable reconciled forecasts
# Fix all the quarterly forecasts of the second variable.
imm_mat <- expand.grid(i = 2, k = 1, j = 1:4)
immreco <- ctrec(base = base, cons_mat = C, agg_order = m, comb = "wlsv",
res = res, immutable = imm_mat)

# Non negative reconciliation
base[2,7] <- -2*base[2,7] # Making negative one of the quarterly base forecasts for variable X
nnreco <- ctrec(base = base, cons_mat = C, agg_order = m, comb = "wlsv",
res = res, nn = "osqp")
recoinfo(nnreco, verbose = FALSE)\$info
#>     obj_val   run_time iter      pri_res status status_polish
#> 1 -635.7645 8.3583e-05   25 1.284524e-15      1             1