Optimal Combination Temporal Reconciliation

This function performs forecast reconciliation for a single time series using temporal hierarchies (Athanasopoulos et al., 2017, Nystrup et al., 2020). The reconciled forecasts can be computed 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

terec(base, agg_order, tew = "sum", comb = "ols", res = NULL,
      approach = "proj", nn = NULL, settings = NULL, bounds = NULL,
      immutable = NULL, ...)

Arguments

base

A ($h(k^\ast + m) \times 1$) numeric vector containing the base forecasts to be reconciled, ordered from lowest to highest frequency; $m$ is the maximum aggregation order, $k^\ast$ is the sum of a chosen subset of the $p - 1$ factors of $m$ (excluding $m$ itself) 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 $m$.

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).

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 or validation errors ordered from the lowest frequency to the highest frequency. This vector is used to compute come covariance matrices.

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 forecasts:

"osqp": quadratic programming optimization (osqp solver, Girolimetto 2025).
"bpv": block principal pivoting algorithm (Wickramasuriya et al., 2020).
"nfca": negative forecasts correction algorithm (Kourentzes and Athanasopoulos, 2021; Girolimetto 2025).
"nnic": iterative non-negative reconciliation with immutable constraints (Girolimetto 2025).
"sntz": heuristic "set-negative-to-zero" (Di Fonzo and Girolimetto, 2023; Girolimetto 2025).

settings

A list of control parameters.

nn = "osqp" An object of class osqpSettings specifying settings for the osqp solver. For details, refer to the osqp documentation (Stellato et al., 2020)
nn = "bpv"
- ptype = "fixed": permutation method: "random" or "fixed"
- par = 10: the number of full exchange rules that may be attempted
- tol = sqrt(.Machine$double.eps): the tolerance criteria
- gtol = sqrt(.Machine$double.eps): the gradient tolerance criteria
- itmax = 100: the maximum number of algorithm iterations
nn = "nfca" and nn = "nnic"
- tol = sqrt(.Machine$double.eps): the tolerance criteria
- itmax = 100: the maximum number of algorithm iterations
nn = "sntz"
- type = "bu": the type of set-negative-to-zero heuristic: "bu" for bottom-up, "tdp" for top-down proportional, "tdsp" for top-down square proportional, "tdvw" for top-down variance weighted (the res param is used). See Girolimetto (2025) for details.
- tol = sqrt(.Machine$double.eps): the tolerance identification of negative values

bounds

A matrix (see set_bounds) with 4 columns ($k,j,lower,upper$), such that

Column 1 represents the temporal aggregation order ($k = m,\dots,1$).
Column 2 represents the temporal forecast horizon ($j = 1,\dots,m/k$).
Columns 3 and 4 indicates the lower and upper bounds, respectively.

immutable

A matrix with 2 columns ($k,j$), such that

Column 1 represents the temporal aggregation order ($k = m,\dots,1$).
Column 2 represents the temporal forecast horizon ($j = 1,\dots,m/k$).

For example, when working with a quarterly time series:

t(c(4, 1)) - Fix the one step ahead annual forecast.
t(c(1, 2)) - Fix the two step ahead quarterly forecast.

...

Arguments passed on to tecov

mse: If TRUE (default) the errors used to compute the covariance matrix are not mean-corrected.
shrink_fun: Shrinkage function of the covariance matrix, shrink_estim (default)

Value

An object of class foreco (see foreco-class) with a ($h(k^\ast+m) \times 1$) numeric vector of temporal reconciled forecasts.

References

Athanasopoulos, G., Hyndman, R.J., Kourentzes, N. and Petropoulos, F. (2017), Forecasting with Temporal Hierarchies, European Journal of Operational Research, 262, 1, 60-74. doi:10.1016/j.ejor.2017.02.046

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

Girolimetto, D. (2025), Non-negative forecast reconciliation: Optimal methods and operational solutions. Forecasting, 7(4), 64; doi:10.3390/forecast7040064

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

Kourentzes, N. and Athanasopoulos, G. (2021) Elucidate structure in intermittent demand series. European Journal of Operational Research, 288, 141-152. doi:10.1016/j.ejor.2020.05.046

Nystrup, P., Lindström, E., Pinson, P. and Madsen, H. (2020), Temporal hierarchies with autocorrelation for load forecasting, European Journal of Operational Research, 280, 1, 876-888. doi:10.1016/j.ejor.2019.07.061

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

Wickramasuriya, S. L., Turlach, B. A., and Hyndman, R. J. (2020). Optimal non-negative forecast reconciliation. Statistics and Computing, 30(5), 1167–1182. doi:10.1007/s11222-020-09930-0

Examples

set.seed(123)
# (7 x 1) base forecasts vector (simulated), m = 4
base <- rnorm(7, rep(c(20, 10, 5), c(1, 2, 4)))
# (70 x 1) in-sample residuals vector (simulated)
res <- rnorm(70)

m <- 4 # from quarterly to annual temporal aggregation
reco <- terec(base = base, agg_order = m, comb = "wlsv", res = res)

# Immutable reconciled forecast
# E.g. fix all the quarterly forecasts
imm_q <- expand.grid(k = 1, j = 1:4)
immreco <- terec(base = base, agg_order = m, comb = "wlsv",
                 res = res, immutable = imm_q)

# Non negative reconciliation
base[7] <- -base[7] # Making negative one of the quarterly base forecasts
nnreco <- terec(base = base, agg_order = m, comb = "wlsv",
                res = res, nn = "osqp")
summary(nnreco)
#> ✔ Temporal point forecast reconciliation
#> 
#> ── Method 
#> • Function used: `terec`
#> • Covariance approximation approach: `wlsv`
#> • Output: unknown object
#> 
#> ── Structure 
#> • Temporal orders (k): 4, 2, and 1
#> • Forecast horizons (h) per k: 1, 2, and 4
#> • Non-negative forecasts (check): `TRUE`
#> 
#> ── Non-negative reconciliation diagnostics 
#>     obj_val   run_time iter     prim_res status status_polish
#> 1 -421.8914 3.9623e-05   25 9.816134e-16      1             1
#> 
#> ── Reconciled forecasts 
#>   k-4 h-1   k-2 h-1   k-2 h-2   k-1 h-1 
#> 19.265312 10.043448  9.221864  4.992334 
#> ... (3 more elements)
#> Use `print(x, n_row)` to see more elements.