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Optimal combination temporal reconciliation

Source: R/reco_opt.R
terec.Rd

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

Arguments

base

A (\(N(k^\ast + m) \times 1\)) 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 \(m\).

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.

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 forecasts:

  • "osqp": quadratic programming optimization (osqp solver).

  • "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

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. mimeo

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

See also

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

Temporal framework: teboot(), tebu(), tecov(), telcc(), temo(), temvn(), tesmp(), tetd(), tetools()

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")
recoinfo(nnreco, verbose = FALSE)$info
#>     obj_val   run_time iter      pri_res status status_polish
#> 1 -421.8914 3.6373e-05   25 8.587788e-16      1             1