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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.


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



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.


Highest available sampling frequency per seasonal cycle (max. order of temporal aggregation, \(m\)), or a vector representing a subset of \(p\) factors of \(m\).


A string specifying the reconciliation method. For a complete list, see [tecov].


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.


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).


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.


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).


An object of class osqpSettings specifying settings for the osqp solver. For details, refer to the osqp documentation (Stellato et al., 2020).


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


A matrix with two columns (\(k,j\)), such that

Column 1

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

Column 2

Indicates 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


If TRUE (default) the residuals used to compute the covariance matrix are not mean-corrected.


Shrinkage function of the covariance matrix, [shrink_estim] (default)


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


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

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

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

See also

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

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


# (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 3.9291e-05   25 8.587788e-16      1             1