terec function

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.

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

• 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 list("osqp")

    for projection approach.

  • "strc_osqp": Numerical solution using list("osqp")

    for structural approach.

• nn: A string specifying the algorithm to compute non-negative reconciled forecasts:

  • "osqp": quadratic programming optimization (list("osqp") solver).
  • "sntz": heuristic "set-negative-to-zero" (Di Fonzo and Girolimetto, 2023).

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

• bounds: 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.

• immutable: 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

  • 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)

Returns

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

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

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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.2307/2982515")

Di Fonzo, T. and Girolimetto, D. (2023), Spatio-temporal reconciliation of solar forecasts, Solar Energy, 251, 13–29. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1016/j.csda.2011.03.006")

Nystrup, P., , E., Pinson, P. and Madsen, H. (2020), Temporal hierarchies with autocorrelation for load forecasting, European Journal of Operational Research, 280, 1, 876-888. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1007/s12532-020-00179-2")

See Also

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

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