ctrec function

Optimal combination cross-temporal reconciliation

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.

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 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 ($n(k^\ast + m) \times 2$) 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).

Returns

A ($n \times h(k^\ast+m)$) numeric matrix of cross-temporal reconciled forecasts.

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

References

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. (2023a), Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives, International Journal of Forecasting, 39, 1, 39-57. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1007/s12532-020-00179-2")

See Also

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

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