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, 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 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.
"sntz": heuristic "set-negative-to-zero" (Di Fonzo and Girolimetto, 2023).

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" It includes: ptype for permutation method ("random" or "fixed", default), par for the number of full exchange rules that may be attempted (10, default), tol for the tolerance in convergence criteria (sqrt(.Machine$double.eps), default), gtol for the gradient tolerance in convergence criteria (sqrt(.Machine$double.eps), default), itmax for the maximum number of algorithm iterations (100, default)

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 lower 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 residuals 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

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

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 4.0623e-05   25 8.587788e-16      1             1