ctrec: Optimal combination cross-temporal reconciliation
In FoReco: Forecast Reconciliation

ctrec

R Documentation

Optimal combination cross-temporal reconciliation

Description

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.

Usage

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 osqp for projection approach. "`strc_osqp`": Numerical solution using osqp for structural approach.
`nn`	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).
`settings`	An object of class `osqpSettings` specifying settings for the osqp solver. For details, refer to the 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).

Value

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

References

Byron, R.P. (1978), The estimation of large social account matrices, Journal of the Royal Statistical Society, Series A, 141, 3, 359-367. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s12532-020-00179-2")}

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

FoReco documentation built on Sept. 14, 2024, 9:07 a.m.