FoReco (version 0.1.1)

tcsrec: Heuristic first-temporal-then-cross-sectional cross-temporal forecast reconciliation

Description

The cross-temporal forecast reconciliation procedure by Kourentzes and Athanasopoulos (2019) can be viewed as an ensemble forecasting procedure which exploits the simple averaging of different forecasts. First, for each time series the forecasts at any temporal aggregation order are reconciled using temporal hierarchies (thfrec), then time-by-time cross-sectional reconciliation is performed (htsrec). The projection matrices obtained at this step are then averaged and used to cross-sectionally reconcile the forecasts obtained at step 1, by this way fulfilling both cross-sectional and temporal constraints.

Usage

tcsrec(basef, m, C, thf_comb, hts_comb, Ut, nb, res, W, Omega,
       mse = TRUE, corpcor = FALSE, avg="KA", nn= FALSE,
       settings = osqpSettings(verbose = FALSE, eps_abs = 1e-5,
       eps_rel = 1e-5, polish_refine_iter = 100, polish = TRUE))

Arguments

basef

(n x h(k* + m)) matrix of base forecasts to be reconciled; n is the total number of variables, m is the highest frequency, k* is the sum of (p-1) factors of m, excluding m, and h is the forecast horizon. Each row identifies, a time series, and the forecasts are ordered as [lowest_freq' ... highest_freq']'.

m

Highest available sampling frequency per seasonal cycle (max. order of temporal aggregation).

C

(na x nb) cross-sectional (contemporaneous) matrix mapping the bottom level series into the higher level ones.

thf_comb

Type of the ((k* + m) x (k* + m)) covariance matrix to be used in the temporal reconciliation, see more in comb param of thfrec.

hts_comb

Type of the (n x n) covariance matrix to be used in the cross-sectional reconciliation, see more in comb param of htsrec.

Ut

Zero constraints cross-sectional (contemporaneous) kernel matrix \(\textbf{U}'\textbf{Y} = \mathbf{0}\) spanning the null space valid for the reconciled forecasts. It can be used instead of parameter C, but needs nb (n = na + nb). If the hierarchy admits a structural representation, Ut has dimension (na x n).

nb

Number of bottom time series; if C is present, nb is not used.

res

(n x N(k* + m)) matrix containing the residuals at all the temporal frequencies ordered [lowest_freq' ... highest_freq']' (columns) for each variable (row), needed to estimate the covariance matrix when hts_comb = {"wls", "shr", "sam"} and/or hts_comb = {"wlsv", "wlsh", "acov", "strar1", "sar1", "har1", "shr", "sam"}. The row must be in the same order as basef.

W

This option permits to directly enter the covariance matrix in the cross-sectional reconciliation, see more in W param of htsrec.

Omega

This option permits to directly enter the covariance matrix in the reconciliation through temporal hierarchies, see more in Omega param of thfrec.

mse

Logical value: TRUE (default) calculates the covariance matrix of the in-sample residuals (when necessary) according to the original hts and thief formulation: no mean correction, T as denominator.

corpcor

Logical value: TRUE if corpcor (Sch<U+00E4>fer et al., 2017) must be used to shrink the sample covariance matrix according to Sch<U+00E4>fer and Strimmer (2005), otherwise the function uses the same implementation as package hts.

avg

If avg = "KA" (default), the final projection matrix M is the one proposed by Kourentzes and Athanasopoulos (2019), otherwise it is calculated as simple average of all the involved projection matrices at step 2 of th procedure (see Di Fonzo and Girolimetto, 2020).

nn

Logical value, TRUE if non-negative reconciled forecasts are wished. Warning, the two-step heuristic reconciliation allows non negativity constraints only in the first step. This means that non-negativity is not guaranteed in the final reconciled values.

settings

Settings for osqp (object osqpSettings). The default options are: verbose = FALSE, eps_abs = 1e-5, eps_rel = 1e-5, polish_refine_iter = 100 and polish = TRUE. For details, see the osqp documentation (Stellato et al., 2019).

Value

The function returns a list with two elements:

recf

(n x h(k* + m)) reconciled forecasts matrix.

M

Projection matrix (projection approach).

Details

This function performs a two-step cross-temporal forecast reconciliation using the covariance matrices chosen by the user. If the combinations used by Kourentzes and Athanasopoulos (2019) are wished, thf_comb must be set equal to either "struc" or "wlsv", and hts_comb equal to either "shr" or "wls".

References

Di Fonzo, T., Girolimetto, D. (2020), Cross-Temporal Forecast Reconciliation: Optimal Combination Method and Heuristic Alternatives, Department of Statistical Sciences, University of Padua, arXiv:2006.08570.

Kourentzes, N., Athanasopoulos, G. (2019), Cross-temporal coherent forecasts for Australian tourism, Annals of Tourism Research, 75, 393-409.

Sch<U+00E4>fer, J.L., Opgen-Rhein, R., Zuber, V., Ahdesmaki, M., Duarte Silva, A.P., Strimmer, K. (2017), Package `corpcor', R package version 1.6.9 (April 1, 2017), https://CRAN.R-project.org/package= corpcor.

Sch<U+00E4>fer, J.L., Strimmer, K. (2005), A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics, Statistical Applications in Genetics and Molecular Biology, 4, 1.

Stellato, B., Banjac, G., Goulart, P., Bemporad, A., Boyd, S. (2018). OSQP: An Operator Splitting Solver for Quadratic Programs, arXiv:1711.08013.

Stellato, B., Banjac, G., Goulart, P., Boyd, S., Anderson, E. (2019), OSQP: Quadratic Programming Solver using the 'OSQP' Library, R package version 0.6.0.3 (October 10, 2019), https://CRAN.R-project.org/package=osqp.

Examples

# NOT RUN {
data(FoReco_data)
obj <- tcsrec(FoReco_data$base, m = 12, C = FoReco_data$C, thf_comb = "acov",
              hts_comb = "shr", res = FoReco_data$res)

# }