FoReco (version 0.1.1)

thfrec: Forecast reconciliation through temporal hierarchies (temporal reconciliation)

Description

Forecast reconciliation of one time series through temporal hierarchies (Athanasopoulos et al., 2017). The reconciled forecasts are calculated either through a projection approach (Byron, 1978), or the equivalent structural approach by Hyndman et al. (2011). Moreover, the classic bottom-up approach is available.

Usage

thfrec(basef, m, comb, res, mse = TRUE, corpcor = FALSE, Omega,
       type = "M", sol = "direct", nn = FALSE, keep = "list",
       settings = osqpSettings(verbose = FALSE, eps_abs = 1e-5,
       eps_rel = 1e-5, polish_refine_iter = 100, polish=TRUE))

Arguments

basef

Vector of base forecasts to be reconciled, containing the forecasts at all the needed temporal frequencies ordered as [lowest_freq' ... highest_freq']'.

m

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

comb

Type of the reconciliation. Except for bottom up, all other options correspond to a different ((k* + m) x (k* + m)) covariance matrix, k* is the sum of (p-1) factors of m (excluding m):

• bu (Bottom-up);

• ols (Identity);

• struc (Structural variances);

• wlsv (Series variances);

• wlsh (Hierarchy variances);

• acov (Auto-covariance matrix);

• strar1 (Structural Markov);

• sar1 (Series Markov);

• har1 (Hierarchy Markov);

• shr (Shrunk cross-covariance matrix);

• sam (Sample cross-covariance matrix);

• omega use your personal matrix Omega in param Omega.

res

vector containing the in-sample residuals at all the temporal frequencies ordered as basef, i.e. [lowest_freq' ... highest_freq']', needed to estimate the covariance matrix when comb = {"wlsv", "wlsh", "acov", "strar1", "sar1", "har1", "shr", "sam"}.

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.

Omega

This option permits to directly enter the covariance matrix:

1. Omega must be a p.d. ((k* + m) x (k* + m)) matrix;

2. if comb is different from "omega", Omega is not used.

type

Approach used to compute the reconciled forecasts: "M" for the projection approach with matrix M (default), or "S" for the structural approach with summing matrix S.

sol

Solution technique for the reconciliation problem: either "direct" (default) for the direct solution or "osqp" for the numerical solution (solving a linearly constrained quadratic program using solve_osqp).

nn

Logical value: TRUE if non-negative reconciled forecasts are wished.

keep

Return a list object of the reconciled forecasts at all levels.

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

If the parameter keep is equal to "recf", then the function returns only the reconciled forecasts vector, otherwise (keep="all") it returns a list that mainly depends on what type of representation (type) and methodology (sol) have been used:

recf

(h(k* + m) x 1) reconciled forecasts vector.

Omega

Covariance matrix used for forecast reconciliation.

nn_check

Number of negative values (if zero, there are no values below zero).

rec_check

Logical value: has the hierarchy been respected?

M (type="M" and type="direct")

Projection matrix (projection approach)

G (type="S" and type="direct")

Projection matrix (structural approach).

S (type="S" and type="direct")

Temporal summing matrix, R.

info (type="osqp")

matrix with some useful indicators (columns) for each forecast horizon h (rows): run time (run_time) number of iteration, norm of primal residual (pri_res), status of osqp's solution (status) and polish's status (status_polish).

Only if comb = "bu", the function returns recf, S and M.

Details

In case of non-negativity constraints, there are two ways:

1. sol = "direct" and nn = TRUE: the base forecasts will be reconciled at first without non-negativity constraints, then, if negative reconciled values are present, the "osqp" solver is used.

2. sol = "osqp" and nn = TRUE: the base forecasts will be reconciled through the "osqp" solver.

References

Athanasopoulos, G., Hyndman, R.J., Kourentzes, N., Petropoulos, F. (2017), Forecasting with Temporal Hierarchies, European Journal of Operational Research, 262, 1, 60-74.

Byron, R.P. (1978), The estimation of large social accounts matrices, Journal of the Royal Statistical Society A, 141, 3, 359-367.

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.

Hyndman, R.J., Ahmed, R.A., Athanasopoulos, G., Shang, H.L. (2011), Optimal combination forecasts for hierarchical time series, Computational Statistics & Data Analysis, 55, 9, 2579-2589.

Nystrup, P., Lindstr<U+00F6>m, E., Pinson, P., Madsen, H. (2020), Temporal hierarchies with autocorrelation for load forecasting, European Journal of Operational Research, 280, 1, 876-888.

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)
# top ts base forecasts ([lowest_freq' ...  highest_freq']')
topbase <- FoReco_data$base[1, ]
 # top ts residuals ([lowest_freq' ...  highest_freq']')
topres <- FoReco_data$res[1, ]
obj <- thfrec(topbase, m = 12, comb = "acov", res = topres)

# }