Iterative procedure which produces cross-temporally reconciled
forecasts by alternating forecast reconciliation along one single dimension
(either cross-sectional or temporal) at each iteration step. Each iteration
consists in the first two steps of the heuristic KA procedure, so the forecasts are
reconciled by alternating cross-sectional (contemporaneous) reconciliation, and
reconciliation through temporal hierarchies in a cyclic fashion.
The choice of the dimension along with the first reconciliation step in each
iteration is performed is up to the user (param start_rec), and there is
no particular reason why one should perform the temporal reconciliation first, and
the cross-sectional reconciliation then.
The iterative procedure allows the user to get non-negative reconciled forecasts.
iterec(basef, m, C, thf_comb, hts_comb, Ut, nb, res, W,
Omega, mse = TRUE, corpcor = FALSE,type = "M",
sol = "direct", nn = FALSE, maxit = 100, tol = 1e-5,
start_rec = "thf", note = TRUE, plot = "mti",
settings = osqpSettings(verbose = FALSE, eps_abs = 1e-5,
eps_rel = 1e-5, polish_refine_iter = 100, polish = TRUE))(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']'.
Highest available sampling frequency per seasonal cycle (max. order of temporal aggregation).
(na x nb) cross-sectional (contemporaneous) matrix mapping the bottom
level series into the higher level ones.
Type of the ((k* + m) x (k* + m)) covariance matrix to be used in
the temporal reconciliation, see more in comb param of thfrec.
Type of the (n x n) covariance matrix to be used in the
cross-sectional reconciliation, see more in comb param of htsrec.
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 in this case nb (n = na + nb) is needed. If
the hierarchy admits a structural representation, Ut has dimension (na x n).
Number of bottom time series; if C is present, nb is not used.
(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.
This option permits to directly enter the covariance matrix in the
cross-sectional reconciliation, see more in W param of htsrec.
This option permits to directly enter the covariance matrix in the
reconciliation through temporal hierarchies, see more in Omega param of thfrec.
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.
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.
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.
Solution technique for the reconciliation problem: either "direct" for the direct
solution or "osqp" for the numerical solution (solving a linearly constrained quadratic
program using solve_osqp).
Logical value: TRUE if non-negative reconciled forecasts are wished.
Max number of iteration (100, default).
Convergence tolerance (1e-5, default).
Dimension along with the first reconciliation step in each
iteration is performed: it start from temporal reconciliation with "thf" (default),
from cross-sectional with "hts" and it does both reconciliation with "auto".
If note = TRUE (default) the function writes some notes to the console,
otherwise no note is produced (also no plot).
Some useful plots: "mti" (default) marginal trend inconsistencies,
"pat" step by step inconsistency pattern for each iteration, "distf" distance
forecasts iteration i and i-1, "all" all the plots.
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).
iterec returns a list with:
recf(n x h(k* + m)) matrix of reconciled forecasts.
d_csCross-sectional incoherence at each iteration.
d_teTemporal incoherence at each iteration.
start_recStarting coherence dimension (thf or hts).
tolTolerance.
flagControl code (see details).
timeElapsed time.
distIf start_rec = "auto", matrix of distances of the forecasts reconciled
from the base.
The above procedure can be seen as an extension of the well known iterative proportional fitting procedure (Deming and Stephan, 1940, Johnston and Pattie, 1993), also known as RAS method (Miller and Blair, 2009), to adjust the internal cell values of a two-dimensional matrix until they sum to some predetermined row and column totals. In that case the adjustment follows a proportional adjustment scheme, whereas in the cross-temporal reconciliation framework each adjustment step is made according to the penalty function associated to the single-dimension reconciliation procedure adopted.
Control status of iterative reconciliation:
-2Temporal/Cross-sectional reconciliation does not work.
-1Convergence not achieved (maximum iteration limit reached).
0Convergence achieved.
+1Convergence achieved: incoherence has increased in the next iteration (at least one time).
+2Convergence achieved: incoherence has increased in the next two or more iteration (at least one time).
+3The forecasts are already reconciled.
Deming, E., Stephan, F.F. (1940), On a least squares adjustment of a sampled frequency table when the expected marginal totals are known, The Annals of Mathematical Statistics, 11, 4, 427<U+2013>444.
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.
Johnston, R.J., Pattie, C.J. (1993), Entropy-Maximizing and the Iterative Proportional Fitting Procedure, The Professional Geographer, 45, 3, 317<U+2013>322.
Kourentzes, N., Athanasopoulos, G. (2019), Cross-temporal coherent forecasts for Australian tourism, Annals of Tourism Research, 75, 393-409.
Miller, R.E., Blair, P.D. (2009), Input-output analysis: foundations and extensions, 2nd edition, New York, Cambridge University Press.
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.
# NOT RUN {
data(FoReco_data)
obj <- iterec(FoReco_data$base, note=FALSE,
m = 12, C = FoReco_data$C, thf_comb = "acov",
hts_comb = "shr", res = FoReco_data$res, start_rec = "thf")
# }
# NOT RUN {
# }
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