Optimal (in least squares sense) combination cross-temporal forecast reconciliation. The reconciled forecasts are calculated either through a projection approach (Byron, 1978), or the equivalent structural approach by Hyndman et al. (2011).
octrec(basef, m, C, comb, res, Ut, nb, W, Sstruc,
mse = TRUE, corpcor = FALSE, 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))(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 reconciliation, it corrispond to a different covariance
matrix (n(k* + m) x n(k* + m)), k* is the sum of (p-1)
factors of m (exclude m as factors) and n is the number
of variables:
ols (Identity);
struc (Cross-temporal summing matrix, use the Sstruc param to reduce computation time);
wlsh (Hierarchy variances matrix);
wlsv (Series variances matrix);
bdshr (Shrunk cross-covariance matrix, cross-sectional framework);
bdsam (Sample cross-covariance matrix, cross-sectional framework);
acov (Series auto-covariance matrix);
Sshr (Series shrunk cross-covariance matrix);
Ssam (Series cross-covariance matrix);
shr (Shrunk cross-covariance matrix);
sam (Sample cross-covariance matrix);
w use your personal matrix W in param W.
(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 comb =
{"sam", "wlsv", "wlsh", "acov", "Ssam",
"Sshr", "Sshr1", "shr"}.
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.
This option permits to directly enter the covariance matrix:
W must be a p.d. (n(k* + m) x n(k* + m)) matrix;
if comb is different from "w", W is not used.
Cross-temporal summing matrix (structural representation)\(,\; \check{\textbf{S}}\); can be obtained through the function ctf_tools.
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" (default) 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.
Return a list object of the reconciled forecasts at all levels.
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).
If the parameter keep is equal to "recf", then the function
returns only the (n x h(k* + m)) reconciled forecasts matrix, otherwise (keep="all")
it returns a list that mainly depends on what type of representation (type)
and methodology (sol) have been used:
recf(n x h(k* + m)) reconciled forecasts matrix.
OmegaCovariance matrix used for reconciled forecasts (vec(Y') representation).
WCovariance matrix used for reconciled forecasts (vec(Y) representation).
nn_checkNumber of negative values (if zero, there are no values below zero).
rec_checkLogical 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")Cross-temporal summing matrix\(, \; \textbf{Q}\check{\textbf{S}}\) (vec(Y') representation).
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).
In case of non-negativity constraints, there are two ways:
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.
sol = "osqp" and nn = TRUE: the base forecasts will be
reconciled through the "osqp" solver.
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.
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 <- octrec(FoReco_data$base, m = 12, C = FoReco_data$C,
comb = "bdshr", res = FoReco_data$res)
# }
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