zbMATH — the first resource for mathematics

Optimal non-negative forecast reconciliation. (English) Zbl 1452.62706
Summary: The sum of forecasts of disaggregated time series is often required to equal the forecast of the aggregate, giving a set of coherent forecasts. The least squares solution for finding coherent forecasts uses a reconciliation approach known as MinT, proposed by S. L. Wickramasuriya et al. [J. Am. Stat. Assoc. 114, No. 526, 804–819 (2019; Zbl 1420.62402)]. The MinT approach and its variants do not guarantee that the coherent forecasts are non-negative, even when all of the original forecasts are non-negative in nature. This has become a serious issue in applications that are inherently non-negative such as with sales data or tourism numbers. While overcoming this difficulty, we reconsider the least squares minimization problem with non-negativity constraints to ensure that the coherent forecasts are strictly non-negative. The constrained quadratic programming problem is solved using three algorithms. They are the block principal pivoting (BPV) algorithm, projected conjugate gradient (PCG) algorithm, and scaled gradient projection algorithm. A Monte Carlo simulation is performed to evaluate the computational performances of these algorithms as the number of time series increases. The results demonstrate that the BPV algorithm clearly outperforms the rest, and PCG is the second best. The superior performance of the BPV algorithm can be partially attributed to the alternative representation of the weight matrix in the MinT approach. An empirical investigation is carried out to assess the impact of imposing non-negativity constraints on forecast reconciliation over the unconstrained method. It is observed that slight gains in forecast accuracy have occurred at the most disaggregated level. At the aggregated level, slight losses are also observed. Although the gains or losses are negligible, the procedure plays an important role in decision and policy implementation processes.
62M30 Inference from spatial processes
62M20 Inference from stochastic processes and prediction
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
Full Text: DOI
[1] Bai, J., Inferential theory for factor models of large dimensions, Econometrica, 71, 1, 135171 (2003) · Zbl 1136.62354
[2] Barzilai, J.; Borwein, JM, Two-point step size gradient methods, IMA J. Numer. Anal., 8, 1, 141-148 (1988) · Zbl 0638.65055
[3] Berry, MW; Browne, M.; Langville, AN; Paul Pauca, V.; Plemmons, RJ, Algorithms and applications for approximate nonnegative matrix factorization, Comput. Stat. Data Anal., 52, 155-173 (2007) · Zbl 1452.90298
[4] Bertero, M., Scaled gradient projection methods for astronomical imaging, EAS Publ. Ser., 59, 325-356 (2013)
[5] Birgin, EG; Martinez, JM; Raydan, M., Inexact spectral projected gradient methods on convex sets, IMA J. Numer. Anal., 23, 4, 539-559 (2003) · Zbl 1047.65042
[6] Bonettini, S.; Zanella, R.; Zanni, L., A scaled gradient projection method for constrained image deblurring, Inverse Probl., 25, 1, 23 (2009) · Zbl 1155.94011
[7] Chatfield, C., Time-Series Forecasting (2000), New York: Chapman and Hall, New York
[8] Chen, D.; Plemmons, RJ; Bultheel, A.; Cools, R., Non-negativity constraints in numerical analysis, The Birth of Numerical Analysis, 109-140 (2009), New Jersey: World Scientific Publishing, New Jersey
[9] Figueiredo, MAT; Nowak, RD; Wright, SJ, Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems, IEEE J. Sel. Top. Signal Process., 1, 4, 586-597 (2007)
[10] Hyndman, R.J.: forecast: forecasting functions for time series and linear models. R package version 8.5 (2019). https://pkg.robjhyndman/forecast. Accessed Feb 2019
[11] Hyndman, RJ; Khandakar, Y., Automatic time series forecasting: the forecast package for R, J. Stat. Softw., 27, 3, 1-22 (2008)
[12] Hyndman, RJ; Ahmed, RA; Athanasopoulos, G.; Shang, HL, Optimal combination forecasts for hierarchical time series, Comput. Stat. Data Anal., 55, 2579-2589 (2011) · Zbl 06917716
[13] Hyndman, RJ; Lee, AJ; Wang, E., Fast computation of reconciled forecasts for hierarchical and grouped time series, Comput. Stat. Data Anal., 97, 16-32 (2016) · Zbl 06918489
[14] Johnson, NL; Kotz, S.; Balakrishnan, N.; Barnett, V., Continuous univariate distributions, Wiley Series in Probability and Mathematical Statistics (1994), New York: Wiley, New York
[15] Júdice, JJ; Pires, FM, Bard-type methods for the linear complementarity problem with symmetric positive definite matrices, IMA J. Manag. Math., 2, 1, 5168 (1989)
[16] Judice, JJ; Pires, FM, A block principal pivoting algorithm for large-scale strictly monotone linear complementarity problems, Comput. Oper. Res., 21, 5, 587-596 (1994) · Zbl 0802.90106
[17] Karjalainen, EJ; Karjalainen, UP, Component reconstruction in the primary space of spectra and concentrations. Alternating regression and related direct methods, Anal. Chim. Acta, 250, 169-179 (1991)
[18] Kim, J.; Park, H., Fast nonnegative matrix factorization: an active-set-like method and comparisons, SIAM J. Sci. Comput., 33, 6, 3261-3281 (2011) · Zbl 1232.65068
[19] Kostreva, MM, Block pivot methods for solving the complementarity problem, Linear Algebra Appl., 21, 207-215 (1978) · Zbl 0395.65032
[20] Lawson, CL; Hanson, RJ, Solving Least Squares Problems (1974), New Jersey: Prentice-Hall, New Jersey
[21] Microsoft Corporation and Steve Weston: doParallel: foreach parallel adaptor for the ‘parallel’ package. R package version 1.0.15 (2019a). https://CRAN.R-project.org/package=doParallel. Accessed Feb 2019
[22] Microsoft Corporation and Steve Weston: foreach: provides foreach looping construct. R package version 1.4.7 (2019b). https://CRAN.R-project.org/package=foreach. Accessed Feb 2019
[23] Murty, KG, Note on a Bard-type scheme for solving the complementarity problem, OPSEARCH, 11, 2-3, 123-130 (1974)
[24] Nocedal, J.; Wright, SJ; Mikosch, TV; Resnick, SI; Robinson, SM, Numerical Optimization (2006), New York: Springer, New York
[25] Turlach, B.A., Weingessel, A.: quadprog: functions to solve quadratic programming problems. R package version 1.5-7 (2019). https://CRAN.R-project.org/package=quadprog. Accessed Feb 2019
[26] Turlach, BA; Wright, SJ, Quadratic programming, Wiley Interdiscip. Rev. Comput. Stat., 7, 2, 153-159 (2015)
[27] Wickramasuriya, SL; Athanasopoulos, G.; Hyndman, RJ, Optimal forecast reconciliation of hierarchical and grouped time series through trace minimization, J. Am. Stat. Assoc., 114, 526, 804-819 (2019) · Zbl 1420.62402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.