Efficient probabilistic reconciliation of forecasts for real-valued and count time series. (English) Zbl 1529.62043
Summary: Hierarchical time series are common in several applied fields. The forecasts for these time series are required to be coherent, that is, to satisfy the constraints given by the hierarchy. The most popular technique to enforce coherence is called reconciliation, which adjusts the base forecasts computed for each time series. However, recent works on probabilistic reconciliation present several limitations. In this paper, we propose a new approach based on conditioning to reconcile any type of forecast distribution. We then introduce a new algorithm, called Bottom-Up Importance Sampling, to efficiently sample from the reconciled distribution. It can be used for any base forecast distribution: discrete, continuous, or in the form of samples, providing a major speedup compared to the current methods. Experiments on several temporal hierarchies show a significant improvement over base probabilistic forecasts.
MSC:
| 62-08 | Computational methods for problems pertaining to statistics |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62M20 | Inference from stochastic processes and prediction |
Keywords:
forecast reconciliation; probabilistic reconciliation; temporal hierarchies; importance samplingReferences:
| [1] | Agapiou, S.; Papaspiliopoulos, O.; Sanz-Alonso, D.; Stuart, AM, Importance sampling: intrinsic dimension and computational cost, Stat. Sci., 32, 405-431 (2017) · Zbl 1442.62026 · doi:10.1214/17-STS611 |
| [2] | Athanasopoulos, G.; Hyndman, RJ; Kourentzes, N.; Petropoulos, F., Forecasting with temporal hierarchies, Eur. J. Oper. Res., 262, 1, 60-74 (2017) · Zbl 1403.62154 · doi:10.1016/j.ejor.2017.02.046 |
| [3] | Azzimonti, D., Rubattu, N., Zambon, L., Corani, G.: bayesRecon: Probabilistic Reconciliation via Conditioning, (2023). R package version 0.1.2 |
| [4] | Billingsley, P., Probability and measure (2008), New York: Wiley, New York |
| [5] | Chen, Y-C, A tutorial on kernel density estimation and recent advances, Biostat. Epidemiol., 1, 1, 161-187 (2017) · doi:10.1080/24709360.2017.1396742 |
| [6] | Çinlar, E., Probability and stochastics (2011), Berlin: Springer, Berlin · Zbl 1226.60001 |
| [7] | Corani, G., Azzimonti, D., Augusto, J.P., Zaffalon, M.: Probabilistic reconciliation of hierarchical forecast via Bayes’ rule. In Proc. European Conf. On Machine Learning and Knowledge Discovery in Database ECML/PKDD, vol. 3, pp. 211-226, (2020) |
| [8] | Corani, G.; Azzimonti, D.; Rubattu, N., Probabilistic reconciliation of count time series, Int. J. Forecast. (2023) · doi:10.1016/j.ijforecast.2023.04.003 |
| [9] | Di Fonzo, T.; Girolimetto, D., Cross-temporal forecast reconciliation: optimal combination method and heuristic alternatives, Int. J. Forecast., 39, 39-57 (2021) · doi:10.1016/j.ijforecast.2021.08.004 |
| [10] | Di Fonzo, T.; Girolimetto, D., Forecast combination-based forecast reconciliation: insights and extensions, Int. J. Forecast. (2022) · doi:10.1016/j.ijforecast.2022.07.001 |
| [11] | Elvira, V.; Martino, L., Advances in importance sampling (2021), New York: Wiley, New York · doi:10.1002/9781118445112.stat08284 |
| [12] | Gneiting, T., Quantiles as optimal point forecasts, Int. J. Forecast., 27, 2, 197-207 (2011) · doi:10.1016/j.ijforecast.2009.12.015 |
| [13] | Haario, H., Saksman, E., Tamminen, J.: An adaptive metropolis algorithm. Bernoulli, pp. 223-242, (2001) · Zbl 0989.65004 |
| [14] | Haughton, J.; Khandker, SR, Handbook on poverty+ inequality (2009), Washington, D.C.: World Bank Publications, Washington, D.C. |
| [15] | Hoffman, MD; Gelman, A., The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo, J. Mach. Learn. Res., 15, 1, 1593-1623 (2014) · Zbl 1319.60150 |
| [16] | Hollyman, R.; Petropoulos, F.; Tipping, ME, Understanding forecast reconciliation, Eur. J. Oper. Res., 294, 1, 149-160 (2021) · Zbl 1487.62113 · doi:10.1016/j.ejor.2021.01.017 |
| [17] | Hyndman, R., Another look at forecast-accuracy metrics for intermittent demand, Foresight Int. J. Appl. Forecast., 4, 4, 43-46 (2006) |
| [18] | Hyndman, R., Athanasopoulos, G.: Forecasting: principles and practice, 3rd edition,. OTexts: Melbourne, Australia, (2021). OTexts.com/fpp3 |
| [19] | Hyndman, R.; Koehler, AB; Ord, JK; Snyder, RD, Forecasting with exponential smoothing: the state space approach (2008), Berlin: Springer, Berlin · Zbl 1211.62165 · doi:10.1007/978-3-540-71918-2 |
| [20] | Hyndman, R.J.: expsmooth: Data sets from “Exponential smoothing: a state space approach” by Hyndman, Koehler, Ord and Snyder (Springer, 2008), (2018). URL http://pkg.robjhyndman.com/expsmooth. R package version 2.4 |
| [21] | Hyndman, RJ; Ahmed, RA; Athanasopoulos, G.; Shang, HL, Optimal combination forecasts for hierarchical time series, Comput. Stat. Data Anal., 55, 9, 2579-2589 (2011) · Zbl 1464.62095 · doi:10.1016/j.csda.2011.03.006 |
| [22] | Jeon, J.; Panagiotelis, A.; Petropoulos, F., Probabilistic forecast reconciliation with applications to wind power and electric load, Eur. J. Oper. Res., 279, 2, 364-379 (2019) · doi:10.1016/j.ejor.2019.05.020 |
| [23] | Kahn, H.: Random sampling (Monte Carlo) techniques in neutron attenuation problems. I. Nucleonics (US) Ceased publication, 6, (1950) |
| [24] | Kolassa, S., Evaluating predictive count data distributions in retail sales forecasting, Int. J. Forecast., 32, 3, 788-803 (2016) · doi:10.1016/j.ijforecast.2015.12.004 |
| [25] | Kolassa, S., Do we want coherent hierarchical forecasts, or minimal MAPEs or MAEs? (We won’t get both!), Int. J. Forecast., 39, 4, 1512-1517 (2023) · doi:10.1016/j.ijforecast.2022.11.006 |
| [26] | Kourentzes, N.; Athanasopoulos, G., Elucidate structure in intermittent demand series, Eur. J. Oper. Res., 288, 1, 141-152 (2021) · Zbl 1487.90029 · doi:10.1016/j.ejor.2020.05.046 |
| [27] | Kullback, S.; Leibler, RA, On information and sufficiency, Annals Math. Stat., 22, 1, 79-86 (1951) · Zbl 0042.38403 · doi:10.1214/aoms/1177729694 |
| [28] | Liboschik, T., Fokianos, K., Fried, R.: tscount: an R package for analysis of count time series following generalized linear models. J. Stat. Softw. 82(5), 1-51 (2017) |
| [29] | Makridakis, S.; Spiliotis, E.; Assimakopoulos, V., The M5 competition: background, organization, and implementation, Int. J. Forecast., 38, 4, 1325-36 (2021) · doi:10.1016/j.ijforecast.2021.07.007 |
| [30] | Martino, L., Elvira, V., Louzada, F.: Effective sample size for importance sampling based on discrepancy measures. Signal Process. 131, 386-401 (2017). doi:10.1016/j.sigpro.2016.08.025 |
| [31] | Panagiotelis, A.; Gamakumara, P.; Athanasopoulos, G.; Hyndman, RJ, Probabilistic forecast reconciliation: properties, evaluation and score optimisation, Eur. J. Oper. Res., 306, 2, 693-706 (2023) · Zbl 1541.62245 · doi:10.1016/j.ejor.2022.07.040 |
| [32] | Panaretos, VM; Zemel, Y., Statistical aspects of Wasserstein distances, Annual Rev. Stat. Appl., 6, 405-431 (2019) · doi:10.1146/annurev-statistics-030718-104938 |
| [33] | Pearl, J.: Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan kaufmann, Massachusetts (1988) |
| [34] | Salinas, D.; Flunkert, V.; Gasthaus, J.; Januschowski, T., DeepAR: probabilistic forecasting with autoregressive recurrent networks, Int. J. Forecast., 36, 3, 1181-1191 (2020) · doi:10.1016/j.ijforecast.2019.07.001 |
| [35] | Salvatier, J., Wiecki, T.V., Fonnesbeck, C.: Probabilistic programming in Python using PyMC3. PeerJ Comput. Sci. 2, e55 (2016) |
| [36] | Smith, A.F., Gelfand, A.E.: Bayesian statistics without tears: a sampling-resampling perspective. Am. Stat. 46(2), 84-88 (1992) |
| [37] | Syntetos, A.A., Boylan, J.E.: The accuracy of intermittent demand estimates. Int. J. Forecast. 21(2), 303-314 (2005). (ISSN 0169-2070) |
| [38] | Székely, GJ; Rizzo, ML, Energy statistics: a class of statistics based on distances, J. Stat. Plan. Inference, 143, 8, 1249-1272 (2013) · Zbl 1278.62072 · doi:10.1016/j.jspi.2013.03.018 |
| [39] | Taieb, SB; Taylor, JW; Hyndman, RJ, Hierarchical probabilistic forecasting of electricity demand with smart meter data, J. Am. Stat. Assoc., 116, 533, 27-43 (2021) · Zbl 1457.62286 · doi:10.1080/01621459.2020.1736081 |
| [40] | Wickramasuriya, SL, Probabilistic forecast reconciliation under the Gaussian framework, J. Bus. Econ. Stat. (2023) · Zbl 1531.62201 · doi:10.1080/07350015.2023.2181176 |
| [41] | Wickramasuriya, SL; Athanasopoulos, G.; Hyndman, RJ, Optimal forecast reconciliation for hierarchical and grouped time series through trace minimization, J. Am. Stat. Assoc., 114, 526, 804-819 (2019) · Zbl 1420.62402 · doi:10.1080/01621459.2018.1448825 |
| [42] | Wickramasuriya, SL; Turlach, BA; Hyndman, RJ, Optimal non-negative forecast reconciliation, Stat. Comput., 30, 5, 1167-1182 (2020) · Zbl 1452.62706 · doi:10.1007/s11222-020-09930-0 |
| [43] | Yang, M., Zamba, G., Cavanaugh, J.: ZIM: Zero-Inflated Models (ZIM) for Count Time Series with Excess Zeros, (2018). URL https://CRAN.R-project.org/package=ZIM. R package version 1.1.0 |
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