×

Temporal hierarchies with autocorrelation for load forecasting. (English) Zbl 1430.62210

Summary: We propose four different estimators that take into account the autocorrelation structure when reconciling forecasts in a temporal hierarchy. Combining forecasts from multiple temporal aggregation levels exploits information differences and mitigates model uncertainty, while reconciliation ensures a unified prediction that supports aligned decisions at different horizons. In previous studies, weights assigned to the forecasts were given by the structure of the hierarchy or the forecast error variances without considering potential autocorrelation in the forecast errors. Our first estimator considers the autocovariance matrix within each aggregation level. Since this can be difficult to estimate, we propose a second estimator that blends autocorrelation and variance information, but only requires estimation of the first-order autocorrelation coefficient at each aggregation level. Our third and fourth estimators facilitate information sharing between aggregation levels using robust estimates of the cross-correlation matrix and its inverse. We compare the proposed estimators in a simulation study and demonstrate their usefulness through an application to short-term electricity load forecasting in four price areas in Sweden. We find that by taking account of auto- and cross-covariances when reconciling forecasts, accuracy can be significantly improved uniformly across all frequencies and areas.

MSC:

62M20 Inference from stochastic processes and prediction
62M09 Non-Markovian processes: estimation

References:

[1] Agrawal, A.; Verschueren, R.; Diamond, S.; Boyd, S., A rewriting system for convex optimization problems, Journal of Control and Decision, 5, 1, 42-60 (2018)
[2] Amemiya, T.; Wu, R. Y., The effect of aggregation on prediction in the autoregressive model, Journal of the American Statistical Association, 67, 339, 628-632 (1972) · Zbl 0258.62054
[3] Athanasopoulos, G.; Ahmed, R. A.; Hyndman, R. J., Hierarchical forecasts for Australian domestic tourism, International Journal of Forecasting, 25, 1, 146-166 (2009)
[4] Athanasopoulos, G.; Hyndman, R. J.; Kourentzes, N.; Petropoulos, F., Forecasting with temporal hierarchies, European Journal of Operational Research, 262, 1, 60-74 (2017) · Zbl 1403.62154
[5] Athanasopoulos, G.; Hyndman, R. J.; Song, H.; Wu, D. C., The tourism forecasting competition, International Journal of Forecasting, 27, 3, 822-844 (2011)
[6] Banerjee, O.; Ghaoui, L. E.; d’Aspremont, A., Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data, Journal of Machine Learning Research, 9, 485-516 (2008) · Zbl 1225.68149
[7] Bien, J.; Bunea, F.; Xiao, L., Convex banding of the covariance matrix, Journal of the American Statistical Association, 111, 514, 834-845 (2016)
[8] Boyd, S.; Vandenberghe, L., Convex optimization (2004), Cambridge University Press: Cambridge University Press New York · Zbl 1058.90049
[9] Clemen, R. T., Combining forecasts: A review and annotated bibliography, International Journal of Forecasting, 5, 4, 559-583 (1989)
[10] Clements, A.; Hurn, A.; Li, Z., Forecasting day-ahead electricity load using a multiple equation time series approach, European Journal of Operational Research, 251, 2, 522-530 (2016) · Zbl 1346.62159
[11] Diamond, S.; Boyd, S., CVXPY: A python-embedded modeling language for convex optimization, Journal of Machine Learning Research, 17, 83, 1-5 (2016) · Zbl 1360.90008
[12] Diebold, F. X.; Mariano, R. S., Comparing predictive accuracy, Journal of Business & Economic Statistics, 13, 3, 253-263 (1995)
[13] van Erven, T.; Cugliari, J., Game-theoretically optimal reconciliation of contemporaneous hierarchical time series forecasts, (Antoniadis, A.; Poggi, J.-M.; Brossat, X., Modeling and stochastic learning for forecasting in high dimensions. Modeling and stochastic learning for forecasting in high dimensions, Lecture notes in statistics, 217 (2015), Springer: Springer Cham), 297-317 · Zbl 1323.62005
[14] Fan, S.; Hyndman, R. J., Short-term load forecasting based on a semi-parametric additive model, IEEE Transactions on Power Systems, 27, 1, 134-141 (2012)
[15] Friedman, J.; Hastie, T.; Tibshirani, R., Sparse inverse covariance estimation with the graphical lasso, Biostatistics, 9, 3, 432-441 (2008) · Zbl 1143.62076
[16] Gamakumara, P.; Panagiotelis, A.; Athanasopoulos, G.; Hyndman, R. J., Probabilisitic forecasts in hierarchical time series, Working Paper 11/18 (2018), Monash University
[17] Ghysels, E.; Santa-Clara, P.; Valkanov, R., The MIDAS touch: Mixed data sampling regression models, Working paper (2004), UNC and UCLA
[18] Gould, P. G.; Koehler, A. B.; Ord, J. K.; Snyder, R. D.; Hyndman, R. J.; Vahid-Araghi, F., Forecasting time series with multiple seasonal patterns, European Journal of Operational Research, 191, 1, 207-222 (2008) · Zbl 1142.62070
[19] Gross, C. W.; Sohl, J. E., Disaggregation methods to expedite product line forecasting, Journal of Forecasting, 9, 3, 233-254 (1990)
[20] Hahn, H.; Meyer-Nieberg, S.; Pickl, S., Electric load forecasting methods: Tools for decision making, European Journal of Operational Research, 199, 3, 902-907 (2009) · Zbl 1176.90291
[21] Hall, S. G.; Mitchell, J., Combining density forecasts, International Journal of Forecasting, 23, 1, 1-13 (2007)
[22] Harvey, D.; Leybourne, S.; Newbold, P., Testing the equality of prediction mean squared errors, International Journal of Forecasting, 13, 2, 281-291 (1997)
[23] Hyndman, R. J.; Ahmed, R. A.; Athanasopoulos, G.; Shang, H. L., Optimal combination forecasts for hierarchical time series, Computational Statistics & Data Analysis, 55, 9, 2579-2589 (2011) · Zbl 1464.62095
[24] Hyndman, R. J.; Khandakar, Y., Automatic time series forecasting: The forecast package for R, Journal of Statistical Software, 27, 3, 1-22 (2008)
[25] Hyndman, R. J.; Koehler, A. B., Another look at measures of forecast accuracy, International Journal of Forecasting, 22, 4, 679-688 (2006)
[26] Hyndman, R. J.; Koehler, A. B.; Ord, J. K.; Snyder, R. D., Forecasting with exponential smoothing: The state space approach (2008), Springer: Springer Berlin · Zbl 1211.62165
[27] Hyndman, R. J.; Lee, A. J.; Wang, E., Fast computation of reconciled forecasts for hierarchical and grouped time series, Computational Statistics & Data Analysis, 97, 16-32 (2016) · Zbl 1468.62086
[28] Jeon, J., Panagiotelis, A., & Petropoulos, F. (2018). Reconciliation of probabilistic forecasts with an application to wind power. arXiv:1808.02635; Jeon, J., Panagiotelis, A., & Petropoulos, F. (2018). Reconciliation of probabilistic forecasts with an application to wind power. arXiv:1808.02635
[29] Kourentzes, N.; Athanasopoulos, G., Cross-temporal coherent forecasts for Australian tourism, Annals of Tourism Research, 75, 393-409 (2019)
[30] Kourentzes, N.; Petropoulos, F.; Trapero, J. R., Improving forecasting by estimating time series structural components across multiple frequencies, International Journal of Forecasting, 30, 2, 291-302 (2014)
[31] Ledoit, O.; Wolf, M., A well-conditioned estimator for large-dimensional covariance matrices, Journal of Multivariate Analysis, 88, 2, 365-411 (2004) · Zbl 1032.62050
[32] Livera, A. M.D.; Hyndman, R. J.; Snyder, R. D., Forecasting time series with complex seasonal patterns using exponential smoothing, Journal of the American Statistical Association, 106, 496, 1513-1527 (2011) · Zbl 1234.62123
[33] Madsen, H., Time series analysis (2008), Chapman & Hall: Chapman & Hall London · Zbl 1143.62053
[34] Nystrup, P.; Madsen, H.; Lindström, E., Long memory of financial time series and hidden Markov models with time-varying parameters, Journal of Forecasting, 36, 8, 989-1002 (2017) · Zbl 1397.60104
[35] Petropoulos, F.; Kourentzes, N., Forecast combinations for intermittent demand, Journal of the Operational Research Society, 66, 6, 914-924 (2015)
[36] Rostami-Tabar, B.; Babai, M. Z.; Syntetos, A.; Ducq, Y., Demand forecasting by temporal aggregation, Naval Research Logistics, 60, 6, 479-498 (2013) · Zbl 1410.91373
[37] Schäfer, J.; Strimmer, K., A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics, Statistical Applications in Genetics and Molecular Biology, 4, 1, Article 32 pp. (2005)
[38] Sedoc, J., Rodu, J., Foster, D., & Ungar, L. (2018). Multiscale hidden Markov models for covariance prediction.; Sedoc, J., Rodu, J., Foster, D., & Ungar, L. (2018). Multiscale hidden Markov models for covariance prediction.
[39] Silvestrini, A.; Veredas, D., Temporal aggregation of univariate and multivariate time series models: A survey, Journal of Economic Surveys, 22, 3, 458-497 (2008)
[40] Taieb, S. B., Sparse and smooth adjustments for coherent forecasts in temporal aggregation of time series, (Anava, O.; Khaleghi, A.; Cuturi, M.; Kuznetsov, V.; Rakhlin, A., Proceedings of the time series workshop at NIPs 2016. Proceedings of the time series workshop at NIPs 2016, Proceedings of machine learning research, 55 (2017)), 16-26
[41] Taieb, S. B.; Taylor, J. W.; Hyndman, R. J., Coherent probabilistic forecasts for hierarchical time series, (Precup, D.; Teh, Y. W., Proceedings of the 34th international conference on machine learning. Proceedings of the 34th international conference on machine learning, Proceedings of machine learning research, 70 (2017)), 3348-3357
[42] Taieb, S. B., Taylor, J. W., & Hyndman, R. J. (2017b). Hierarchical probabilistic forecasting of electricity demand with smart meter data.; Taieb, S. B., Taylor, J. W., & Hyndman, R. J. (2017b). Hierarchical probabilistic forecasting of electricity demand with smart meter data.
[43] Taylor, J. W., Short-term electricity demand forecasting using double seasonal exponential smoothing, Journal of the Operational Research Society, 54, 8, 799-805 (2003) · Zbl 1097.91541
[44] Taylor, J. W., Exponentially weighted methods for forecasting intraday time series with multiple seasonal cycles, International Journal of Forecasting, 26, 4, 627-646 (2010)
[45] Taylor, J. W., Short-term load forecasting with exponentially weighted methods, IEEE Transactions on Power Systems, 27, 1, 458-464 (2012)
[46] Tiao, G. C., Asymptotic behaviour of temporal aggregates of time series, Biometrika, 59, 3, 525-531 (1972) · Zbl 0263.62051
[47] Tibshirani, R.; Saunders, M.; Rosset, S.; Zhu, J.; Knight, K., Sparsity and smoothness via the fused lasso, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67, 1, 91-108 (2005) · Zbl 1060.62049
[48] Timmermann, A., Forecast combinations, (Elliott, G.; Granger, C. W.J.; Timmermann, A., Chapter 4: Handbook of economic forecasting, 1 (2006), Elsevier: Elsevier Amsterdam), 135-196
[49] Wickramasuriya, S. L.; Athanasopoulos, G.; Hyndman, R. J., Optimal forecast reconciliation for hierarchical and grouped time series through trace minimization, Journal of the American Statistical Association, 114, 526, 804-819 (2019) · Zbl 1420.62402
[50] Yang, D.; Quan, H.; Disfani, V. R.; Liu, L., Reconciling solar forecasts: Geographical hierarchy, Solar Energy, 146, 276-286 (2017)
[51] Yang, D.; Quan, H.; Disfani, V. R.; Rodríguez-Gallegos, C. D., Reconciling solar forecasts: Temporal hierarchy, Solar Energy, 158, 332-346 (2017)
[52] Yuan, M.; Lin, Y., Model selection and estimation in the Gaussian graphical model, Biometrika, 94, 1, 19-35 (2007) · Zbl 1142.62408
[53] Zhang, Y.; Dong, J., Least squares-based optimal reconciliation method for hierarchical forecasts of wind power generation, IEEE Transactions on Power Systems, 1 (2018)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.