Approaches for multi-step density forecasts with application to aggregated wind power. (English) Zbl 1202.62129

Ann. Appl. Stat. 4, No. 3, 1311-1341 (2010); correction ibid. 4, No. 4, 2205 (2010).
Summary: The generation of multi-step density forecasts for non-Gaussian data mostly relies on Monte Carlo simulations which are computationally intensive. Using aggregated wind power in Ireland, we study two approaches of multi-step density forecasts which can be obtained from simple iterations so that intensive computations are avoided. In the first approach, we apply a logistic transformation to normalize the data approximately and describe the transformed data using ARIMA-GARCH models so that multi-step forecasts can be iterated easily. In the second approach, we describe the forecast densities by truncated normal distributions which are governed by two parameters, namely, the conditional mean and conditional variance. We apply exponential smoothing methods to forecast the two parameters simultaneously. Since the underlying model of exponential smoothing is Gaussian, we are able to obtain multi-step forecasts of the parameters by simple iterations and thus generate forecast densities as truncated normal distributions. We generate forecasts for wind power from 15 minutes to 24 hours ahead. Results show that the first approach generates superior forecasts and slightly outperforms the second approach under various proper scores. Nevertheless, the second approach is computationally more efficient and gives more robust results under different lengths of training data. It also provides an attractive alternative approach since one is allowed to choose a particular parametric density for the forecasts, and is valuable when there are no obvious transformations to normalize the data.


62M20 Inference from stochastic processes and prediction
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P12 Applications of statistics to environmental and related topics
65C05 Monte Carlo methods


FinTS; expsmooth
Full Text: DOI arXiv


[1] Berrocal, V. J., Raftery, A. E. and Gneiting, T. (2008). Probabilistic quantitative precipitation field forecasting using a two-stage spatial model. Ann. Appl. Statist. 2 1170-1193. · Zbl 1168.62086
[2] Bjørnar Bremnes, J. (2006). A comparison of a few statistical models for making quantile wind power forecasts. Wind Energy 9 3-11.
[3] Brown, B. G., Katz, R. W. and Murphy, A. H. (1984). Time series models to simulate and forecast wind speed and wind power. Journal of Applied Meteorology 23 1184-1195.
[4] Brown, R. G. and Meyer, R. F. (1961). The fundamental theorem of exponential smoothing. Oper. Res. 9 673-685. · Zbl 0102.35702
[5] Christoffersen, P. F. and Diebold, F. X. (1997). Optimal prediction under asymmetric loss. Econometric Theory 13 808-817. · Zbl 04543048
[6] Costa, A., Crespo, A., Navarro, J., Lizcano, G., Madsen, H. and Feitona, E. (2008). A review on the young history of wind power short-term prediction. Renewable & Sustainable Energy Reviews 12 1725-1744.
[7] Cressie, N. A. C. (1993). Statistics for Spatial Data . Wiley, New York. · Zbl 0799.62002
[8] Davies, N., Pemberton, J. and Petruccelli, J. D. (1988). An automatic procedure for identification, estimation and forecasting univariate self exiting threshold autoregressive models. J. Roy. Statist. Soc. Ser. D 37 199-204.
[9] Diebold, F. X., Gunther, T. A. and Tay, A. S. (1998). Evaluating density forecasts: With applications to financial risk management. Internat. Econom. Rev. 39 863-883.
[10] Doherty, R. and O’Malley, M. (2005). A new approach to quantify reserve demand in systems with significant installed wind capacity. IEEE Transactions on Power Systems 20 587-595.
[11] Gardner, E. S. (2006). Exponential smoothing: The state of the art. J. Forecast. 4 1-28.
[12] Giebel, G., Kariniotakis, G. and Brownsword, R. (2003). The state-of-the-art in short-term prediction of wind power-A literature overview. EU project Anemos, Deliverable Report D1.1.
[13] Gneiting, T., Genton, M. G. and Guttorp, P. (2007). Geostatistical space-time models, stationarity, separability and full symmetry. In Statistical Methods for Spatio-Temporal Systems. Monographs on Statistics and Applied Probability 107 151-175. Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1282.86019
[14] Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc. 102 359-378. · Zbl 1284.62093
[15] Gneiting, T., Raftery, A. E., Westveld, A. H. and Goldman, T. (2005). Calibrated probabilistic forecasting using ensemble model output statistics and minimum CRPS estimation. Monthly Weather Review 133 1098-1118.
[16] Gneiting, T., Larson, K., Westrick, K., Genton, M. G. and Aldrich, E. (2006). Calibrated probabilistic forecasting at the stateline wind energy center: The regime-switching spacetime method. J. Amer. Statist. Assoc. 101 968-979. · Zbl 1120.62341
[17] Hering, A. S. and Genton, M. G. (2010). Powering up with space-time wind forecasting. J. Amer. Statist. Assoc. 105 92-104. · Zbl 1397.62484
[18] Higdon, D. M. (1998). A process-convolution approach to modeling temperatures in the North Atlantic Ocean. Journal of Ecological and Environmental Statistics 5 173-190.
[19] Hyndman, R. J., Koehler, A. B., Ord, J. K. and Snyder, R. D. (2008). Forecasting with Exponential Smoothing: The State Space Approach . Springer, Berlin. · Zbl 1211.62165
[20] Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika 36 149-176. · Zbl 0033.07204
[21] Jorgenson, D. W. (1967). Seasonal adjustment of data for econometric analysis. J. Amer. Statist. Assoc. 62 137-140.
[22] Landberg, L., Giebel, G., Nielsen, H. A., Nielsen, T. and Madsen, H. (2003). Short-term prediction-An overview. Wind Energy 6 273-280.
[23] Lau, A. (2010). Probabilistic wind power forecasts: From aggregated approach to spatiotemporal models. Ph.D. thesis, Mathematical Institute, Univ. Oxford.
[24] Ledolter, J. and Box, G. E. P. (1978). Conditions for the optimality of exponential smoothing forecast procedures. Metrika 25 77-93. · Zbl 0377.62057
[25] Manzan, S. and Zerom, D. (2008). A bootstrap-based non-parametric forecast density. International Journal of Forecasting 24 535-550.
[26] Milligan, M., Schwartz, M. and Wan, Y. (2004). Statistical wind power forecasting for U.S. wind farms. In The 17th Conference on Probability and Statistics in the Atmospheric Sciences/2004 American Meteorological Society Annual Meeting, Seattle, Washington, January 11-15, 2004 .
[27] Moeanaddin, R. and Tong, H. (1990). Numerical evaluation of distributions in nonlinear autoregression. J. Time Ser. Anal. 11 33-48. · Zbl 0691.62084
[28] Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59 347-370. · Zbl 0722.62069
[29] Patton, A. J. and Timmermann, A. (2007). Properties of optimal forecasts under asymmetric loss and nonlinearity. J. Econometrics 140 884-918. · Zbl 1247.91144
[30] Pinson, P., Chevallier, C. and Kariniotakis, G. (2007). Trading wind generation with short-term probabilistic forecasts of wind power. IEEE Transactions on Power Systems 22 1148-1156.
[31] Pinson, P. and Madsen, H. (2009). Ensemble-based probabilistic forecasting at Horns Rev. Wind Energy 12 137-155.
[32] Sanchez, I. (2006). Short-term prediction of wind energy production. International Journal of Forecasting 22 43-56.
[33] Sanso, B. and Guenni, L. (1999). Venezuelan rainfall data analysed by using a Bayesian space-time model. Appl. Statist. 48 345-362. · Zbl 0939.62124
[34] Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging . Springer, New York. · Zbl 0924.62100
[35] Stein, M. L. (2009). Spatial interpolation of high-frequency monitoring data. Ann. Appl. Statist. 3 272-291. · Zbl 1160.62094
[36] Taylor, J. W. (2003). Short-term electricity demand forecasting using double seasonal exponential smoothing. Journal of the Operational Research Society 54 799-805. · Zbl 1097.91541
[37] Taylor, J. W. (2004). Volatility forecasting with smooth transition exponential smoothing. International Journal of Forecasting 20 273-286.
[38] Taylor, J. W., McSharry, P. E. and Buizza, R. (2009). Wind power density forecasting using wind ensemble predictions and time series models. IEEE Transactions on Energy Conversion 24 775-782.
[39] Tsay, R. S. (2005). Analysis of Financial Time Series , 2nd ed. Wiley, Hoboken. · Zbl 1086.91054
[40] Weigend, A. S. and Shi, S. (2000). Predicting daily probability distributions of S&P500 returns. J. Forecast. 19 375-392.
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.