Generalized  exponential smoothing in prediction of hierarchical time series

Daniel  Kosiorowski; Dominik  Mielczarek; Jerzy P.  Rydlewski; Małgorzata  Snarska

doi:https://doi.org/10.21307/stattrans-2018-019

Generalized exponential smoothing in prediction of hierarchical time series

Daniel Kosiorowski Department of Statistics, Cracow University of Economics, Krakow, Poland. ´ , Dominik Mielczarek AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland. , Jerzy P. Rydlewski AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland. , Małgorzata Snarska Department of Financial Markets, Cracow University of Economics, Krakow, Poland. Statistics in Transition new series, vol. 19, 2018, 2, pages: 331-350 Published online: 1 June 2018 https://doi.org/10.21307/stattrans-2018-019

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ABSTRACT

Shang and Hyndman (2017) proposed a grouped functional time series forecasting approach as a combination of individual forecasts obtained using the generalized least squares method. We modify their methodology using a generalized exponential smoothing technique for the most disaggregated functional time series in order to obtain a more robust predictor. We discuss some properties of our proposals based on the results obtained via simulation studies and analysis of real data related to the prediction of demand for electricity in Australia in 2016

KEYWORDS

functional time series, hierarchical time series, forecast reconciliation, depth for functional data.

REFERENCES

AUE, A., DUBABART NORINHO, D., HORMANN, S., (2015). On the prediction of ¨stationary functional time series, Journal of the American Statistical Associa tion, 110 (509), pp. 378–392.

BESSE, P. C., CARDOT, H., STEPHENSON, D. B., (2000). Autoregressive fore casting of some functional climatic variations, Scandinavian Journal of Statis tics, 27 (4), pp. 673–687.

BOSQ, D. (2000). Linear processes in function spaces. Springer.

DIDERICKSEN, D., KOKOSZKA, P., ZHANG, X. (2012). Empirical properties of forecasts with the functional autoregressive model, Computational Statistics,27 (2), pp. 285–298.

FEBRERO-BANDE, M. O., DE LA FUENTE, M., (2012). Statistical computing in functional data analysis: the R package fda.usc, Journal of Statistical Soft ware, 51 (4), pp. 1–28.

HORVATH, L., KOKOSZKA, P., (2012). Inference for functional data with applica tions, Springer-Verlag.

HORMANN S., KOKOSZKA, P., (2012). Functional Time Series, in Handbook of ¨Statistics: Time Series Analysis – Methods and Applications, 30, pp. 157–186.

HYNDMAN, R. J., AHMED R. A., ATHANASOPOULOS, G., SHANG, H. L., (2011).Optimal combination forecasts for hierarchical time series, Computational Statis tics & Data Analysis, 55 (9), pp. 2579–2589.

HYNDMAN, R.J., KOEHLER, A.B., ORD, J. K., SNYDER, R. D., (2008). Forecast ing with exponential smoothing – the state space approach, Springer-Verlag.

HYNDMAN, R. J., SHANG, H., L., (2009). Forecasting functional time series, Jour nal of the Korean Statistical Society, 38 (3), pp. 199–221.

HYNDMAN, R. J., ULLAH, M., (2007). Robust forecasting of mortality and fertility rates: A functional data approach, Computational Statistics & Data Analysis,51 (10), pp. 4942–4956.

HYNDMAN, R. J., KOEHLER, A. B., SNYDER, R.D., GROSE, S., (2002). A statespace framework for automatic forecasting using exponential smoothing meth ods, International Journal of Forecasting, 18 (3), pp. 439–454.

KAHN, K. B., (1998). Revisiting top-down versus bottom-up forecasting, The Jour nal of Business Forecasting Methods & Systems, 17 (2), pp. 14–19.

KOHN, R., (1982). When is an aggregate of a time series efficiently forecast by its past, Journal of Econometrics, 18 (3), pp. 337–349.

KOSIOROWSKI, D., ZAWADZKI, Z., (2018). DepthProc: An R package for robust exploration of multidimensional economic phenomena, arXiv: 1408.4542.

KOSIOROWSKI, D., (2014). Functional regression in short term prediction of eco nomic time series, Statistics in Transition, 15 (4), pp. 611–626.

KOSIOROWSKI, D. (2016). Dilemmas of robust analysis of economic data streams,Journal of Mathematical Sciences (Springer), 218 (2), pp. 167–181.

KOSIOROWSKI, D., RYDLEWSKI, J. P., SNARSKA, M., (2017a). Detecting a structural change in functional time series using local Wilcoxon statistic, Sta tistical Papers, pp. 1–22, URL http://dx.doi.org/10.1007/ s00362-017-0891-y.

KOSIOROWSKI, D., MIELCZAREK, D., RYDLEWSKI, J. P., (2017b). Double func tional median in robust prediction of hierarchical functional time series – an application to forecasting of the Internet service users behaviour, available at:arXiv:1710.02669v1.

KOSIOROWSKI, D., RYDLEWSKI, J.P., ZAWADZKI Z., (2018a). Functional out liers detection by the example of air quality monitoring, Statistical Review (in Polish, forthcoming).

KOSIOROWSKI, D., MIELCZAREK, D., RYDLEWSKI, J. P., (2018b). Forecasting of a Hierarchical Functional Time Series on Example of Macromodel for the Day and Night Air Pollution in Silesia Region - A Critical Overview, Central Eu ropean Journal of Economic Modelling and Econometrics, 10 (1), pp. 53–73.

KOSIOROWSKI, D., MIELCZAREK, D., RYDLEWSKI, J. P., (2018c). Outliers in Functional Time Series – Challenges for Theory and Applications of Robust Statistics, In M. Papiez & S. ˙ Smiech (eds.), The ´ 12th Professor Aleksander Zelias International Conference on Modelling and Forecasting of Socio-Economic ´ Phenomena, Conference Proceedings, Cracow: Foundation of the CracowUniversity of Economics, pp. 209–218.

KRZYSKO, M., DEREGOWSKI, K., G ´ ORECKI, T., WOŁY ´ NSKI, W., (2013). Kernel ´ and functional principal component analysis, Multivariate Statistical Analysis 2013 Conference, plenary lecture.

LOPEZ-PINTADO, S., ROMO, J., (2009). On the concept of depth for functional ´ data, Journal of the American Statistical Association, 104, pp. 718–734.

LOPEZ-PINTADO, S., J ´ ORNSTEN, R., (2007). Functional analysis via extensions ¨ of the band depth, IMS Lecture Notes–Monograph Series Complex Datasets and Inverse Problems: Tomography, Networks and Beyond, Vol. 54, pp. 103–120, Institute of Mathematical Statistics.

NAGY, S., GIJBELS, I., OMELKA, M., HLUBINKA, D., (2016). Integrated depth for functional data: Statistical properties and consistency, ESIAM Probability and Statistics, 20, pp. 95–130.

NAGY, S. GIJBELS, I., HLUBINKA, D., (2017). Depth-Based Recognition of Shape Outlying Functions, Journal of Computational and Graphical Statistics, DOI:10.1080/10618600.2017.1336445.

NIETO-REYES, A., BATTEY, H., (2016). A topologically valid definition of depth for functional data, Statistical Science 31 (1), pp. 61–79.

PAINDAVEINE, D., G. VAN BEVER, G., (2013). From depth to local depth: a focus on centrality, Journal of the American Statistical Asssociation, Vol. 108, No.503, Theory and Methods, pp. 1105–1119.

RAMSAY, J.O., G. HOOKER, G., GRAVES, S., (2009). Functional data analysis with R and Matlab, Springer-Verlag.

SGUERA, C., GALEANO, P., LILLO, R. E., (2016). Global and local functional depths, arXiv 1607.05042v1.

SHANG, H., L., HYNDMAN, R. J., (2017). Grouped functional time series forecast ing: an application to age-specific mortality rates, Journal of Computational and Graphical Statistics, 26(2), pp. 330–343.

SHANG, H., L., (2018). Bootstrap methods for stationary functional time series,Statistics and Computing, 28(1), pp. 1–10.

WEALE, M., (1988). The reconciliation of values, volumes and prices in the na tional accounts, Journal of the Royal Statistical Society A, 151(1),pp. 211–221.

VAKILI, K., SCHMITT, E., (2014). Finding multivariate outliers with FastPCS, Com putational Statistics & Data Analysis, 69, pp. 54–66.

VINOD, H.D., LOPEZ-DE-LACALLE, J. L., (2009). Maximum entropy bootstrap for ´time series: the meboot R package, Journal of Statistical Software, 29 (5).

ZUO, Y., SERFLING, R., (2000). Structural properties and convergence results for contours of sample statistical depth functions, Annals of Statistics, 28 (2), pp.483–499.Australian Energy Market Operator, https://www.aemo.com.au