## Abstract

This article presents short-term predictions using neural networks tuned by energy associated to series based-predictor filter for complete and incomplete datasets. A benchmark of high roughness time series from Mackay Glass (MG), Logistic (LOG), Henon (HEN) and some univariate series chosen from NN3 Forecasting Competition are used. An average smoothing technique is assumed to complete the data missing in the dataset. The Hurst parameter estimated through wavelets is used to estimate the roughness of the real and forecasted series. The validation and horizon of the time series is presented by the 15 values ahead. The performance of the proposed filter shows that even a short dataset is incomplete, besides a linear smoothing technique employed; the prediction is almost fair by means of SMAPE index. Although the major result shows that the predictor system based on energy associated to series has an optimal performance from several chaotic time series, in particular, this method among other provides a good estimation when the short-term series are taken from one point observations.

## References

[1] Little, R.J.A. and D.B. Rubin, Statistical Analysis with Missing Data. John Wiley Publishers Company, 2002.

[2] Tsikriktsis, N., A review of techniques for treating missing data in OM survey research, Journal of Operations Management 24, 2005, pp. 53-62.

[3] Maravall A, Pea D., Missing observations and additive outliers in time series models. In: Mariano RS (ed) Advances in statistical analysis and statistical computing. JAI Press, Stanford, 1986.

[4] Mitat Uysal, Reconstruction of Time Series Data with Missing Values, Journal of Applied Sciences, 7 (6): ISSN 1812-5654, 2007, pp. 922-925 .

[5] Kornelsen, K., & Coulibaly, P., Comparison of Interpolation, Statistical, and Data-Driven Methods for Imputation of Missing Values in a Distributed Soil Moisture Dataset. Journal of Hydrologic Engineering, 19(1), 2012, pp. 26-43.

[6] Kharin, Yu.S. and Huryn, A.S., Statistical analysis and forecasting of autoregressive time series under missing values. Bulletin of the International Statistic Institute, 1, 2003, pp. 612-613.

[7] Hong, B. and CH. Chen, Radial basis function neural network-based nonparametric estimation approach for missing data reconstruction of nonstationary series. IEEE Int. Conf. Neural Networks and Signal Processing Nanjing, China, December, 14-17, 2003, pp. 75-78.

[8] Coulibaly P. Comparison of neural network methods for infilling missing daily weather records. Journal of Hydrology. v. 341, 2007, pp. 27-41.

[9] Kidson, J.W., and K.E. Trenberth, Effects of missing data on estimates of monthly mean general circulation statistics, J. Climate, 1, 1988, pp. 1261-1275.

[10] Vincent, L. A., and D. W. Gullet., Canadian historical and homogeneous temperature datasets for climate change analyses. International Journal of Climatology. 19, 1999, 1375-1388.

[11] Rodrguez Rivero, Cristian; Patio, Hector Daniel; Pucheta, Julian Antonio, Short-term rainfall time series prediction with incomplete data, in Neural Networks (IJCNN), 2015 International Joint Conference on, 2015, pp. 1-6, 10.1109/IJCNN.2015.7280315.

[12] Wasito, I.: 2003, Least Squares Algorithms with Nearest Neighbour Techniques for Imputing Missing Data Values, PhD thesis, University of London.

[13] Nelwamondo, F. V., Mohamed, S. and Marwala, T.: n.d., Missing data: A comparison of neural networks and expectation maximization techniques, Current Science 93(11), 2007.

[14] Cristian Rodrguez Rivero, Julin Pucheta, Sergio Laboret, Daniel Patio, Vctor Sauchelli. Forecasting short time series with missing data by means of energy associated of series. Applied Mathematics, 6, 2015, pp. 1611-1619, http://dx.doi.org/10.4236/am.2015.69143.

[15] Richman, M. B.; Trafalis, T. B.; Adrianto I.; Missing data imputation through machine learning algorithms, in Artificial Intelligence Methods in the Environmental Sciences. Ed. by H. Sue Ellen, P. Antonello, M. Caren. Springer Netherlands Press, 2009, pp.153-169, doi:10.1007/978-1-4020-9119-3 7.

[16] Haviluddina, Ahmad Jawahir, Comparing of ARIMA and RBFNN for short-term forecasting, International Journal of Advances in Intelligent Informatics, Vol. 1, No 1, 2015, pp. 15-22.

[17] Zhang G. P., Time series forecasting using a hybrid ARIMA and neural network model. Neurocomputing, 50, 2003, pp. 159-175.

[18] Schneider, T., Analysis of incomplete climate data: estimation of mean values and covariance matrices and imputation of missing values. J. Climate, 14, 2001, pp. 853-871.

[19] C. Rodrguez Rivero, M. Herrera, J. Pucheta, J. Baumgartner, D. Patio and V. Sauchelli. High roughness time series forecasting based on energy associated of series, Journal of Communication and Computer, USA, David Publishing Company, Vol. 9 No. 5, 2012, pp. 576-586, ISSN 1548-7709.

[20] C. Rodriguez Rivero, J. Pucheta, H. Patio, J. Baumgartner, S. Laboret and V. Sauchelli. Analysis of a Gaussian Process and Feed-Forward Neural Networks based Filter for Forecasting Short Rainfall Time Series, 2013, International Joint Conference on Neural Networks. doi: 10.1109/IJCNN.2013.6706741.

[21] Kohn, R., & Ansley, C. F., Estimation, Prediction, and Interpolation for ARIMA Models with Missing Data, Journal of the American Statistical Association, 81, 1986, pp. 751-761.

[22] Jones R. H., Maximum Likelihood Fitting of ARMA Models to Time Series with Missing Observations, Technometrics, 22(3), 1980, pp. 389-395.

[23] Shumway R., and D. Stoffer, An approach to time series smoothing and forecasting using the EM algorithm, Journal of Time Series Analysis, 3, 1982, pp. 253-264.

[24] Tresp V., & Hofmann R., Nonlinear Time-Series Prediction with Missing and Noisy Data. Neural Computation, 10, 1998, pp. 731-747.

[25] Mandelbrot B. B., The Fractal Geometry of Nature, Freeman, San Francisco, CA, 1983.

[26] Dieker, T. Simulation of fractional Brownian motion. MSc theses, University of Twente, Amsterdam, The Netherlands, 2004.

[27] C. Rodrguez Rivero, J. Pucheta, J. Baumgartner, M. Herrera, D. Patio y B. Kuchen. A NN-based model for time series forecasting in function of energy associated of series, Proc. of the International Conference on Applied, Numerical and Computational Mathematics (ICANCM11), Barcelona, Spain, 2011, pp. 80-86, ISBN 978-1-61804-030-5.

[28] Pucheta J., Patio H. D., Kuchen B., A Statistically Dependent Approach for the Monthly Rainfall Forecast from One Point Observations, In Proc. of the Second IFIP Conference on Computer and Computing Technologies in Agriculture (CCTA2008) 2008, Beijing, China.

[29] Glass L. and M. C. Mackey. From Clocks to Chaos, The Rhythms of Life. Princeton University Press, Princeton, NJ, 1988.

[30] Robert M. May, Simple mathematical models with very complicated dynamics, Nature, vol.261, 1976, pp. 459-467.

[31] Henon M., A two-dimensional mapping with a strange attractor. Communications in Mathematical Physics. Vol. 50, 1976, pp. 69-77.

[32] Abry, P.; P. Flandrin, M.S. Taqqu, D. Veitch., Self-similarity and long-range dependence through the wavelet lens. Theory and applications of longrange dependence, Birkhuser, 2003, pp. 527-556.

[33] Bardet, J.-M.; G. Lang, G. Oppenheim, A. Philippe, S. Stoev, M.S. Taqqu. Semi-parametric estimation of the long-range dependence parameter: a survey. Theory and applications of longrange dependence, Birkhuser, 2003, pp. 557-577.

[34] S. F. Crone, M. Hibon, and K. Nikolopoulos, Advances in forecasting with neural networks? Empirical evidence from the NN3 competition on time series prediction, International Journal of Forecasting, vol. 27, 2011, pp. 635-660.

[35] Rubin D. B., Multiple Imputation for Nonresponse in Surveys, New York:Wiley, 1987.

[36] Schafer J., Analysis of Incomplete Multivariate Data, Chapman & Hall, 1997.

[37] Armstrong J.S. (Ed.) Principles offorecasting: Handbook for researchers and practitioners. Kluwer, 2001.

[38] Makridakis, S. & M. Hibon, The M3-Competition: Results, conclusions and implications, International Journal of Forecasting, 16, 2000, pp. 451-476.

[39] Shang Zhaowei, Zhang Lingfeng, Ma Shangjun, Fang Bin, Zhang Taiping. Incomplete Time Series Prediction Using Max-Margin Classification of Data with Absent Features Hindawi Publishing Corporation. Mathematical Problems in Engineering, Volume 2010, Article ID 513810, doi:10.1155/2010/513810.

[40] C. Zecchin, A. Facchinetti, G. Sparacino, G. De Nicolao, C. Cobelli, A new neural network approach for short-term glucose prediction using continuous glucose monitoring time-series and meal information, 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2011, pp. 5653-5656.

[41] L.P. Wang and X.J. Fu, Data Mining with Computational Intelligence, Springer, Berlin, 2005.

[42] Y. Ren, P. N. Suganthan, N. Srikanth, G. Amaratunga, Random Vector Functional Link Network for Short-term Electricity Load Demand Forecasting, Information Sciences, 2016.

[43] L. P. Wang and Shekhar Gupta, Neural networks and wavelet de-noising for stock trading and prediction, Time Series Analysis, Modeling and Applications, Witold Pedrycz and Shyi Ming Chen (eds.), Springer, 2013, pp. 229-247.

[44] L. P. Wang, K.K. Teo, and Z.P. Lin, Predicting time series with wavelet packet neural networks, 2001 IEEE International Joint Conference on Neural Networks (IJCNN 2001), 2001, pp. 1593-1597.

[45] Dong-Chul Park, ”A Time Series Data Prediction Scheme Using Bilinear Recurrent Neural Network,” 2010 International Conference on Information Science and Applications (ICISA), 2010, pp. 1-7.

[46] M. Zhu and L. P. Wang, Intelligent trading using support vector regression and multilayer perceptrons optimized with genetic algorithms, in The 2010 International Joint Conference on Neural Networks (IJCNN), 2010, pp. 1-5.

[47] K.K. Teo, L.P. Wang, Z.P. Lin, Wavelet packet multi-layer perceptron for chaotic time series prediction: effects of weight initialization, Computational Science - ICCS 2001, Proceedings Pt 2, Volume: 2074, 2001, pp. 310-317.

[48] J. Pucheta, D. Patio and B. Kuchen. Neural Networks-Based Time Series Prediction Using Long and Short Term Dependence in the Learning Process”. In proc. of the 2007 International Symposium on Forecasting, New York 2007.

[49] C.-N. Ko, C.-M. Lee, Short-term load forecasting using SVR (support vector regression)-based radial basis function neural network with dual extended kalman filter, Energy 49, 2013, pp. 413-422.

[50] R.-A. Hooshmand, H. Amooshahi, M. Parastegari, A hybrid intelligent algorithm based short-term load forecasting approach, International Journal of Electrical Power & Energy Systems 45, 2013, pp. 313-324.

[51] Kamal S. Selim and Gihan A. Elanany, A New Method for Short Multivariate Fuzzy Time Series Based on Genetic Algorithm and Fuzzy Clustering, Advances in Fuzzy Systems, vol. 2013, Article ID 494239, 2013, doi:10.1155/2013/494239.