Time Series Forecasting Using Wavelet-Least Squares Support Vector Machines and Wavelet Regression Models for Monthly Stream Flow Data

Affiliation(s)

Faculty of Science, Department of Mathematics, Universiti Teknologi Malaysia, Skudai, Malaysia.

Faculty of Science, Department of Mathematics, Universiti Teknologi Malaysia, Skudai, Malaysia.

ABSTRACT

This study explores the least square support vector and wavelet
technique (WLSSVM) in the monthly stream flow forecasting. This is a
new hybrid technique. The 30 days periodic predicting statistics used in this
study are derived from the subjection of this model to the river flow data of
the Jhelum and Chenab rivers. The root mean square error (RMSE), mean absolute
error (RME) and correlation (R) statistics are used for evaluating the accuracy
of the WLSSVM and WR models. The accuracy of the WLSSVM model is compared with
LSSVM, WR and LR models. The two rivers surveyed are in the Republic of
Pakistan and cover an area encompassing 39,200 km^{2} for the Jhelum River and 67,515 km^{2} for the Chenab River. Using
discrete wavelets, the observed data has been decomposed into sub-series. These
have then appropriately been used as inputs in the least square support vector
machines for forecasting the hydrological variables. The resultant observation
from this comparison indicates the WLSSVM is more accurate than the LSSVM, WR
and LR models in river flow forecasting.

Cite this paper

S. Pandhiani and A. Shabri, "Time Series Forecasting Using Wavelet-Least Squares Support Vector Machines and Wavelet Regression Models for Monthly Stream Flow Data,"*Open Journal of Statistics*, Vol. 3 No. 3, 2013, pp. 183-194. doi: 10.4236/ojs.2013.33021.

S. Pandhiani and A. Shabri, "Time Series Forecasting Using Wavelet-Least Squares Support Vector Machines and Wavelet Regression Models for Monthly Stream Flow Data,"

References

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[14] D. Hanbay, “An Expert System Based on Least Square Support Vector Machines for Diagnosis of Valvular Heart Disease,” Expert Systems with Applications, Vol. 36, No. 4, 2009, pp. 8368-8374.

[15] Y. W. Kang, J. Li, C. Y. Guang, H.-Y. Tu, J. Li and J. Yang, “Dynamic Temperature Modeling of an SOFC Using Least Square Support Vector Machines,” Journal of Power Sources, Vol. 179, No. 2, 2008, pp. 683-692. doi:10.1016/j.jpowsour.2008.01.022

[16] B. Krishna, Y. R. Satyaji Rao and P. C. Nayak, “Time Series Modeling of River Flow Using Wavelet Neutral Networks,” Journal of Water Resources and Protection, Vol. 3, No. 1, 2011, pp. 50-59. doi:10.4236/jwarp.2011.31006

[17] L. C. Simith, D. L. Turcotte and B. Isacks, “Streamflow Characterization and Feature Detection Using a Discrete Wavelet Transform,” Hydrological Processes, Vol. 12, No. 2, 1998, pp. 233-249. doi:10.1002/(SICI)1099-1085(199802)12:2<233::AID-HYP573>3.0.CO;2-3

[18] D. Wang and J. Ding, “Wavelet Network Model and Its Application to the Prediction of Hydrology,” Nature and Science, Vol. 1, No. 1, 2003, pp. 67-71.

[19] D. Labat, R. Ababou and A. Mangin, “Rainfall-Runoff Relations for Karastic Springs: Part II. Continuous Wave let and Discrete Orthogonal Multiresolution Analysis,” Journal of Hydrology, Vol. 238, No. 3-4, pp. 2000, pp. 149-178. doi:10.1016/S0022-1694(00)00322-X

[20] S. G. Mallat, “A Theory for Multi Resolution Signal De composition: The Wavelet Representation,” IEEE Trans actions on Pattern Analysis and Machine Intelligence, Vol. 11, No. 7, 1998, pp. 674-693.

[21] A. Grossman and J. Morlet, “Decomposition of Harley Functions into Square Integral Wavelets of Constant Shape,” SIAM Journal on Mathematical Analysis, Vol. 15, No. 4, 1984, pp. 723-736. doi:10.1137/0515056

[22] I. Daubechies, “Orthogonal Bases of Compactly Supported Wavelets,” Communications on Pure and Applied Mathematics, Vol. 41, No. 7, 1988, pp. 909-996. doi:10.1002/cpa.3160410705

[23] E. Foufoula-Georgiou and P. E Kumar, “Wavelets in Geophysics,” Academic, San Diego and London, 1994.

[24] R. M. Rao and A. S. Bopardikar, “Wavelet Transforms: Introduction to Theory and Applications,” Addison Wesley Longman, Inc., Reading, 1998, 310 p.

[25] M. Kucuk and N. Agiralio?lu, “Wavelet Regression Technique for Streamflow Prediction,” Journal of Applied Statistics, Vol. 33, No. 9, 2006, pp. 943-960. doi:10.1080/02664760600744298

[26] O. Kisi, “Wavelet Regression Model as an Alternative to Neural Networks for Monthly Streamflow Forecasting,” Hydrological Processes, Vol. 23, No. 25, 2009, pp. 3583-3597. doi:10.1002/hyp.7461

[27] O. Kisi, “Wavelet Regression Model for Short-Term Streamflow Forecasting,” Journal of Hydrology, Vol. 389, No. 3-4, 2010, pp. 344-353. doi:10.1016/j.jhydrol.2010.06.013

[28] P. Y. Ma, “A Fresh Engineering Approach for the Fore cast of Financial Index Volatility and Hedging Strategies,” PhD thesis, Quebec University, Montreal, 2006.

[29] M. Firat, “Comparison of Artificial Intelligence Techniques for River Flow Forecasting,” Hydrology and Earth System Sciences, Vol. 12, No. 1, 2008, pp. 123-139.

[30] R. Samsudin, S. Ismail and A. Shabri, “A Hybrid Model of Self-Organizing Maps (SOM) and Least Square Support Vector Machine (LSSVM) for Time-Series Fore casting,” Expert Systems with Applications, Vol. 38, No. 8, 2011, pp. 10574-10578.

[31] M. T. Gencoglu and M. Uyar, “Prediction of Flashover Voltage of Insulators Using Least Square Support Vector Machines,” Expert Systems with Applications, Vol. 36, No. 7, 2009, pp. 10789-10798. doi:10.1016/j.eswa.2009.02.021

[32] L. Liu and W. Wang, “Exchange Rates Forecasting with Least Squares Support Vector Machines,” International Conference on Computer Science and Software Engineering, Wuhan, 12-14 December 2008, pp. 1017-1019.

[1] H. E. Hurst, “Long Term Storage Capacity of Reservoirs,” Transactions of ASCE, Vol. 116, 1961, pp. 770-799.

[2] N. C. Matalas, “Mathematical Assessment of Symmetric Hydrology,” Water Resources Research, Vol. 3, No. 4, 1967, pp. 937-945. doi:10.1029/WR003i004p00937

[3] G. E. P. Box and G. M. Jenkins, “Time Series Analysis Forecasting and Control,” Holden Day, San Francisco, 1970.

[4] J. W. Delleur, P. C. Tao and M. L. Kavvas, “An Evaluation of the Practicality and Complexity of Some Rainfall and Runoff Time Series Model,” Water Resources Research, Vol. 12, No. 5, 1976, pp. 953-970. doi:10.1029/WR012i005p00953

[5] V. Vapnik, “The Nature of Statistical Learning Theory,” Springer Verlag, Berlin, 1995. doi:10.1007/978-1-4757-2440-0

[6] P. S. Yu, S. T. Chen and I. F. Chang, “Support Vector Regression for Real-Time Flood Stage Forecasting,” Journal of Hydrology, Vol. 328, No. 3-4, 2006, pp. 704-716. doi:10.1016/j.jhydrol.2006.01.021

[7] Y. B. Dibike, S. Velickov, D. P. Solomatine and M. B. Abbott, “Model Induction with Support Vector Machines: Introduction and Applications,” Journal of Computing in Civil Engineering, Vol. 15, No. 3, 2001, pp. 208-216. doi:10.1061/(ASCE)0887-3801(2001)15:3(208)

[8] A. Elshorbagy, G. Corzo, S. Srinivasulu and D. P. Solomatine, “Experimental Investigation of the Predictive Capabilities of Data Driven Modeling Techniques in Hydro logy, Part 1: Concepts and Methodology,” Hydrology and Earth System Sciences Discussions, Vol. 6, 2009, pp. 7055-7093.

[9] A. Elshorbagy, G. Corzo, S. Srinivasulu and D. P. Solomatine, “Experimental Investigation of the Predictive Capabilities of Data Driven Modeling Techniques in Hydro logy, Part2: Application,” Hydrology and Earth System Sciences Discussions, Vol. 6, 2009, pp. 7095-7142.

[10] T. Asefa, M. Kemblowski, M. McKee and A. Khalil, “Multi-Time Scale Stream Flow Predictions: The Support Vector Machines Approach,” Journal of Hydrology, Vol. 318, No. 1-4, 2006, pp. 7-16.

[11] J. Y. Lin, C. T. Cheng and K. W. Chau, “Using Support Vector Machines for Long-Term Discharge Prediction,” Hydrological Sciences Journal, Vol. 51, No. 4, 2006, pp. 599-612. doi:10.1623/hysj.51.4.599

[12] W. C. Wang, K. W. Chau, C. T. Cheng and L. Qiu, “A Comparison of Performance of Several Artificial Intelligence Methods for Forecasting Monthly Discharge Time Series,” Journal of Hydrology, Vol. 374, No. 3-4, 2009, pp. 294-306. doi:10.1016/j.jhydrol.2009.06.019

[13] J. A. K. Suykens and J. Vandewalle, “Least Squares Support Vector Machine Classifiers,” Neural Processing Letters, Vol. 9, No. 3, 1999, pp. 293-300. doi:10.1023/A:1018628609742

[14] D. Hanbay, “An Expert System Based on Least Square Support Vector Machines for Diagnosis of Valvular Heart Disease,” Expert Systems with Applications, Vol. 36, No. 4, 2009, pp. 8368-8374.

[15] Y. W. Kang, J. Li, C. Y. Guang, H.-Y. Tu, J. Li and J. Yang, “Dynamic Temperature Modeling of an SOFC Using Least Square Support Vector Machines,” Journal of Power Sources, Vol. 179, No. 2, 2008, pp. 683-692. doi:10.1016/j.jpowsour.2008.01.022

[16] B. Krishna, Y. R. Satyaji Rao and P. C. Nayak, “Time Series Modeling of River Flow Using Wavelet Neutral Networks,” Journal of Water Resources and Protection, Vol. 3, No. 1, 2011, pp. 50-59. doi:10.4236/jwarp.2011.31006

[17] L. C. Simith, D. L. Turcotte and B. Isacks, “Streamflow Characterization and Feature Detection Using a Discrete Wavelet Transform,” Hydrological Processes, Vol. 12, No. 2, 1998, pp. 233-249. doi:10.1002/(SICI)1099-1085(199802)12:2<233::AID-HYP573>3.0.CO;2-3

[18] D. Wang and J. Ding, “Wavelet Network Model and Its Application to the Prediction of Hydrology,” Nature and Science, Vol. 1, No. 1, 2003, pp. 67-71.

[19] D. Labat, R. Ababou and A. Mangin, “Rainfall-Runoff Relations for Karastic Springs: Part II. Continuous Wave let and Discrete Orthogonal Multiresolution Analysis,” Journal of Hydrology, Vol. 238, No. 3-4, pp. 2000, pp. 149-178. doi:10.1016/S0022-1694(00)00322-X

[20] S. G. Mallat, “A Theory for Multi Resolution Signal De composition: The Wavelet Representation,” IEEE Trans actions on Pattern Analysis and Machine Intelligence, Vol. 11, No. 7, 1998, pp. 674-693.

[21] A. Grossman and J. Morlet, “Decomposition of Harley Functions into Square Integral Wavelets of Constant Shape,” SIAM Journal on Mathematical Analysis, Vol. 15, No. 4, 1984, pp. 723-736. doi:10.1137/0515056

[22] I. Daubechies, “Orthogonal Bases of Compactly Supported Wavelets,” Communications on Pure and Applied Mathematics, Vol. 41, No. 7, 1988, pp. 909-996. doi:10.1002/cpa.3160410705

[23] E. Foufoula-Georgiou and P. E Kumar, “Wavelets in Geophysics,” Academic, San Diego and London, 1994.

[24] R. M. Rao and A. S. Bopardikar, “Wavelet Transforms: Introduction to Theory and Applications,” Addison Wesley Longman, Inc., Reading, 1998, 310 p.

[25] M. Kucuk and N. Agiralio?lu, “Wavelet Regression Technique for Streamflow Prediction,” Journal of Applied Statistics, Vol. 33, No. 9, 2006, pp. 943-960. doi:10.1080/02664760600744298

[26] O. Kisi, “Wavelet Regression Model as an Alternative to Neural Networks for Monthly Streamflow Forecasting,” Hydrological Processes, Vol. 23, No. 25, 2009, pp. 3583-3597. doi:10.1002/hyp.7461

[27] O. Kisi, “Wavelet Regression Model for Short-Term Streamflow Forecasting,” Journal of Hydrology, Vol. 389, No. 3-4, 2010, pp. 344-353. doi:10.1016/j.jhydrol.2010.06.013

[28] P. Y. Ma, “A Fresh Engineering Approach for the Fore cast of Financial Index Volatility and Hedging Strategies,” PhD thesis, Quebec University, Montreal, 2006.

[29] M. Firat, “Comparison of Artificial Intelligence Techniques for River Flow Forecasting,” Hydrology and Earth System Sciences, Vol. 12, No. 1, 2008, pp. 123-139.

[30] R. Samsudin, S. Ismail and A. Shabri, “A Hybrid Model of Self-Organizing Maps (SOM) and Least Square Support Vector Machine (LSSVM) for Time-Series Fore casting,” Expert Systems with Applications, Vol. 38, No. 8, 2011, pp. 10574-10578.

[31] M. T. Gencoglu and M. Uyar, “Prediction of Flashover Voltage of Insulators Using Least Square Support Vector Machines,” Expert Systems with Applications, Vol. 36, No. 7, 2009, pp. 10789-10798. doi:10.1016/j.eswa.2009.02.021

[32] L. Liu and W. Wang, “Exchange Rates Forecasting with Least Squares Support Vector Machines,” International Conference on Computer Science and Software Engineering, Wuhan, 12-14 December 2008, pp. 1017-1019.