Time Series Modeling of River Flow Using Wavelet Neural Networks

ABSTRACT

A new hybrid model which combines wavelets and Artificial Neural Network (ANN) called wavelet neural network (WNN) model was proposed in the current study and applied for time series modeling of river flow. The time series of daily river flow of the Malaprabha River basin (Karnataka state, India) were analyzed by the WNN model. The observed time series are decomposed into sub-series using discrete wavelet transform and then appropriate sub-series is used as inputs to the neural network for forecasting hydrological variables. The hybrid model (WNN) was compared with the standard ANN and AR models. The WNN model was able to provide a good fit with the observed data, especially the peak values during the testing period. The benchmark results from WNN model applications showed that the hybrid model produced better results in estimating the hydrograph properties than the latter models (ANN and AR).

A new hybrid model which combines wavelets and Artificial Neural Network (ANN) called wavelet neural network (WNN) model was proposed in the current study and applied for time series modeling of river flow. The time series of daily river flow of the Malaprabha River basin (Karnataka state, India) were analyzed by the WNN model. The observed time series are decomposed into sub-series using discrete wavelet transform and then appropriate sub-series is used as inputs to the neural network for forecasting hydrological variables. The hybrid model (WNN) was compared with the standard ANN and AR models. The WNN model was able to provide a good fit with the observed data, especially the peak values during the testing period. The benchmark results from WNN model applications showed that the hybrid model produced better results in estimating the hydrograph properties than the latter models (ANN and AR).

Cite this paper

nullB. Krishna, Y. Rao and P. Nayak, "Time Series Modeling of River Flow Using Wavelet Neural Networks,"*Journal of Water Resource and Protection*, Vol. 3 No. 1, 2011, pp. 50-59. doi: 10.4236/jwarp.2011.31006.

nullB. Krishna, Y. Rao and P. Nayak, "Time Series Modeling of River Flow Using Wavelet Neural Networks,"

References

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[12] P. Hettiarachchi, M. J. Hall and A. W. Minns, “The Extrapolation of Artificial Neural Networks for the Modeling of Rainfall-Runoff Relationships,” Journal of Hydroinformatics, Vol. 7, No. 4, 2005, pp. 291-296.

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[15] P. C. Nayak, Y. R. Satyaji Rao and K. P. Sudheer, “Gro- undwater Level Forecasting in a Shallow Aquifer Using Artificial Neural Network Approach,” Water Resources Management, Vol. 20, No. 1, 2006, pp. 77-90. doi:10.1007/s11269-006-4007-z

[16] B. Krishna, Y. R. Satyaji Rao and T. Vijaya, “Modeling Groundwater Levels in an Urban Coastal Aquifer Using Artificial Neural Networks,” Hydrological Processes, Vol. 22, No. 12, 2008, pp. 1180-1188. doi:10.1002/hyp.6686

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[18] S. R. Massel, “Wavelet Analysis for Processing of Ocean Surface Wave Records,” Ocean Engineering, Vol. 28, 2001, pp. 957-987. doi:10.1016/S0029-8018(00)00044-5

[19] M. C. Huang, “Wave Parameters and Functions in Wa- velet Analysis,” Ocean Engineering, Vol. 31, No. 1, 2004, pp. 111-125. doi:10.1016/S0029-8018(03)00047-7

[20] L. C. Smith, D. Turcotte and B. L. Isacks, “Stream Flow 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

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

[22] P. Saco and P. Kumar, “Coherent Modes in Multiscale Variability of Stream Flow over the United States,” Water Resources Research, Vol. 36, No. 4, 2000, pp. 1049- 1067.doi:10.1029/1999WR900345

[23] P. Kumar and E. Foufoula-Georgiou, “A Multicomponent Decomposition of Spatial Rainfall Fields: Segregation of Large- and Small-Scale Features Using Wavelet Transforms,” Water Resources Research, Vol. 29, No. 8, 1993, pp. 2515-2532. doi:10.1029/93WR00548

[24] P. Kumar, “Role of Coherent Structure in the Stochastic Dynamic Variability of Precipitation,” Journal of Geophysical Research, Vol. 101, 1996, pp. 393-404. doi:10.1029/96JD01839

[25] K. Fraedrich, J. Jiang, F.-W. Gerstengarbe and P. C. Werner, “Multiscale Detection of Abrupt Climate Cha- nges: Application to the River Nile Flood,” International Journal of Climatology, Vol. 17, No. 12, 1997, pp. 1301- 1315.doi:10.1002/(SICI)1097-0088(199710)17:12<1301::AID-JOC196>3.0.CO;2-W

[26] R. H. Compagnucci, S. A. Blanco, M. A. Filiola and P. M. Jacovkis, “Variability in Subtropical Andean Argentinian Atuel River: A Wavelet Approach,” Environmetrics, Vol. 11, No. 3, 2000, pp. 251-269. doi:10.1002/(SICI)1099-095X(200005/06)11:3<251::AID-ENV405>3.0.CO;2-0

[27] S. Tantanee, S. Patamatammakul, T. Oki, V. Sriboonlue and T. Prempree, “Coupled Wavelet-Autoregressive Mo- del for Annual Rainfall Prediction,” Journal of Environmental Hydrology, Vol. 13, No. 18, 2005, pp. 1-8.

[28] P. Coulibaly, “Wavelet Analysis of Variability in Annual Canadian Stream Flows,” Water Resources Research, Vol. 40, 2004.

[29] F. Xiao, X. Gao, C. Cao and J. Zhang, “Short-Term Prediction on Parameter-Varying Systems by Multiwavelets Neural Network,” Lecture Notes in Computer Science, S- pringer-Verlag, Vol. 3630, No. 3611, 2005, pp. 139- 146.

[30] D. J. Wu, J. Wang and Y. Teng, “Prediction of Underground Water Levels Using Wavelet Decompositions and Transforms,” Journal of Hydro-Engineering, Vol. 5, 2004, pp. 34-39.

[31] A. Aussem and F. Murtagh, “Combining Neural Network Forecasts on Wavelet Transformed Series,” Connection Science, Vol. 9, No. 1, 1997, pp. 113-121. doi:10.1080/095400997116766

[32] T. Partal and O. Kisi, “Wavelet and Neuro-Fuzzy Conjunction Model for Precipitation Forecasting,” Journal of Hydrology, Vol. 342, No. 1-2, 2007, pp. 199-212. doi:10.1016/j.jhydrol.2007.05.026

[33] A. Grossmann and J. Morlet, “Decomposition of Hardy Functions into Square Integrable Wavelets of Constant shape,” SIAM Journal on Mathematical Analysis, Vol. 15, No. 4, 1984, pp. 723-736. doi:10.1137/0515056

[34] A. Antonios and E. V. Constantine, “Wavelet Exploratory Analysis of the FTSE ALL SHARE Index,” Preprint submitted to Economics Letters University of Durham, Durham, 2003.

[35] D. Benaouda, F. Murtagh, J. L. Starck and O. Renaud, “Wavelet-Based Nonlinear Multiscale Decomposition M- odel for Electricity Load Forecasting,” Neurocomputing, Vol. 70, No. 1-3, 2006, pp. 139-154. doi:10.1016/j.neucom.2006.04.005

[36] W. McCulloch and W. Pitts, “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Bulletin of Mathematical Biophysics, Vol. 5, 1943, pp. 115-133. doi:10.1007/BF02478259

[37] D. E. Rumelhart, G. E. Hinton and R. J. Williams, “Lear- ning Representations by Back-Propagating Errors,” Nature, Vol. 323, No. 9, 1986, pp. 533-536. doi:10.1038/323533a0

[38] M. T. Hagan and M. B. Menhaj, “Training Feed forward Networks with Marquardt Algorithm,” IEEE Transactions on Neural Networks, Vol. 5, No. 6, 1994, pp. 989- 993. doi:10.1109/72.329697

[39] P. Coulibaly, F. Anctil, P. Rasmussen and B. Bobee, “A Recurrent Neural Networks Approach Using Indices of Low-Frequency Climatic Variability to Forecast Regional Annual Runoff,” Hydrological Processes, Vol. 14, No. 15, 2000, pp. 2755-2777. doi:10.1002/1099-1085(20001030)14:15<2755::AID-HYP90>3.0.CO;2-9

[1] H. Raman and N. Sunil Kumar, “Multivariate Modeling of Water Resources Time Series Using Artificial Neural Networks,” Journal of Hydrological Sciences, Vol. 40, No. 4, 1995, pp. 145-163. doi:10.1080/02626669509491401

[2] H. R. Maier and G. C. Dandy, “Determining Inputs for Neural Network Models of Multivariate Time Series,” Microcomputers in Civil Engineering, Vol. 12, 1997, pp. 353-368.

[3] M. C. Deo and K. Thirumalaiah, “Real Time Forecasting Using Neural Networks: Artificial Neural Networks in Hydrology,” In: R. S. Govindaraju and A. Ramachandra Rao, Kluwer Academic Publishers, Dordrecht, 2000, pp. 53-71.

[4] B. Fernandez and J. D. Salas, “Periodic Gamma Autoregressive Processes for Operational Hydrology,” Water Resources Research, Vol. 22, No. 10, 1986, pp. 1385- 1396. doi:10.1029/WR022i010p01385

[5] S. L. S. Jacoby, “A Mathematical Model for Non-Linear Hydrologic Systems,” Journal of Geophysics Research, Vol. 71, No. 20, 1966, pp. 4811-4824.

[6] J. Amorocho and A. Brandstetter, “A Critique of Current Methods of Hydrologic Systems Investigations,” Eos Transactions of AGU, Vol. 45, 1971, pp. 307-321.

[7] S. Ikeda, M. Ochiai and Y. Sawaragi, “Sequential GMDH Algorithm and Its Applications to River Flow Prediction,” IEEE Transactions of System Management and Cy- bernetics, Vol. 6, No. 7, 1976, pp. 473-479. doi:10.1109/TSMC.1976.4309532

[8] ASCE Task Committee, “Artificial Neural Networks in hydrology-I: Preliminary Concepts,” Journal of Hydrolo- gic Engineering, Vol. 5, No. 2, 2000(a), pp. 115-123.

[9] ASCE Task Committee, “Artificial Neural Networks in Hydrology-II: Hydrologic Applications,” Journal of Hydrologic Engineering, Vol. 5, No. 2, 2000(b), pp. 124- 137.

[10] D. W. Dawson and R. Wilby, “Hydrological Modeling Using Artificial Neural Networks,” Progress in Physical Geograpgy, Vol. 25, No. 1, 2001, pp. 80-108.

[11] S. Birikundavy, R. Labib, H. T. Trung and J. Rousselle, “Performance of Neural Networks in Daily Stream Flow Forecasting,” Journal of Hydrologic Engineering, Vol. 7, No. 5, 2002, pp. 392-398. doi:10.1061/(ASCE)1084-0699(2002)7:5(392)

[12] P. Hettiarachchi, M. J. Hall and A. W. Minns, “The Extrapolation of Artificial Neural Networks for the Modeling of Rainfall-Runoff Relationships,” Journal of Hydroinformatics, Vol. 7, No. 4, 2005, pp. 291-296.

[13] E. J. Coppola, M. Poulton, E. Charles, J. Dustman and F. Szidarovszky, “Application of Artificial Neural Networks to Complex Groundwater Problems,” Journal of Natural Resources Research, Vol. 12, No. 4, 2003(a), pp. 303- 320.

[14] E. J. Coppola, F. Szidarovszky, M. Poulton and E. Charles, “Artificial Neural Network Approach for Predicting Transient Water Levels in a Multilayered Groundwater System under Variable State, Pumping and Climate Conditions,” Journal of Hydrologic Engineering, Vol. 8, No. 6, 2003(b), pp. 348-359.

[15] P. C. Nayak, Y. R. Satyaji Rao and K. P. Sudheer, “Gro- undwater Level Forecasting in a Shallow Aquifer Using Artificial Neural Network Approach,” Water Resources Management, Vol. 20, No. 1, 2006, pp. 77-90. doi:10.1007/s11269-006-4007-z

[16] B. Krishna, Y. R. Satyaji Rao and T. Vijaya, “Modeling Groundwater Levels in an Urban Coastal Aquifer Using Artificial Neural Networks,” Hydrological Processes, Vol. 22, No. 12, 2008, pp. 1180-1188. doi:10.1002/hyp.6686

[17] 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.

[18] S. R. Massel, “Wavelet Analysis for Processing of Ocean Surface Wave Records,” Ocean Engineering, Vol. 28, 2001, pp. 957-987. doi:10.1016/S0029-8018(00)00044-5

[19] M. C. Huang, “Wave Parameters and Functions in Wa- velet Analysis,” Ocean Engineering, Vol. 31, No. 1, 2004, pp. 111-125. doi:10.1016/S0029-8018(03)00047-7

[20] L. C. Smith, D. Turcotte and B. L. Isacks, “Stream Flow 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

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

[22] P. Saco and P. Kumar, “Coherent Modes in Multiscale Variability of Stream Flow over the United States,” Water Resources Research, Vol. 36, No. 4, 2000, pp. 1049- 1067.doi:10.1029/1999WR900345

[23] P. Kumar and E. Foufoula-Georgiou, “A Multicomponent Decomposition of Spatial Rainfall Fields: Segregation of Large- and Small-Scale Features Using Wavelet Transforms,” Water Resources Research, Vol. 29, No. 8, 1993, pp. 2515-2532. doi:10.1029/93WR00548

[24] P. Kumar, “Role of Coherent Structure in the Stochastic Dynamic Variability of Precipitation,” Journal of Geophysical Research, Vol. 101, 1996, pp. 393-404. doi:10.1029/96JD01839

[25] K. Fraedrich, J. Jiang, F.-W. Gerstengarbe and P. C. Werner, “Multiscale Detection of Abrupt Climate Cha- nges: Application to the River Nile Flood,” International Journal of Climatology, Vol. 17, No. 12, 1997, pp. 1301- 1315.doi:10.1002/(SICI)1097-0088(199710)17:12<1301::AID-JOC196>3.0.CO;2-W

[26] R. H. Compagnucci, S. A. Blanco, M. A. Filiola and P. M. Jacovkis, “Variability in Subtropical Andean Argentinian Atuel River: A Wavelet Approach,” Environmetrics, Vol. 11, No. 3, 2000, pp. 251-269. doi:10.1002/(SICI)1099-095X(200005/06)11:3<251::AID-ENV405>3.0.CO;2-0

[27] S. Tantanee, S. Patamatammakul, T. Oki, V. Sriboonlue and T. Prempree, “Coupled Wavelet-Autoregressive Mo- del for Annual Rainfall Prediction,” Journal of Environmental Hydrology, Vol. 13, No. 18, 2005, pp. 1-8.

[28] P. Coulibaly, “Wavelet Analysis of Variability in Annual Canadian Stream Flows,” Water Resources Research, Vol. 40, 2004.

[29] F. Xiao, X. Gao, C. Cao and J. Zhang, “Short-Term Prediction on Parameter-Varying Systems by Multiwavelets Neural Network,” Lecture Notes in Computer Science, S- pringer-Verlag, Vol. 3630, No. 3611, 2005, pp. 139- 146.

[30] D. J. Wu, J. Wang and Y. Teng, “Prediction of Underground Water Levels Using Wavelet Decompositions and Transforms,” Journal of Hydro-Engineering, Vol. 5, 2004, pp. 34-39.

[31] A. Aussem and F. Murtagh, “Combining Neural Network Forecasts on Wavelet Transformed Series,” Connection Science, Vol. 9, No. 1, 1997, pp. 113-121. doi:10.1080/095400997116766

[32] T. Partal and O. Kisi, “Wavelet and Neuro-Fuzzy Conjunction Model for Precipitation Forecasting,” Journal of Hydrology, Vol. 342, No. 1-2, 2007, pp. 199-212. doi:10.1016/j.jhydrol.2007.05.026

[33] A. Grossmann and J. Morlet, “Decomposition of Hardy Functions into Square Integrable Wavelets of Constant shape,” SIAM Journal on Mathematical Analysis, Vol. 15, No. 4, 1984, pp. 723-736. doi:10.1137/0515056

[34] A. Antonios and E. V. Constantine, “Wavelet Exploratory Analysis of the FTSE ALL SHARE Index,” Preprint submitted to Economics Letters University of Durham, Durham, 2003.

[35] D. Benaouda, F. Murtagh, J. L. Starck and O. Renaud, “Wavelet-Based Nonlinear Multiscale Decomposition M- odel for Electricity Load Forecasting,” Neurocomputing, Vol. 70, No. 1-3, 2006, pp. 139-154. doi:10.1016/j.neucom.2006.04.005

[36] W. McCulloch and W. Pitts, “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Bulletin of Mathematical Biophysics, Vol. 5, 1943, pp. 115-133. doi:10.1007/BF02478259

[37] D. E. Rumelhart, G. E. Hinton and R. J. Williams, “Lear- ning Representations by Back-Propagating Errors,” Nature, Vol. 323, No. 9, 1986, pp. 533-536. doi:10.1038/323533a0

[38] M. T. Hagan and M. B. Menhaj, “Training Feed forward Networks with Marquardt Algorithm,” IEEE Transactions on Neural Networks, Vol. 5, No. 6, 1994, pp. 989- 993. doi:10.1109/72.329697

[39] P. Coulibaly, F. Anctil, P. Rasmussen and B. Bobee, “A Recurrent Neural Networks Approach Using Indices of Low-Frequency Climatic Variability to Forecast Regional Annual Runoff,” Hydrological Processes, Vol. 14, No. 15, 2000, pp. 2755-2777. doi:10.1002/1099-1085(20001030)14:15<2755::AID-HYP90>3.0.CO;2-9