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Optimization of wavelet neural networks with the firefly algorithm for approximation problems

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Abstract

The effectiveness of swarm intelligence has been proven to be at the heart of various optimization problems. In this study, a recently developed nature-inspired algorithm, specifically the firefly algorithm (FA), is integrated in the learning strategy of wavelet neural networks (WNNs). The FA, which systematically optimizes the initial location of the translation parameters for WNNs, has reduced the number of hidden nodes while simultaneously improved the generalization capability of WNNs significantly. The applicability of the proposed model was demonstrated through empirical simulations for function approximation study, with both synthetic and real-world data. Performance assessment demonstrated its enhancement over the K-means clustering and random initialization approaches, as well as to the other neural network models reported in the literature, whereby a noteworthy decrease in the approximation error was observed.

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References

  1. Haykin S (1998) Neural networks: a comprehensive foundation. Prentice Hall PTR, Englewood Cliff, p 842

    Google Scholar 

  2. Zhang Q, Benveniste A (1992) Wavelet networks. IEEE Trans Neural Netw 3:889–898

    Article  Google Scholar 

  3. Zainuddin Z, Ong P (2011) Reliable multiclass cancer classification of microarray gene expression profiles using an improved wavelet neural network. Expert Syst Appl 38:13711–13722

    Google Scholar 

  4. Zainuddin Z, Pauline O (2011) Modified wavelet neural network in function approximation and its application in prediction of time-series pollution data. Appl Soft Comput 11:4866–4874

    Article  Google Scholar 

  5. Wanrosli WD, Zainuddin Z, Ong P, Rohaizu R (2013) Optimization of cellulose phosphate synthesis from oil palm lignocellulosics using wavelet neural networks. Ind Crops Prod 50:611–617

    Article  Google Scholar 

  6. Zainuddin Z, Ong P (2012) An effective and novel wavelet, neural network approach in classifying type 2 diabetics. Neural Netw World 22:407–428

    Article  Google Scholar 

  7. Zainuddin Z, Ong P (2013) Design of wavelet neural networks based on symmetry fuzzy C-means for function approximation. Neural Comput Appl 23:247–259

    Article  Google Scholar 

  8. Wang G, Guo L, Duan H (2013) Wavelet neural network using multiple wavelet functions in target threat assessment. Sci World J 2013:7

    Google Scholar 

  9. Oussar Y, Dreyfus G (2000) Initialization by selection for wavelet network training. Neurocomputing 34:131–143

    Article  MATH  Google Scholar 

  10. Hwang K, Mandayam S, Udpa SS, Udpa L, Lord W, Atzal M (2000) Characterization of gas pipeline inspection signals using wavelet basis function neural networks. NDT E Int 33:531–545

    Article  Google Scholar 

  11. Wei M, Jin S, Wang L, Zhou Y (2004) Defect characteristic prediction of pipeline by means of wavelet neural network based on the hierarchical clustering algorithm. In: American Society of Mechanical Engineers (ed) 2004 international pipeline conference, ASME, Alberta, Canada

  12. Lin C-J (2006) Wavelet neural networks with a hybrid learning approach. J Inf Sci Eng 22:1367–1387

    Google Scholar 

  13. Han M, Yin J (2008) The hidden neurons selection of the wavelet networks using support vector machines and ridge regression. Neurocomputing 72:471–479

    Article  Google Scholar 

  14. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks. IEEE Publisher

  15. Dorigo M, Maniezzo V, Colorni A (1991) The ant system: an autocatalytic optimizing process. No. 91-016, Technical report, pp 163–183

  16. Karaboga D, Basturk B (2008) On the performance of artificial bee colony (ABC) algorithm. Appl Soft Comput 8:687–697

    Article  Google Scholar 

  17. Wang G, Guo L, Wang H, Duan H, Liu L, Li J (2014) Incorporating mutation scheme into krill herd algorithm for global numerical optimization. Neural Comput Appl 24:853–871

    Article  Google Scholar 

  18. Wang G-G, Guo L, Gandomi AH, Hao G-S, Wang H (2014) Chaotic krill herd algorithm. Inf Sci 274:17–34

    Article  MathSciNet  Google Scholar 

  19. Wang G-G, Gandomi AH, Alavi AH (2014) Stud krill herd algorithm. Neurocomputing 128:363–370

    Article  Google Scholar 

  20. Fong S, Deb S, Yang X-S (2015) A heuristic optimization method inspired by wolf preying behavior. Neural Comput Appl 26:1725–1738

    Article  Google Scholar 

  21. Wang G-G, Deb S, Cui Z (2015) Monarch butterfly optimization. Neural Comput Appl 1–20. doi:10.1007/s00521-015-1923-y

  22. Simon D (2008) Biogeography-based optimization. Evolutionary computation. IEEE Trans 12:702–713

    Google Scholar 

  23. Wang G-G, Gandomi AH, Alavi AH (2014) An effective krill herd algorithm with migration operator in biogeography-based optimization. Appl Math Model 38:2454–2462

    Article  MathSciNet  Google Scholar 

  24. Yang X-S, Deb S (2010) Engineering optimisation by cuckoo search. Int J Math Model Numer Optim 1:330–343

    MATH  Google Scholar 

  25. Wang G-G, Gandomi A, Zhao X, Chu H (2014) Hybridizing harmony search algorithm with cuckoo search for global numerical optimization. Soft Comput 1–13. doi:10.1007/s00500-014-1502-7

  26. Wang G-G, Deb S, Gandomi A, Zhang Z, Alavi A (2015) Chaotic cuckoo search. Soft Comput 1–14. doi:10.1007/s00500-015-1726-1

  27. Yang X-S (2009) Firefly algorithms for multimodal optimization. In: Watanabe O, Zeugmann T (eds) Stochastic algorithms: foundations and applications. Springer, Berlin, pp 169–178

    Chapter  Google Scholar 

  28. Senthilnath J, Omkar SN, Mani V (2011) Clustering using firefly algorithm: performance study. Swarm Evol Comput 1:164–171

    Article  Google Scholar 

  29. Yang X-S (2010) Firefly algorithm, Lévy flights and global optimization. In: Bramer M, Ellis R, Petridis M (eds) Research and development in intelligent systems XXVI. Springer, London, pp 209–218

    Chapter  Google Scholar 

  30. Kanimozhi T, Latha K (2015) An integrated approach to region based image retrieval using firefly algorithm and support vector machine. Neurocomputing 151:1099–1111

    Article  Google Scholar 

  31. Massan SUR, Wagan AI, Shaikh MM, Abro R (2015) Wind turbine micrositing by using the firefly algorithm. Appl Soft Comput 27:450–456

    Article  Google Scholar 

  32. Wang G-G, Guo L, Duan H, Wang H (2014) A new improved firefly algorithm for global numerical optimization. J Comput Theor Nanosci 11:477–485

    Article  Google Scholar 

  33. Wang GG, Guo L, Duan H, Liu L, Wang H (2012) A modified firefly algorithm for UCAV path planning. Int J Hybrid Inf Technol 5:123–133

    Google Scholar 

  34. Upadhyay P, Kar R, Mandal D, Ghoshal SP (2014) A new design method based on firefly algorithm for IIR system identification problem. J King Saud Univ Eng Sci. doi:10.1016/j.jksues.2014.03.001

    Google Scholar 

  35. Kulluk S, Ozbakir L, Baykasoglu A (2012) Training neural networks with harmony search algorithms for classification problems. Eng Appl Artif Intell 25:11–19

    Article  Google Scholar 

  36. Cao J, Lin Z, Huang G-B (2010) Composite function wavelet neural networks with extreme learning machine. Neurocomputing 73:1405–1416

    Article  Google Scholar 

  37. Chauhan N, Ravi V, Karthik Chandra D (2009) Differential evolution trained wavelet neural networks: application to bankruptcy prediction in banks. Expert Syst Appl 36:7659–7665

    Article  Google Scholar 

  38. Paramjeet, Ravi V (2011) Bacterial foraging trained wavelet neural networks: application to bankruptcy prediction in banks. Int J Data Anal Tech Strateg 3:261–280

    Article  Google Scholar 

  39. MathWorks (2010) Matlab. The MathWorks Inc. MA

  40. Singh M, Srivastava S, Hanmandlu M, Gupta JRP (2009) Type-2 fuzzy wavelet networks (T2FWN) for system identification using fuzzy differential and Lyapunov stability algorithm. Appl Soft Comput 9:977–989

    Article  Google Scholar 

  41. Chen J, Bruns DD (1995) WaveARX neural network development for system identification using a systematic design synthesis. Ind Eng Chem Res 34:4420–4435

    Article  Google Scholar 

  42. Karatepe E, Alcı M (2005) A new approach to fuzzy wavelet system modeling. Int J Approx Reason 40:302–322

    Article  MathSciNet  Google Scholar 

  43. Wang J, Xiao J, Peng H, Gao X (2005) Constructing fuzzy wavelet network modeling. Int J Inf Technol 11:68–74

    Google Scholar 

  44. Srivastava S, Singh M, Hanmandlu M, Jha AN (2005) New fuzzy wavelet neural networks for system identification and control. Appl Soft Comput 6:1–17

    Article  Google Scholar 

  45. Ho DWC, Zhang PA, Xu JH (2001) Fuzzy wavelet networks for function learning. IEEE Trans Fuzzy Syst 9:200–211

    Article  Google Scholar 

  46. Tzeng S-T (2010) Design of fuzzy wavelet neural networks using the GA approach for function approximation and system identification. Fuzzy Sets Syst 161:2585–2596

    Article  MathSciNet  Google Scholar 

  47. Yao S, Wei CJ, He ZY (1996) Evolving wavelet neural networks for function approximation. Electron Lett 32:360

    Article  Google Scholar 

  48. Mackey M, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197:287–289

    Article  Google Scholar 

  49. Wu X, Wang Y (2012) Extended and Unscented Kalman filtering based feedforward neural networks for time series prediction. Appl Math Model 36:1123–1131

    Article  MathSciNet  MATH  Google Scholar 

  50. Castro JR, Castillo O, Melin P, Rodríguez-Díaz A (2009) A hybrid learning algorithm for a class of interval type-2 fuzzy neural networks. Inf Sci 179:2175–2193

    Article  MATH  Google Scholar 

  51. Fu Y-Y, Wu C-J, Ko C-N, Jeng J-T, Lai L-C (2009) ARRBFNs with SVR for prediction of chaotic time series with outliers. Artif Life Robot 14:29–33

    Article  Google Scholar 

  52. Rojas I, Valenzuela O, Rojas F, Guillen A, Herrera LJ, Pomares H, Marquez L, Pasadas M (2008) Soft-computing techniques and ARMA model for time series prediction. Neurocomputing 71:519–537

    Article  MATH  Google Scholar 

  53. Ardalani-Farsa M, Zolfaghari S (2011) Residual analysis and combination of embedding theorem and artificial intelligence in chaotic time series forecasting. Appl Artif Intell 25:45–73

    Article  Google Scholar 

  54. Ebadzadeh MM, Salimi-Badr A (2015) CFNN: correlated fuzzy neural network. Neurocomputing 148:430–444

    Article  Google Scholar 

  55. Melin P, Soto J, Castillo O, Soria J (2012) A new approach for time series prediction using ensembles of ANFIS models. Expert Syst Appl 39:3494–3506

    Article  Google Scholar 

  56. Smith C, Jin Y (2014) Evolutionary multi-objective generation of recurrent neural network ensembles for time series prediction. Neurocomputing 143:302–311

    Article  Google Scholar 

  57. Hernandez JAM, Castaeda FG, Cadenas JAM (2009) An evolving fuzzy neural network based on the mapping of similarities. IEEE Trans Fuzzy Syst 17:1379–1396

    Article  Google Scholar 

  58. Mirzaee H (2009) Linear combination rule in genetic algorithm for optimization of finite impulse response neural network to predict natural chaotic time series. Chaos Solitons Fractals 41:2681–2689

    Article  Google Scholar 

  59. Bhardwaj S, Srivastava S, Gupta JRP (2015) Pattern-similarity-based model for time series prediction. Comput Intell 31:106–131

    Article  MathSciNet  Google Scholar 

  60. Gholipour A, Araabi B, Lucas C (2006) Predicting chaotic time series using neural and neurofuzzy models: a comparative study. Neural Process Lett 24:217–239

    Article  Google Scholar 

  61. Mirzaee H (2009) Long-term prediction of chaotic time series with multi-step prediction horizons by a neural network with Levenberg–Marquardt learning algorithm. Chaos Solitons Fractals 41:1975–1979

    Article  Google Scholar 

  62. Yadav RN, Kalra PK, John J (2007) Time series prediction with single multiplicative neuron model. Appl Soft Comput 7:1157–1163

    Article  Google Scholar 

  63. Chang L-C, Chang F-J, Wang Y-P (2009) Auto-configuring radial basis function networks for chaotic time series and flood forecasting. Hydrol Process 23:2450–2459

    Article  Google Scholar 

  64. Gaxiola F, Melin P, Valdez F (2015) Comparison of neural networks with different membership functions in the type-2 fuzzy weights. In: Angelov P et al (eds) Advances in intelligent systems and computing. Springer, pp 707–713

  65. Sugeno M, Yasukawa T (1993) A fuzzy-logic-based approach to qualitative modeling. IEEE Trans Fuzzy Syst 1:7

    Article  Google Scholar 

  66. Lorenz E (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130–141

    Article  Google Scholar 

  67. Inoussa G, Peng H, Wu J (2012) Nonlinear time series modeling and prediction using functional weights wavelet neural network-based state-dependent AR model. Neurocomputing 86:59–74

    Article  Google Scholar 

  68. Chandra R, Zhang M (2012) Cooperative coevolution of Elman recurrent neural networks for chaotic time series prediction. Neurocomputing 86:116–123

    Article  Google Scholar 

  69. Ardalani-Farsa M, Zolfaghari S (2010) Chaotic time series prediction with residual analysis method using hybrid Elman–NARX neural networks. Neurocomputing 73:2540–2553

    Article  Google Scholar 

  70. Gan M, Peng H, Peng X, Chen X, Inoussa G (2010) A locally linear RBF network-based state-dependent AR model for nonlinear time series modeling. Inf Sci 180:4370–4383

    Article  MathSciNet  Google Scholar 

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Acknowledgments

Financial supports from the Malaysian Government with cooperation of Universiti Sains Malaysia in the form of FRGS Grant 203/PMATHS/6711323 and Universiti Tun Hussein Onn Malaysia (UTHM) in the form of FRGS Grant Vot 1490 are gratefully acknowledged.

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Authors and Affiliations

  1. School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800, Minden, Penang, Malaysia

    Zarita Zainuddin & Pauline Ong

  2. Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), 86400, Parit Raja, Batu Pahat, Johor, Malaysia

    Pauline Ong

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  1. Zarita Zainuddin
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  2. Pauline Ong
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Correspondence to Zarita Zainuddin.

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Zainuddin, Z., Ong, P. Optimization of wavelet neural networks with the firefly algorithm for approximation problems. Neural Comput & Applic 28, 1715–1728 (2017). https://doi.org/10.1007/s00521-015-2140-4

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